4.1: Inventory
A merchandising business manufactures products, marks them up, and sells them to customers. A merchandiser may therefore be either the buyer or the seller in a given transaction.
Inventory is items that are purchased for resale. The process of inventory valuation involves determining the quantities and dollar value of the inventory that a company owns.
The perpetual inventory system is the process of keeping a current running total of inventory, both in number of units on hand and its dollar value, at all times. When product is purchased for resale, inventory immediately increases. When inventory is sold, its total value is immediately reduced.
Items in inventory are not always purchased at the same price; the same items may cost different amounts at different times. Therefore, a business needs a system of deciding which cost to select as its expense amount for Cost of Merchandise Sold when it sells an item.
As a simple example, let’s say a company has purchased 30 identical items for resale to customers. It bought 10 items on 2/2 for $1 each, 10 items on 2/3 for $2 each, and 10 items on 2/4 for $3 each. The total cost of the 30 units in inventory is $60.
| Date | Purchases | ||
| 2/2 | 10 | $1 | $10 |
| 2/3 | 10 | $2 | $20 |
| 2/4 | 10 | $3 | $30 |
| TOTAL | 30 | $60 |
The issue is this: If the company sells ONE item to a customer for $10, the cost of that one item needs to be determined.
| Date | Account | Debit | Credit | ||
| 2/12 | Accounts Receivable | 10 | ▲ Accounts Receivable is an asset account that is increasing . | ||
| Sales | 10 | ▲ Sales is a revenue account that is increasing . |
| Date | Account | Debit | Credit | ||
| 2/12 | Cost of Merchandise Sold | ??? | ▲ Cost of Merchandise Sold is an expense account that is increasing. | ||
| Merchandise Inventory | ??? | ▼ Merchandise Inventory is an asset account that is decreasing . |
The company will select an accepted method of valuing withdrawals from inventory. Three common methods are the following:
FIFO (First-In, First-Out) method withdraws inventory beginning with those units purchased earliest. In the example above, the Cost of Merchandise Sold would be $1, one of the items purchased on 2/2. As a result, the gross profit on the sale would be $9 ($10 - $1).
LIFO (Last-In, First-Out) method withdraws inventory beginning with those units purchased most recently. In the example above, the Cost of Merchandise Sold would be $3, one of the items purchased on 2/4. As a result, the gross profit on the sale would be $7 ($10 - $3).
Average Cost Method uses an average of the cost of all items currently in stock. In the example above, the average is $60/30 units, so the Cost of Merchandise Sold would be $2 per unit. As a result, the gross profit on the salewould be $8 ($10 - $2).
We will be answering the following four questions about inventory for an accounting period:
- What is total sales?
- What is total cost of merchandise sold?
- What is gross profit?
- What is the ending inventory balance?
4.1.1 Perpetual Inventory System
Inventory Grids
By entering transactions into a cost grid, you can organize your data to easily determine Cost of Merchandise Sold amounts and Merchandise Inventory balances after every purchase and sale. The grids show increases in Merchandise Inventory due to purchases, decreases in Merchandise Inventory due to sales, and the running Merchandise Inventory balance.
The following grid organizes the purchases and sales of a merchandiser for one of its products. It is essentially an expanded Merchandise Inventory account ledger. Not only does it show the dollar amount for each transaction and the updated running balance in dollars, but it also keeps track of the number of items bought, sold, and currently in inventory.
Sample Inventory Cost Grid
| Date | Purchases | Cost of Merchandise Sold | Inventory Balance | ||||||
| Units | Cost | Total | Units | Cost | Total | Units | Cost | Total | |
| 6/1 | 10 | $4 | $40 | ||||||
| 6/5 | 1 | $4 | $4 | 9 | $4 | $36 | |||
| 6/10 | 10 | $5 | $50 | 9 | $4 | $36 | |||
| 10 | $5 | $50 |
The Purchases columns show the details about items that were bought on different dates for resale to customers. Entries in the Purchases columns are the same regardless of the inventory valuation method selected. For a purchase, there is a debit to Merchandise Inventory and total inventory increases.
The Cost of Merchandise Sold columns show the detail about the order in which items are withdrawn from inventory for each sale. The amounts in these columns will vary based on whether the method is FIFO, LIFO, or average cost. For a sale, there is a debit to Cost of Merchandise Sold and total inventory decreases.
The Inventory Balance columns keep a running total of the number of items and their costs on each date. Each purchase is added to any inventory balance that already appears there. With a purchase, it is a good practice to first copy down what was in stock on the previous date in the Inventory Balance columns and add the new purchase below that. This clearly shows what is in stock on any given date. Each sale reduces the inventory balance by the cost of merchandise sold amount.
Only costs are entered into the grid; not the price that you sell the merchandise for to customers. If you are given the selling price, you can also determine the amount of sales and gross profit amounts outside of the grid.
There is a journal entry that corresponds to each purchase and sale. One key reason for the grid is that it enables you to determine the amounts for the cost of merchandise sold for each sale.
FIFO under the perpetual inventory system—FIFO (first-In, first-out) is a method of inventory valuation where the cost of the items purchased earliest is used in Cost of Merchandise Sold when one item is resold. The balance in Merchandise Inventory, which includes those items still available for sale, is comprised of the costs of those items purchased most recently.
| 6/1 | The inventory balance that is given is entered. This is carried over from the previous month. |
| 6/5 | One unit is sold. Since all 10 units in stock cost $4, the only choice is a $4 cost for that item in the Cost of Merchandise Sold columns. This is deducted from the inventory balance. |
| 6/10 | Purchases are entered in the Purchases columns and added to the inventory balance. |
| 6/16 | Now it is important to know you are using FIFO. The customer ordered 12 items. You have 19 in stock at two different costs. Under FIFO you usethe “oldest” ones first – the $4 items. You sell all 9 of those and then need 3 items that cost $5 to complete the order. You use two lines in the Cost of Merchandise Sold columns – one for each unit cost. This is deducted from the inventory balance. |
| 6/22 | Purchases are entered in the Purchases columns and added to the inventory balance. |
| 6/30 | The customer ordered 6 items. You have 17 in stock at two different costs. Under FIFO you use the “oldest” ones first – the $5 items. You sell 6 ofthose and enter this in the Cost of Merchandise Sold. This is deducted from the inventory balance. |
Four inventory questions under FIFO:
-
What is total sales?
19 units, $190
(1 + 12 + 6) = 19 units sold x $10 per unit
-
What is total cost of merchandise sold?
$85
($4 + $36 + $15 + $30) from cost of merchandise sold column
-
What is gross profit?
$105
Sales – cost of merchandise sold is $190 - $85
- What is the ending inventory balance? 11 units, $65 6/30 inventory balance amounts in cost grid
LIFO under the perpetual inventory system—LIFO (last-in, first-out) is a method of inventory valuation where the cost of the item purchased most recently is used in Cost of Merchandise Sold when one item is resold. The balance in Merchandise Inventory, which includes those items still available for sale, is comprised of the costs of those items purchased earliest.
| 6/1 | The inventory balance that is given is entered. This is carried over from the previous month. |
| 6/5 | One unit is sold. Since all 10 units in stock cost $4, the only choice is a $4 cost for that item in the Cost of Merchandise Sold columns. This is deducted from the inventory balance. |
| 6/10 | Purchases are entered in the Purchases columns and added to the inventory balance. |
| 6/16 | Now it is important to know you are using LIFO. The customer ordered 12 items. You have 19 in stock at two different costs. Under LIFO you usethe “newest” ones first – the $5 items. You sell all 10 of those and then need 2 items that cost $4 to complete the order. You use two lines in the Cost of Merchandise Sold columns – one for each unit cost. This is deducted from the inventory balance. |
| 6/22 | Purchases are entered in the Purchases columns and added to the inventory balance. |
| 6/30 | The customer ordered 6 items. You have 17 in stock at two different costs. Under LIFO you use the “newest” ones first - the $6 items. You sell 6 of those and enter this in the Cost of Merchandise Sold. This is deducted from the inventory balance. |
Four inventory questions under LIFO:
What is total sales? 19 units, $190
(1 + 12 + 6) = 19 units sold x $10 per unit
-
What is total sales?
19 units, $190
(1 + 12 + 6) = 19 units sold x $10 per unit
-
What is total cost of merchandise sold?
$98
($4 + $50 + $8 + $36) from cost of merchandise sold column
-
What is gross profit?
$92
Sales – cost of merchandise sold is $190 - $98
-
What is the ending inventory balance?
11 units, $52
6/30 inventory balance amounts in cost grid
Average cost under the perpetual inventory system—Average cost is a method of inventory valuation where each time there is a purchase or sale, the dollar value of the remaining inventory on hand is divided by the number of units in stock to arrive at an average cost per unit. Likewise, the cost of merchandise sold is determined by using an average cost per unit.
-
What is total sales?
19 units, $190.00
(1 + 12 + 6) = 19 units sold x $10 per unit
- What is total cost of merchandise sold? $90.70 ($4 + $54.36 + $32.34) from cost of merchandise sold column
-
What is gross profit?
$99.30
Sales – cost of merchandise sold is $190.00 - $90.70
-
What is the ending inventory balance?
11 units, $58.74
6/30 inventory balance amounts in cost grid
The results of the preceding example for both FIFO and LIFO under the perpetual inventory system can be summarized in four questions.
| Four inventory questions | FIFO | LIFO | Average cost |
| 1. What is total sales? (19 units) | $190.00 | $190.00 | $190.00 |
| 2. What is total cost of merchandise sold? (19 units) | 85.00 | 98.00 | 90.70 |
| 3. What is gross profit? | 105.00 | 92.00 | 99.30 |
| 4. What is the ending inventory balance? (11 units) | 65.00 | 52.00 | 58.74 |
Under all three methods, 19 units were sold and total sales were $190. Notice, however, that under FIFO the 19 units COST $85, under LIFO these same 19 units COST $98, and under average cost these same 19 units COST $90.70. This is a $13 difference between the highest and lowest costing method.
Gross profit is also different among the three methods. Because less cost is deducted from sales under the FIFO method, gross profit is $13 higher under FIFO than it is for LIFO.
That $13 difference also appears in the ending inventory balances. Since the cost of merchandise sold was lower under FIFO than it was under LIFO and average cost, the ending inventory balance under FIFO is higher that with the other two methods.
To summarize, there is a $13 difference between FIFO and LIFO in the cost of goods sold and ending inventory amounts. FIFO includes that $13 as part of ending inventory; LIFO considers that $13 to be part of cost of merchandise sold.
NOTE: The pattern above will result when costs are rising over time. In this example, they increased from $4 to $5 to $6. If costs decrease over time, the results will be the opposite: LIFO would include the difference as part of ending inventory and FIFO would consider the difference to be part of cost of merchandise sold.
The results for the average cost method typically fall between those for LIFO and FIFO.
4.1.2 Periodic Inventory System
As was mentioned in the merchandising discussion, some companies do not keep an ongoing running inventory balance as was shown under the perpetual inventory system. Instead, these companies choose to wait until the end of the accounting period, just before financial statements are prepared, to conduct a physical inventory count to determine (1) how much ending inventory they still have in stock (counted) and (2) how much inventory they have sold during the period, which is their cost of merchandise sold (calculated).
Cost of merchandise sold is determined by first calculating cost of merchandise available for sale, which is the beginning inventory value plus purchases during the period.
The following is sample information for a single product for a merchandising company that uses the periodic inventory system in June:
| 6/1 | Beginning inventory | 10 units @ $ 4 = | $ 40 |
| 6/10 | Purchase | 10 units @ $ 5 = | 50 |
| 6/22 | Purchase | 10 units @ $ 6 = | \(\ \underline{60}\) |
Cost of goods available for sale: 30 units at a total cost of $150
| 6/5 | Sale | 1 units @ $ 10 | |
| 6/16 | Sale | 12 units @ $ 10 | |
| 6/30 | Sale | 6 units @ $ 10 |
Total units sold: 19 units
Ending inventory: 11 units (30 units available – 19 units sold from above)
The same three flow methods of withdrawing inventory from stock—FIFO, LIFO, and average cost—are used under the periodic system. The periodic system disregards the dates of the purchases and sales and just looks at the totals of each collectively.
FIFO Under the Periodic Inventory System
Under FIFO, the 19 units sold are drawn from earliest inventory in stock to determine cost of goods sold. The first 10 units are from the beginning inventory and the remaining 8 units are from the 6/10 purchase.
Cost of merchandise sold
= (10 x $4) + (9 x $5) = $40 + $45 =
$85
The
11 units in ending inventory
include the remaining 1 unit from the
6/10 purchase and all 10 units from the 6/22 purchase.
Ending inventory
= (1 x $5) + (10 x $6) = $5 + $60 =
$65
The total cost of goods available for sale during the period, which was 30 units at a total cost of $150 , is split between cost of merchandise sold and ending inventory.
LIFO Under the Periodic Inventory System
Under LIFO, the 18 units sold are drawn from latest inventory in stock to determine cost of goods sold. The first 10 units are from the 6/22 purchase and the remaining 8 units are from the 6/10 purchase.
Cost of merchandise sold
= (10 x $6) + (9 x $5) = $60 + $45 =
$105
The
12 units in ending inventory
include the remaining 2 units from the
6/10 purchase and all 10 units from beginning inventory.
Ending inventory
= (1 x $5) + (10 x $4) = $5 + $40 =
$45
The total cost of goods available for sale during the period, which was 30 units at a total cost of $150 , is split between cost of merchandise sold and ending inventory.
Average Cost Under the Periodic Inventory System
Under average cost, the 30 units available for sale are divided into their total cost, as follows:
$150 / 30 = $5 per unit
The 19 units sold are all costed at $5. Cost of merchandise sold = 19 x $5 = $95 .
The 11 units in ending inventory are all costed at $5. Ending inventory = 11 x $5 = $55 .
The total cost of goods available for sale during the period, which was 30 units at a total cost of $150 , is split between cost of merchandise sold and ending inventory.
The results of the preceding example for both FIFO and LIFO under the periodic inventory system can be summarized in four questions.
| Four inventory questions | FIFO | LIFO | Average cost |
| 1. What is total sales? (19 units) | $190 | $190 | $190.00 |
| 2. What is total cost of merchandise sold? (19 units) | 85 | 105 | 95 |
| 3. What is gross profit? | 105 | 85 | 95 |
| 4. What is the ending inventory balance? (11 units) | 65 | 45 | 55 |
Under all three methods, 19 units were sold and total sales were $190. Notice, however, that under FIFO the 19 units COST $85, under LIFO these same 19 units COST $105, and under average cost these same 19 units COST $95. This is a $30 difference between the highest and lowest costing method.
Gross profit is also different among the three methods. Because less cost is deducted from sales under the FIFO method, gross profit is $30 higher under FIFO that it is for LIFO.
That $30 difference also appears in the ending inventory balances. Since the cost of merchandise sold was lower under FIFO than it was under LIFO and average cost, the ending inventory balance under FIFO is higher that with the other two methods.
To summarize, there is a $30 difference between FIFO and LIFO in the cost of goods sold and ending inventory amounts. FIFO includes that $30 as part of ending inventory; LIFO considers that $30 to be part of cost of merchandise sold.
NOTE: The pattern above will result when costs are rising over time. In this example, they increased from $4 to $5 to $6. If costs decrease over time, the results will be the opposite: LIFO would include the difference as part of ending inventory and FIFO would consider the difference to be part of cost of merchandise sold.
The results for the average cost method typically fall between those for LIFO and FIFO.
Also note that the results for FIFO are the same under the periodic and perpetual inventory systems.
4.1.3 Lower-of-Cost-or-Market Inventory Valuation
A company should follow the principle of conservatism, which means that if there is more than one way to report its financial information, the approach that shows the results in the least favorable light should be presented. In this way,readers of the financial information see the “worst-case scenario” and are not misled into believing the results are more positive than they really are.
The value of a company’s inventory is one of the amounts where this principle should apply. Therefore, after a company has valued its ending inventory by the FIFO, LIFO, or average cost method, it may take an additional step to ensure that the value of the inventory that is reported is not misinterpreted or overstated. Lower-of-cost-or-market is an additional calculation that is used to value inventory if the cost of a product (or products) declines after the item(s) has been purchased for inventory. “Market” can be interpreted as replacement cost, or what the item is selling for today. The company lists all the products it sells and for each product compares the price paid (cost) to the current market value. The lower of the two numbers is used to report the value of a product’s inventory on the balance sheet.
Notice how Merchandise Inventory is presented on the balance sheet when lower-of-cost-or-market is used.
The following example presents inventory data for July 31 for a business that uses the lower-of-cost-or-market basis of inventory valuation. The information in the white cells is given. The gray boxes are the cells that need to be calculated.
| Commodity | Quantity | Unit Cost | Unit Market Price | Total Cost | Total Market | Lower of Cost or Market |
| A | 10 | $ 6 | $ 5 | $ 60 | $ 50 | $ 50 |
| B | 20 | 10 | 11 | 200 | 220 | 200 |
| C | 30 | 9 | 8 | 270 | 240 | 240 |
| D | 40 | 3 | 4 | 120 | 160 | 120 |
| Totals | $650 | $610 |
- Multiply the inventory quantity by the unit cost price to get total cost.
- Multiply the inventory quantity by the unit market price to get total market value.
- For lower of cost or market, take the lower of the two results in each row.
The total purchase price of all of the merchandise combined is $650. The total lower-of-cost-or-market amount for all of the merchandise as of July 31 is $610. The inventory lost $40 of value due to market decline/prices dropping.
A business has two models of cell phones in stock to sell to customers.
It has 200 units of Model #1. Each of those cost the company $100. If the company were to buy these phones today, each unit would cost $110.
It also has 200 units of Model #2, which were purchased two years ago for $100 per unit. The market price for these is currently $60 per unit. It has dropped because these units are somewhat out of date.
If lower-of-cost-or-market is NOT used, the total inventory is valued at $40,000.
| Model #1: | 200 x $100 = $20,000 | (number of units x cost per unit) |
| Model #2: | 200 x $100 = 20,000 | (number of units x cost per unit) |
| Total | $40,000 |
If lower-of-cost-or-market is used, the total inventory is valued at $32,000.
| Model #1: | 200 x $100 = $20,000 | (number of units x cost per unit) |
| Model #2: | 200 x $ 60 = 12,000 | (number of units x market price per unit since it is lower) |
| Total | $32,000 |
The inventory should be reported at $32,000 on the balance sheet even though it was purchased for $40,000. This gives the reader a clearer picture of what the inventory is actually worth.
4.1.4 Physical Inventory Count
Companies using a perpetual inventory system keep a running total of the inventory they have on hand in their record books. At times, a physical inventory count is done to verify that a company actually has the amount of inventory that is indicated in its records.
The company will count/include the items that it owns that are on hand on its premises.
Items may be on the company’s premises that it does not own, and these should not be included in the physical inventory count. These may include:
- Items on consignment from someone else (the company has agreed to sell someone else’s product for them)
- Items in for warranty repair (the company does not re-possess these)
- Items held aside for customers that have been paid for already (ownership has been transferred)
The following items would be owned by the company and should be included:
- Items returned by customers (the company re-possesses these)
- Items held aside for customers that have not been paid for yet (ownership has not yet been transferred)
The company must also count/include items that it owns that are off premise at other locations. These may include:
- Items on consignment to someone else (the other party has agreed to sell the company’s items for them)
- Items out for warranty repair with another company (the other party does not re-possess these)
- Items that the company has purchased that are in transit (i.e., on the UPS truck) if the shipping terms are FOB shipping
- Items that the company has sold that are in transit (i.e., on the UPS truck) if the shipping terms are FOB destination
Whoever is responsible for absorbing the transportation cost (buyer or seller) also owns the merchandise while it is in transit.
Effect of Errors in Physical Inventory Count
To see the effect of an error in the physical inventory count on the financial statements, let’s assume that a business reports what it counts as its Merchandise Inventory amount on the balance sheet.
In the example below, assume that the correct amount of merchandise inventory on hand is $20,000. The amounts in yellow in the excerpts of the following financial statements are correct.
Understating Merchandise Inventory (reporting an amount that is too low)
The financial statements that follow show the effect of understating Merchandise Inventory, where something was missed in the physical inventory count. Only $19,500 rather than $20,000 is reported on the balance sheet.
As a result of this error, (1) Merchandise Inventory is understated, (2) Total assets are understated, (3) Cost of merchandise sold is overstated, (4) Net income is understated, and (5) Retained earnings and total stockholders’ equity (not shown) are understated.
Overstating Merchandise Inventory (reporting an amount that is too high)
The financial statements that follow show the effect of overstating Merchandise Inventory, where something in the physical inventory count was included that should not have been. Instead of $20,000, $20,500 is reported on the balance sheet.
As a result of this error, (1) Merchandise Inventory is overstated, (2) Total assets are overstated, (3) Cost of merchandise sold is understated, (4) Net income is overstated, and (5) Retained earnings and total stockholders’ equity (not shown) are overstated.
| ACCT 2101 Topics—Inventory Valuation | Fact | Journal Entry | Calculate Amount | Format |
| Concept of inventory valuation methods | x | |||
| Calculate cost of merchandise sold under FIFO | x | |||
| Calculate ending inventory under FIFO | x | |||
| Calculate cost of merchandise sold under LIFO | x | |||
| Calculate ending inventory under LIFO | x | |||
| Calculate cost of merchandise sold under average cost method | x | |||
| Calculate ending inventory under average cost method | x | |||
| Journalize purchase of merchandise on account under perpetual system | x | x | ||
| Journalize sale of merchandise on account under perpetual system | x | x | ||
| Calculate gross profit | x | |||
| Calculate lower-of-cost-or-market amounts | x | |||
| Financial statements | x | x | ||
| Physical inventory counts | x | x |


16.6: The Sociocultural Model and Therapy Utilization
Learning Objectives
- Explain how the sociocultural model is used in therapy
- Discuss barriers to mental health services among ethnic minorities
The sociocultural perspective looks at you, your behaviors, and your symptoms in the context of your culture and background. For example, José is an \(18\)-year-old Hispanic male from a traditional family. José comes to treatment because of depression.  He also discloses that he is concerned because his religious background has taught him that homosexuality is wrong. How does his religious and cultural background affect him? How might his cultural background affect how his family reacts if José were to tell them he is gay?
As our society becomes increasingly multiethnic and multiracial, mental health professionals must develop cultural competence (See figure \(\PageIndex{1}\)), which means they must understand and address issues of race, culture, and ethnicity. They must also develop strategies to effectively address the needs of various populations for which Eurocentric therapies have limited application (Sue, 2004). For example, a counselor whose treatment focuses on individual decision making may be ineffective at helping a Chinese client with a collectivist approach to problem solving (Sue, 2004).
Multicultural counseling and therapy aims to offer both a helping role and process that uses modalities and defines goals consistent with the life experiences and cultural values of clients. It strives to recognize client identities to include individual, group, and universal dimensions, advocate the use of universal and culture-specific strategies and roles in the healing process, and balances the importance of individualism and collectivism in the assessment, diagnosis, and treatment of client and client systems (Sue, 2001).
This therapeutic perspective integrates the impact of cultural and social norms, starting at the beginning of treatment. Therapists who use this perspective work with clients to obtain and integrate information about their cultural patterns into a unique treatment approach based on their particular situation (Stewart, Simmons, & Habibpour, 2012). Sociocultural therapy can include individual, group, family, and couples treatment modalities.
Barriers to Treatment
Statistically, ethnic minorities tend to utilize mental health services less frequently than White, middle-class Americans (Alegría et al., 2008; Richman, Kohn-Wood, & Williams, 2007). Why is this so? Perhaps the reason has to do with access and availability of mental health services. Ethnic minorities and individuals of low socioeconomic status (SES) report that barriers to services include lack of insurance, transportation, and time (Thomas & Snowden, 2002). However, researchers have found that even when income levels and insurance variables are taken into account, ethnic minorities are far less likely to seek out and utilize mental health services. And when access to mental health services is comparable across ethnic and racial groups, differences in service utilization remain (Richman et al., 2007).
In a study involving thousands of women, it was found that the prevalence rate of anorexia was similar across different races, but that bulimia nervosa was more prevalent among Hispanic and African American women when compared with non-Hispanic whites (Marques et al., 2011). Although they have similar or higher rates of eating disorders, Hispanic and African American women with these disorders tend to seek and engage in treatment far less than Caucasian women. These findings suggest ethnic disparities in access to care, as well as clinical and referral practices that may prevent Hispanic and African American women from receiving care, which could include lack of bilingual treatment, stigma, fear of not being understood, family privacy, and lack of education about eating disorders.
Perceptions and attitudes toward mental health services may also contribute to this imbalance. A recent study at King’s College, London, found many complex reasons why people do not seek treatment: self-sufficiency and not seeing the need for help, not seeing therapy as effective, concerns about confidentiality, and the many effects of stigma and shame (Clement et al., 2014). And in another study, African Americans exhibiting depression were less willing to seek treatment due to fear of possible psychiatric hospitalization as well as fear of the treatment itself (Sussman, Robins, & Earls, 1987). Instead of mental health treatment, many African Americans prefer to be self-reliant or use spiritual practices (Snowden, 2001; Belgrave & Allison, 2010). For example, it has been found that the Black church plays a significant role as an alternative to mental health services by providing prevention and treatment-type programs designed to enhance the psychological and physical well-being of its members (Blank, Mahmood, Fox, & Guterbock, 2002).
Additionally, people belonging to ethnic groups that already report concerns about prejudice and discrimination are less likely to seek services for a mental illness because they view it as an additional stigma (Gary, 2005; Townes, Cunningham, & Chavez-Korell, 2009; Scott, McCoy, Munson, Snowden, & McMillen, 2011). For example, in one recent study of 462 older Korean Americans (over the age of 60) many participants reported suffering from depressive symptoms. However, 71% indicated they thought depression was a sign of personal weakness, and 14% reported that having a mentally ill family member would bring shame to the family (Jang, Chiriboga, & Okazaki, 2009).
Language differences are a further barrier to treatment. In the previous study on Korean Americans’ attitudes toward mental health services, it was found that there were no Korean-speaking mental health professionals where the study was conducted (Orlando and Tampa, Florida) (Jang et al., 2009). Because of the growing number of people from ethnically diverse backgrounds, there is a need for therapists and psychologists to develop knowledge and skills to become culturally competent (Ahmed, Wilson, Henriksen, & Jones, 2011). Those providing therapy must approach the process from the context of the unique culture of each client (Sue & Sue, 2007).
DIG DEEPER: Treatment Perceptions
By the time a child is a senior in high school, \(20\%\) of his classmates—that is \(1\) in \(5\) —will have experienced a mental health problem (U.S. Department of Health and Human Services, 1999), and \(8\%\) —about \(1\) in \(12\) —will have attempted suicide (Centers for Disease Control and Prevention, 2014). Of those classmates experiencing mental disorders, only \(20\%\) will receive professional help (U.S. Public Health Service, 2000). Why?
It seems that the public has a negative perception of children and teens with mental health disorders. According to researchers from Indiana University, the University of Virginia, and Columbia University, interviews with over \(1,300\) U.S. adults show that they believe children with depression are prone to violence and that if a child receives treatment for a psychological disorder, then that child is more likely to be rejected by peers at school.
Bernice Pescosolido, author of the study, asserts that this is a misconception. However, stigmatization of psychological disorders is one of the main reasons why young people do not get the help they need when they are having difficulties. Pescosolido and her colleagues caution that this stigma surrounding mental illness, based on misconceptions rather than facts, can be devastating to the emotional and social well-being of our nation’s children.
This warning played out as a national tragedy in the 2012 shootings at Sandy Hook Elementary. In her blog, Suzy DeYoung (2013), co-founder of Sandy Hook Promise (the organization parents and concerned others set up in the wake of the school massacre) speaks to treatment perceptions and what happens when children do not receive the mental health treatment they desperately need.
I've become accustomed to the reaction when I tell people where I'm from.
Eleven months later, it's as consistent as it was back in January.
Just yesterday, inquiring as to the availability of a rental house this holiday season, the gentleman taking my information paused to ask, “Newtown, CT? Isn't that where that...that thing happened?
A recent encounter in the Massachusetts Berkshires, however, took me by surprise.
It was in a small, charming art gallery. The proprietor, a woman who looked to be in her 60s, asked where we were from. My response usually depends on my present mood and readiness for the inevitable dialogue. Sometimes it's simply, Connecticut. This time, I replied, Newtown, CT.
The woman's demeanor abruptly shifted from one of amiable graciousness to one of visible agitation.
“Oh my god,” she said wide eyed and open mouthed. “Did you know her?”
. . . .
“Her?” I inquired
That woman,” she replied with disdain, “that woman that raised that monster.”
“That woman's” name was Nancy Lanza. Her son, Adam, killed her with a rifle blast to the head before heading out to kill 20 children and six educators at Sandy Hook Elementary School in Newtown, CT last December 14th.
When Nelba Marquez Greene, whose beautiful 6-year-old daughter, Ana, was killed by Adam Lanza, was recently asked how she felt about “that woman,” this was her reply:
“She's a victim herself. And it's time in America that we start looking at mental illness with compassion, and helping people who need it."
“This was a family that needed help, an individual that needed help and didn't get it. And what better can come of this, of this time in America, than if we can get help to people who really need it?” (pars. 1–7, 10–15)
Fortunately, we are starting to see campaigns related to the destigmatization of mental illness and an increase in public education and awareness. Join the effort by encouraging and supporting those around you to seek help if they need it. To learn more, visit the National Alliance on Mental Illness (NAMI) website ( http://www.nami.org/ ). The nation’s largest nonprofit mental health advocacy and support organization is NAMI.
Summary
The sociocultural perspective looks at you, your behaviors, and your symptoms in the context of your culture and background. Clinicians using this approach integrate cultural and religious beliefs into the therapeutic process. Research has shown that ethnic minorities are less likely to access mental health services than their White middle-class American counterparts. Barriers to treatment include lack of insurance, transportation, and time; cultural views that mental illness is a stigma; fears about treatment; and language barriers.
Glossary
- cultural competence
- therapist’s understanding and attention to issues of race, culture, and ethnicity in providing treatment
Contributors and Attributions
-
Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


6.5: References, Resources and Works Cited
Byers, S. & Stanberry, K. (2020). Business Ethics . OpenStax. 8.2: Diversity and Inclusion in the Workforce - Business LibreTexts
Center for Workforce Information & Analysis. (2021). Economic Review of Pennsylvania 2020 . Work Stats Department of Labor and Industry [www.workstats.dli.pa.gov]
Crossover. (2022). Remote/Hybrid Work Policies at Fortune 500 Companies . Remote work policies of Fortune 500 companies | Crossover
Institute of Medicine Committee on the Robert Wood Johnson Foundation Initiative on the Future of Nursing, at the Institute of Medicine. (2011). The Future of Nursing: Leading Change, Advancing Health . National Academies Press. Institute of Medicine Committee on the Robert Wood Johnson Foundation Initiative on the Future of Nursing [www.ncbi.nlm.nih.gov]
Lee, Jordan. (2018, May 23). The Gig Economy Revisited: Benefits of Independent Work Over Antiquated 9-5 Employment. Medium.com The Gig Economy, Revisited: Benefits of Independent Work Over Antiquated 9–5 Employment | by Jordan Lee | Medium
LinkedIn. (2022). Find jobs . Indeed [www.indeed.com]
Lumen Learning. (2022). Business Communication Skills for Managers. Lumen Learning. Working across Generations | Business Communication Skills for Managers (lumenlearning.com)
Marcus, Jon. (2020, June 2). Urgency of getting people back to work gives new momentum to “microcredentials.” The Hechinger Report. “Microcredentials” gain new momentum due to urgency to get back to work (hechingerreport.org)
Martinez, G., Shaker, E. (2021). Career and Life Planning . LibreTexts. Career and Life Planning (Martinez and Shaker) - Social Sci LibreTexts
National Center for O*NET Development. (2022). 13-1161.01 - Search Marketing Strategists . O*NET OnLine. O*NET Online [www.onetonline.org]
Sodexo. (2017). 2017 Global Workplace Trends . Sodexo 2017 Global Workplace Trends [www.sodexo.com]
The Hartford. (2014). Generations at Work . The Hartford Generations at Work [s0.hfdstatic.com]
Zelevansky, Nora. (2014, December 1). Bridging the Gap at Work: Improving Intergenerational Communication. Coca Cola Journey. Zelevansky Bridging the Gap at Work [www.coca-colacompany.com]


5.6: Electrical Energy and Power
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LEARNING OBJECTIVES
By the end of this section, you will be able to:
- Express electrical power in terms of the voltage and the current
- Describe the power dissipated by a resistor in an electric circuit
- Calculate the energy efficiency and cost effectiveness of appliances and equipment
In an electric circuit, electrical energy is continuously converted into other forms of energy. For example, when a current flows in a conductor, electrical energy is converted into thermal energy within the conductor. The electrical field, supplied by the voltage source, accelerates the free electrons, increasing their kinetic energy for a short time. This increased kinetic energy is converted into thermal energy through collisions with the ions of the lattice structure of the conductor. Previously, we defined power as the rate at which work is done by a force measured in watts. Power can also be defined as the rate at which energy is transferred. In this section, we discuss the time rate of energy transfer, or power, in an electric circuit.
Power in Electric Circuits
Power is associated by many people with electricity. Power transmission lines might come to mind. We also think of light bulbs in terms of their power ratings in watts. What is the expression for electric power ?
Let us compare a 25-W bulb with a 60-W bulb (Figure \(\PageIndex{1a}\)). The 60-W bulb glows brighter than the 25-W bulb. Although it is not shown, a 60-W light bulb is also warmer than the 25-W bulb. The heat and light is produced by from the conversion of electrical energy. The kinetic energy lost by the electrons in collisions is converted into the internal energy of the conductor and radiation. How are voltage, current, and resistance related to electric power?
To calculate electric power, consider a voltage difference existing across a material (Figure \(\PageIndex{2}\)). The electric potential \(V_1\) is higher than the electric potential at \(V_2\), and the voltage difference is negative \(V = V_2 - V_1\). As discussed in Electric Potential , an electrical field exists between the two potentials, which points from the higher potential to the lower potential. Recall that the electrical potential is defined as the potential energy per charge, \(V = \Delta U /q\), and the charge \(\Delta Q\) loses potential energy moving through the potential difference.
If the charge is positive, the charge experiences a force due to the electrical field \(\vec{F} = m\vec{a} = \Delta Q\vec{E}\). This force is necessary to keep the charge moving. This force does not act to accelerate the charge through the entire distance \(\Delta L\) because of the interactions of the charge with atoms and free electrons in the material. The speed, and therefore the kinetic energy, of the charge do not increase during the entire trip across \(\Delta L\), and charge passing through area \(A_2\) has the same drift velocity \(v_d\) as the charge that passes through area \(A_1\). However, work is done on the charge, by the electrical field, which changes the potential energy. Since the change in the electrical potential difference is negative, the electrical field is found to be
\[E = - \dfrac{(V_2 - V_1)}{\Delta L} = \dfrac{V}{\Delta L}. \nonumber \]
The work done on the charge is equal to the electric force times the length at which the force is applied,
\[W = F \Delta L = (\Delta Q E) \Delta L = \left(\Delta Q \dfrac{V}{\Delta L}\right) \Delta L = \Delta Q V = \Delta U. \nonumber \]
The charge moves at a drift velocity \(v_d\) so the work done on the charge results in a loss of potential energy, but the average kinetic energy remains constant. The lost electrical potential energy appears as thermal energy in the material. On a microscopic scale, the energy transfer is due to collisions between the charge and the molecules of the material, which leads to an increase in temperature in the material. The loss of potential energy results in an increase in the temperature of the material, which is dissipated as radiation. In a resistor, it is dissipated as heat, and in a light bulb, it is dissipated as heat and light.
The power dissipated by the material as heat and light is equal to the time rate of change of the work:
\[P = IV = I(IR) = I^2R \nonumber \]
or
\[P = IV = \left(\dfrac{V}{R}\right) V = \dfrac{V^2}{R}. \nonumber \]
If a resistor is connected to a battery, the power dissipated as radiant energy by the wires and the resistor is equal to
\[P = IV = I^2R = \dfrac{V^2}{R}. \nonumber \]
The power supplied from the battery is equal to current times the voltage, \(P = IV\).
Definition: Electric Power
The electric power gained or lost by any device has the form
\[P = IV. \nonumber \]
The power dissipated by a resistor has the form
\[P = I^2 R = \dfrac{V^2}{R}. \nonumber \]
Different insights can be gained from the three different expressions for electric power. For example, \(P = V^2/R\) implies that the lower the resistance connected to a given voltage source, the greater the power delivered. Furthermore, since voltage is squared in \(P = V^2/R\), the effect of applying a higher voltage is perhaps greater than expected. Thus, when the voltage is doubled to a 25-W bulb, its power nearly quadruples to about 100 W, burning it out. If the bulb’s resistance remained constant, its power would be exactly 100 W, but at the higher temperature, its resistance is higher, too.
Example \(\PageIndex{1}\): Calculating Power in Electric Devices
A DC winch motor is rated at 20.00 A with a voltage of 115 V. When the motor is running at its maximum power, it can lift an object with a weight of 4900.00 N a distance of 10.00 m, in 30.00 s, at a constant speed.
- What is the power consumed by the motor?
- What is the power used in lifting the object? Ignore air resistance. (c) Assuming that the difference in the power consumed by the motor and the power used lifting the object are dissipated as heat by the resistance of the motor, estimate the resistance of the motor?
Strategy
- The power consumed by the motor can be found using \(P = IV\).
- The power used in lifting the object at a constant speed can be found using \(P = Fv\), where the speed is the distance divided by the time. The upward force supplied by the motor is equal to the weight of the object because the acceleration is zero. (c) The resistance of the motor can be found using \(P = I^2 R\).
Solution
- The power consumed by the motor is equal to \(P = IV\) and the current is given as 20.00 A and the voltage is 115.00 V: \[P = IV = (20.00 \, A) 115.00 \, V = 2300.00 \, W. \nonumber \]
- The power used lifting the object is equal to \(P = Fv\) where the force is equal to the weight of the object (1960 N) and the magnitude of the velocity is \[v = \dfrac{10.00 \, m}{30.00 \, s} = 0.33 \dfrac{m}{s} \nonumber \] \[P = Fv = (4900 \, N)0.33 \, m/s = 1633.33 \, W. \nonumber \]
- The difference in the power equals \(2300.00 \, W - 1633.33 \, W = 666.67 \, W\) and the resistance can be found using \(P = I^2R\): \[R = \dfrac{P}{I^2} = \dfrac{666.67 \, W}{(20.00 \, A)^2} = 1.67 \, \Omega. \nonumber \]
Significance The resistance of the motor is quite small. The resistance of the motor is due to many windings of copper wire. The power dissipated by the motor can be significant since the thermal power dissipated by the motor is proportional to the square of the current \((P = I^2R)\).
Exercise \(\PageIndex{1}\)
Electric motors have a reasonably high efficiency. A 100-hp motor can have an efficiency of 90% and a 1-hp motor can have an efficiency of 80%. Why is it important to use high-performance motors?
- Answer
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Even though electric motors are highly efficient 10–20% of the power consumed is wasted, not being used for doing useful work. Most of the 10–20% of the power lost is transferred into heat dissipated by the copper wires used to make the coils of the motor. This heat adds to the heat of the environment and adds to the demand on power plants providing the power. The demand on the power plant can lead to increased greenhouse gases, particularly if the power plant uses coal or gas as fuel.
A fuse (Figure \(\PageIndex{3}\)) is a device that protects a circuit from currents that are too high. A fuse is basically a short piece of wire between two contacts. As we have seen, when a current is running through a conductor, the kinetic energy of the charge carriers is converted into thermal energy in the conductor. The piece of wire in the fuse is under tension and has a low melting point. The wire is designed to heat up and break at the rated current. The fuse is destroyed and must be replaced, but it protects the rest of the circuit. Fuses act quickly, but there is a small time delay while the wire heats up and breaks.
Circuit breakers are also rated for a maximum current, and open to protect the circuit, but can be reset. Circuit breakers react much faster. The operation of circuit breakers is not within the scope of this chapter and will be discussed in later chapters. Another method of protecting equipment and people is the ground fault circuit interrupter (GFCI), which is common in bathrooms and kitchens. The GFCI outlets respond very quickly to changes in current. These outlets open when there is a change in magnetic field produced by current-carrying conductors, which is also beyond the scope of this chapter and is covered in a later chapter.
The Cost of Electricity
The more electric appliances you use and the longer they are left on, the higher your electric bill. This familiar fact is based on the relationship between energy and power. You pay for the energy used. Since \(P = \dfrac{dE}{dt}\), we see that
\[E = \int P dt \nonumber \]
is the energy used by a device using power P for a time interval t . If power is delivered at a constant rate, then then the energy can be found by \(E = Pt\). For example, the more light bulbs burning, the greater P used; the longer they are on, the greater t is.
The energy unit on electric bills is the kilowatt-hour \((kW \cdot h)\), consistent with the relationship \(E = Pt\). It is easy to estimate the cost of operating electrical appliances if you have some idea of their power consumption rate in watts or kilowatts, the time they are on in hours, and the cost per kilowatt-hour for your electric utility. Kilowatt-hours, like all other specialized energy units such as food calories, can be converted into joules. You can prove to yourself that \(1 \, kW \cdot h = 3.6 \times 10^6 \, J\).
The electrical energy ( E ) used can be reduced either by reducing the time of use or by reducing the power consumption of that appliance or fixture. This not only reduces the cost but also results in a reduced impact on the environment. Improvements to lighting are some of the fastest ways to reduce the electrical energy used in a home or business. About 20% of a home’s use of energy goes to lighting, and the number for commercial establishments is closer to 40%. Fluorescent lights are about four times more efficient than incandescent lights—this is true for both the long tubes and the compact fluorescent lights (CFLs), e.g., Figure \(\PageIndex{1b}\). Thus, a 60-W incandescent bulb can be replaced by a 15-W CFL, which has the same brightness and color. CFLs have a bent tube inside a globe or a spiral-shaped tube, all connected to a standard screw-in base that fits standard incandescent light sockets. (Original problems with color, flicker, shape, and high initial investment for CFLs have been addressed in recent years.)
The heat transfer from these CFLs is less, and they last up to 10 times longer than incandescent bulbs. The significance of an investment in such bulbs is addressed in the next example. New white LED lights (which are clusters of small LED bulbs) are even more efficient (twice that of CFLs) and last five times longer than CFLs.
Example \(\PageIndex{2\): Calculating the Cost Effectiveness of LED Bulb
The typical replacement for a 100-W incandescent bulb is a 20-W LED bulb. The 20-W LED bulb can provide the same amount of light output as the 100-W incandescent light bulb. What is the cost savings for using the LED bulb in place of the incandescent bulb for one year, assuming $0.10 per kilowatt-hour is the average energy rate charged by the power company? Assume that the bulb is turned on for three hours a day.
Strategy
- Calculate the energy used during the year for each bulb, using \(E = Pt\).
- Multiply the energy by the cost.
Solution
- Calculate the power for each bulb. \[E_{Incandescent} = Pt = 100 \, W \left(\dfrac{1 \, kW}{1000 \, W}\right) \left(\dfrac{3 \, h}{day}\right) (365 \, days) = 109.5 \, kW \cdot h \nonumber \] \[E_{LED} = Pt = 20 \, W \left(\dfrac{1 \, kW}{1000 \, W}\right) \left(\dfrac{3 \, h}{day}\right) (365 \, days) = 21.9 \, kW \cdot h \nonumber \]
- Calculate the cost for each. \[cost_{Incandescent} = 109.5 \, kW \cdot h \left(\dfrac{\$0.10}{kW \cdot h}\right) = \$10.95 \nonumber \] \[cost_{LED} = 21.90 \, kW \cdot h \left(\dfrac{\$0.10}{kW \cdot h}\right) = \$2.19 \nonumber \]
Significance
A LED bulb uses 80% less energy than the incandescent bulb, saving $8.76 over the incandescent bulb for one year. The LED bulb can cost $20.00 and the 100-W incandescent bulb can cost $0.75, which should be calculated into the computation.  The incandescent bulb would last 1.08 years at 3 hours a day and the LED bulb would last 45.66 years. The initial cost of the LED bulb is high, but the cost to the home owner will be $0.69 for the incandescent bulbs versus $0.44 for the LED bulbs per year. (Note that the LED bulbs are coming down in price.) The cost savings per year is approximately $8.50, and that is just for one bulb.
Exercise \(\PageIndex{2}\)
Is the efficiency of the various light bulbs the only consideration when comparing the various light bulbs?
- Answer
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No, the efficiency is a very important consideration of the light bulbs, but there are many other considerations. As mentioned above, the cost of the bulbs and the life span of the bulbs are important considerations. For example, CFL bulbs contain mercury, a neurotoxin, and must be disposed of as hazardous waste. When replacing incandescent bulbs that are being controlled by a dimmer switch with LED, the dimmer switch may need to be replaced. The dimmer switches for LED lights are comparably priced to the incandescent light switches, but this is an initial cost which should be considered. The spectrum of light should also be considered, but there is a broad range of color temperatures available, so you should be able to find one that fits your needs. None of these considerations mentioned are meant to discourage the use of LED or CFL light bulbs, but they are considerations.
Changing light bulbs from incandescent bulbs to CFL or LED bulbs is a simple way to reduce energy consumption in homes and commercial sites. CFL bulbs operate with a much different mechanism than do incandescent lights. The mechanism is complex and beyond the scope of this chapter, but here is a very general description of the mechanism. CFL bulbs contain argon and mercury vapor housed within a spiral-shaped tube. The CFL bulbs use a “ballast” that increases the voltage used by the CFL bulb. The ballast produce an electrical current, which passes through the gas mixture and excites the gas molecules. The excited gas molecules produce ultraviolet (UV) light, which in turn stimulates the fluorescent coating on the inside of the tube. This coating fluoresces in the visible spectrum, emitting visible light. Traditional fluorescent tubes and CFL bulbs had a short time delay of up to a few seconds while the mixture was being “warmed up” and the molecules reached an excited state. It should be noted that these bulbs do contain mercury, which is poisonous, but if the bulb is broken, the mercury is never released. Even if the bulb is broken, the mercury tends to remain in the fluorescent coating. The amount is also quite small and the advantage of the energy saving may outweigh the disadvantage of using mercury.
The CFL light bulbs are being replaced with LED light bulbs, where LED stands for “light-emitting diode.” The diode was briefly discussed as a nonohmic device, made of semiconducting material, which essentially permits current flow in one direction. LEDs are a special type of diode made of semiconducting materials infused with impurities in combinations and concentrations that enable the extra energy from the movement of the electrons during electrical excitation to be converted into visible light. Semiconducting devices will be explained in greater detail in Condensed Matter Physics .
Commercial LEDs are quickly becoming the standard for commercial and residential lighting, replacing incandescent and CFL bulbs. They are designed for the visible spectrum and are constructed from gallium doped with arsenic and phosphorous atoms. The color emitted from an LED depends on the materials used in the semiconductor and the current. In the early years of LED development, small LEDs found on circuit boards were red, green, and yellow, but LED light bulbs can now be programmed to produce millions of colors of light as well as many different hues of white light.
Comparison of Incandescent, CFL, and LED Light Bulbs
The energy savings can be significant when replacing an incandescent light bulb or a CFL light bulb with an LED light. Light bulbs are rated by the amount of power that the bulb consumes, and the amount of light output is measured in lumens. The lumen (lm) is the SI -derived unit of luminous flux and is a measure of the total quantity of visible light emitted by a source. A 60-W incandescent light bulb can be replaced with a 13- to 15-W CFL bulb or a 6- to 8-W LED bulb, all three of which have a light output of approximately 800 lm. A table of light output for some commonly used light bulbs appears in Table \(\PageIndex{1}\).
The life spans of the three types of bulbs are significantly different. An LED bulb has a life span of 50,000 hours, whereas the CFL has a lifespan of 8000 hours and the incandescent lasts a mere 1200 hours. The LED bulb is the most durable, easily withstanding rough treatment such as jarring and bumping. The incandescent light bulb has little tolerance to the same treatment since the filament and glass can easily break. The CFL bulb is also less durable than the LED bulb because of its glass construction. The amount of heat emitted is 3.4 btu/h for the 8-W LED bulb, 85 btu/h for the 60-W incandescent bulb, and 30 btu/h for the CFL bulb. As mentioned earlier, a major drawback of the CFL bulb is that it contains mercury, a neurotoxin, and must be disposed of as hazardous waste. From these data, it is easy to understand why the LED light bulb is quickly becoming the standard in lighting.
| Light Output (lumens) | LED Light Bulbs (watts) | Incandescent Light Bulbs (watts) | CFL Light Bulbs (watts) |
|---|---|---|---|
| 450 | 4−5 | 40 | 9−13 |
| 800 | 6−8 | 60 | 13−15 |
| 1100 | 9−13 | 75 | 18−25 |
| 1600 | 16−20 | 100 | 23−30 |
| 2600 | 25−28 | 150 | 30−55 |
Summary of Relationships
In this chapter, we have discussed relationships between voltages, current, resistance, and power. Figure \(\PageIndex{4}\) shows a summary of the relationships between these measurable quantities for ohmic devices. (Recall that ohmic devices follow Ohm’s law \(V = IR\).) For example, if you need to calculate the power, use the pink section, which shows that \(P = VI, \, P = \dfrac{V^2}{R}\), and \(P = I^2R\).
Which equation you use depends on what values you are given, or you measure. For example if you are given the current and the resistance, use \(P = I^2R\). Although all the possible combinations may seem overwhelming, don’t forget that they all are combinations of just two equations, Ohm’s law \((V = IR)\) and power \((P = IV)\).


3.2: Development of Parents
How Parents Develop
Check-in Time!
Have you ever thought about parenting? What are some of your ideas about what makes a "good enough" parent?
Parenting is a complex process in which parents and children both impact one another. There are many reasons that parents behave the way they do. The multiple influences on parenting are still being explored. Proposed influences on parental behavior include:
- parent characteristics
- child characteristics
- contextual and sociocultural characteristics
Parent Characteristics
Parents bring unique traits and qualities to the parenting relationship that affect their decisions as parents. These characteristics include a parent’s age, gender identity, personality, developmental history, beliefs, knowledge about parenting and child development, and mental and physical health. Parents’ personalities also affect parenting behaviors. Parents who are more agreeable, conscientious, and outgoing are warmer and provide more structure to their children. Parents who are more agreeable, less anxious, and less negative also support their children’s autonomy more than parents who are anxious and less agreeable. Parents who have these personality traits appear to be better able to respond to their children positively and provide a more consistent, structured environment for their children.
Parents’ developmental histories, or their experiences as children, can also affect their parenting strategies. Parents may learn parenting practices from their own parents. Fathers whose own parents provided monitoring, consistent and age-appropriate discipline, and warmth are more likely to provide this constructive parenting to their own children. Patterns of negative parenting and ineffective discipline also appear from one generation to the next. However, parents who are dissatisfied with their primary caregivers’ approach may be more likely to change their parenting methods when they have children.
Child Characteristics
Parenting is bidirectional. Not only do parents and caregivers affect their children, but children influence their parents/primary caregivers as well. Child characteristics, such as gender identity, birth order, temperament, and health status, can affect child-rearing behaviors and roles. For example, an infant with an easy temperament may enable caregivers to feel more effective, as they are easily able to soothe the child and elicit smiling and cooing. On the other hand, a cranky or fussy infant can elicit fewer positive reactions from caregivers and may result in parents feeling less effective in the role. Over time, parents of more challenging children may become more punitive and less patient with their children. Many parents who have a fussy, difficult child have been found to be less satisfied with their relationships and have greater challenges in balancing work and family roles. Thus, child temperament is one of the child characteristics that influences how caregivers behave with their children.
Another child characteristic is the child’s gender identity. Some parents assign different household chores to their children based on their child’s gender identity. For example, older research has shown girls are more often responsible for caring for younger siblings and household chores, whereas boys are more likely to be asked to perform chores outside the home, such as mowing the lawn. Research has also demonstrated that some parents talk differently with their children based on their child’s gender identity, such as providing more scientific explanations to their sons and using more emotion words with their daughters. These cultural practices impact the way in which parents see their children and the children see their role in the family and in society.
Contextual Factors and Sociocultural Characteristics
The parent-child relationship does not occur in isolation. Sociocultural characteristics, including economic hardship, religion, politics, neighborhoods, schools, and social support, can also influence parenting. Parents who experience economic hardship tend to be more easily frustrated, depressed, and sad, and these emotional characteristics can affect their parenting skills.
Culture can also impact parenting behaviors in fundamental ways. Although promoting the development of skills necessary to function effectively in one’s community, to the best of one’s abilities, is a universal goal of parenting, the specific skills necessary vary widely from culture to culture. Thus, parents have different goals for their children that mostly depend on their culture. In our early learning programs in America, there is a tendency to focus on children's independence. This is not so for all cultures. Some cultures see the value of interdependence. This is often referred to as collectivism. This is where the family places the group over the self. This is an important concept to think about as the children and families you will serve, have differing goals for their children that must be acknowledged and valued.
These differences in parental goals are influenced by the culture of the family and their sense of belonging in their communities. The roles of an early learning professional is to acknowledge the complexities of parenting and to provide support that is culturally appropriate for each of the children and families they serve. It is crucial that early learning professionals acknowledge those complexities and do not think of parenting as a one size fits all.
The Six Stages of Parenthood - Ellen Galinsky
In the 80's, Ellen Galinsky, a researcher who studies changing family dynamics, was one of the first scholars to emphasize the development of parents themselves, how they respond to their children’s development, and how they grow as parents. Parenthood is an experience that transforms one’s identity as one takes on new roles. Children’s growth and development force parents to change their roles. They must develop new skills and abilities in response to children’s development. Galinsky identified six stages of parenthood that focus on different tasks and goals:
Stage 1:
The Image-Making Stage (prospective parents) - They enter this stage when the family is planning for a child. They are considering what it means to be parents and their plans for the future. They (a) think about and form images about their roles as parents, (b) contemplate what will emerge as a result of parenthood, and (c) prepare for changes associated adding an infant to their family. Prospective parents develop ideas about what it will be like to be a parent and what type of parent they want to be. Individuals may evaluate their relationships with their own parents as a model for their upcoming roles as parents.
Stage 2:
The Nurturing Stage (infancy) - This is the stage that occurs after the birth of the baby or after a baby may join a family through adoption. A parent’s main goal during this stage is to develop an attachment relationship with their baby. Parents must adapt their romantic relationships, their relationships with their other children, and their relationships with their own parents to include the new infant. Some parents feel attached to their baby immediately, but for other parents, this occurs more gradually. Parents may have imagined their infant in specific ways, but they now have to reconcile those images with their actual baby. In incorporating their relationship with their child into their other relationships, parents often have to reshape their conceptions of themselves and their identity. Parenting responsibilities are the most demanding during infancy because infants are completely dependent on caregiving.
Stage 3 :
The Authority Stage (toddler and pre-school) - During this stage, parents make decisions about how much authority to exert over their children’s behavior. Parents must establish rules to guide their child’s behavior and development. They have to decide how strictly they should enforce rules and what to do when rules are broken. Parents create rules and figure out how to effectively guide their child’s behavior.
Stage 4:
The Interpretive Stage (middle childhood) - During this stage, parents interpret their children’s experiences as they are increasingly exposed to the world outside of the family. Parents answer their children’s questions, provide explanations, and determine what behaviors and values to teach. They decide what experiences to provide their children in terms of schooling, neighborhood, and extracurricular activities. By this time, parents have experience in the parenting role and often reflect on their strengths and weaknesses as parents, review their images of parenthood, and determine how realistic they have been. Parents have to negotiate how involved to be with their children, when to step in, and when to encourage children to make choices independently.
Stage 5:
The Interdependent Stage (adolescence) - During this stage, parents must redefine their authority and renegotiate their relationship with their adolescent as the children increasingly make decisions independent of parental control and authority. On the other hand, parents do not permit their adolescent children to have complete autonomy over their decision-making and behavior, and thus adolescents and parents must adapt their relationship to allow for greater negotiation and discussion about rules and limits.
Stage 6:
The Departure Stage (early adulthood) - During the departure stage of parenting, parents evaluate their entire parenting experience.  This stage forms a transition to a new era in parents’ lives. This stage usually spans a long time period from when the oldest child moves away (and often returns) until the youngest child leaves. The parenting role must be redefined as a less central role in a parent’s identity.
Cultural Considerations
Despite the interest in the development of parents, little research has examined developmental changes in parents’ experiences and behaviors over time. Thus, it is not clear whether these theoretical stages are generalizable to parents of different races, ages, cultures, and religions, nor do we have empirical data on the factors that influence individual differences in these stages. In evaluating these ideas, early learning professionals must consider the many differences that exist among parents and use these ideas as more of a guide in understanding parents points of view.
References:
"Parenting and Family Diversity" by Diana Lang is licensed under CC BY-NC-SA 4.0
"Child, Family, and Community" by Rebecca Laff and Wendy Ruiz is licensed under CC BY 4.0


6.2: Wave functions
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Learning Objectives
By the end of this section, you will be able to:
- Describe the statistical interpretation of the wavefunction
- Use the wavefunction to determine probabilities
- Calculate expectation values of position, momentum, and kinetic energy
In the preceding chapter, we saw that particles act in some cases like particles and in other cases like waves. But what does it mean for a particle to “act like a wave”? What precisely is “waving”? What rules govern how this wave changes and propagates? How is the wavefunction used to make predictions? For example, if the amplitude of an electron wave is given by a function of position and time, \(\Psi \, (x,t)\), defined for all x , where exactly is the electron? The purpose of this chapter is to answer these questions.
Using the Wavefunction
A clue to the physical meaning of the wavefunction \(\Psi \, (x,t)\) is provided by the two-slit interference of monochromatic light (Figure \(\PageIndex{1}\)) that behave as electromagnetic waves . The wavefunction of a light wave is given by E ( x , t ), and its energy density is given by \(|E|^2\), where E is the electric field strength. The energy of an individual photon depends only on the frequency of light, \(\epsilon_{photon} = hf\), so \(|E|^2\) is proportional to the number of photons. When light waves from \(S_1\) interfere with light waves from \(S_2\) at the viewing screen (a distance D away), an interference pattern is produced (\(\PageIndex{1a}\)). Bright fringes correspond to points of constructive interference of the light waves, and dark fringes correspond to points of destructive interference of the light waves (\(\PageIndex{1b}\)).
Suppose the screen is initially unexposed to light. If the screen is exposed to very weak light, the interference pattern appears gradually (Figure \(\PageIndex{1c}\), left to right). Individual photon hits on the screen appear as dots. The dot density is expected to be large at locations where the interference pattern will be, ultimately, the most intense. In other words, the probability (per unit area) that a single photon will strike a particular spot on the screen is proportional to the square of the total electric field, \(|E|^2\) at that point. Under the right conditions, the same interference pattern develops for matter particles, such as electrons.
The square of the matter wave \(|\Psi|^2\) in one dimension has a similar interpretation as the square of the electric field \(|E|^2\). It gives the probability that a particle will be found at a particular position and time per unit length, also called the probability density . The probability (\(P\)) a particle is found in a narrow interval ( x , x + dx ) at time t is therefore
\[P(x,x + dx) = |\Psi \, (x,t)|^2 dx. \label{7.1} \]
(Later, we define the magnitude squared for the general case of a function with “imaginary parts.”) This probabilistic interpretation of the wavefunction is called the Born interpretation . Examples of wavefunctions and their squares for a particular time \(t\) are given in Figure \(\PageIndex{2}\).
If the wavefunction varies slowly over the interval \(\Delta x\), the probability a particle is found in the interval is approximately
\[P(x,x + \Delta x) \approx |\Psi \, (x,t)|^2 \delta x. \label{7.2} \]
Notice that squaring the wavefunction ensures that the probability is positive. (This is analogous to squaring the electric field strength—which may be positive or negative—to obtain a positive value of intensity.) However, if the wavefunction does not vary slowly, we must integrate:
\[P(x,x + \Delta x) = \int_x^{x + \Delta x} |\Psi \, (x,t)|^2 dx. \label{7.3} \]
This probability is just the area under the function \(|Ψ(x,t)|^2\) between \(x\) and \(x+Δx\). The probability of finding the particle “somewhere” (the normalization condition) is
\[P(-\infty, +\infty) = \int_{-\infty}^{\infty} |\Psi \, (x,t)|^2 dx = 1.\label{7.4} \]
For a particle in two dimensions, the integration is over an area and requires a double integral; for a particle in three dimensions, the integration is over a volume and requires a triple integral. For now, we stick to the simple one-dimensional case.
Example \(\PageIndex{1A}\): Where Is the Ball? (Part I)
A ball is constrained to move along a line inside a tube of length \(L\). The ball is equally likely to be found anywhere in the tube at some time \(t\). What is the probability of finding the ball in the left half of the tube at that time? (The answer is 50%, of course, but how do we get this answer by using the probabilistic interpretation of the quantum mechanical wavefunction?)
Strategy
The first step is to write down the wavefunction.  The normalization condition can be used to find the value of the function and a simple integration over half of the box yields the final answer.
Solution
The wavefunction of the ball can be written as \(\Psi \, (x,t) = C(0 < x < L)\), where \(C\) is a constant, and \(\Psi \, (x,t) = 0\) otherwise. We can determine the constant C by applying the normalization condition (we set \(t = 0\) to simplify the notation):
\[P(\infty, +\infty) = \int_{-\infty}^{\infty} |C|^2 dx = 1. \nonumber \]
This integral can be broken into three parts: (1) negative infinity to zero, (2) zero to L , and (3) L to infinity. The particle is constrained to be in the tube, so \(C=0\) outside the tube and the first and last integrations are zero. The above equation can therefore be written
\[P(x = 0, L) = \int_0^L |C|^2 dx = 1. \nonumber \]
The value C does not depend on x and can be taken out of the integral, so we obtain
\[|C|^2 \int_0^L dx = 1. \nonumber \]
Integration gives
\[C = \sqrt{\dfrac{1}{L}}. \nonumber \]
To determine the probability of finding the ball in the first half of the box (\(0 < x < L\)), we have
\[\begin{align} P(x = 0, L/2) &= \int_0^{L/2} \left|\sqrt{\dfrac{1}{L}}\right|^2dx \nonumber \\[5pt] &= \left(\dfrac{1}{L}\right)\dfrac{L}{2} \nonumber \\[5pt] &= 0.50. \end{align} \nonumber \]
Significance
The probability of finding the ball in the first half of the tube is 50%, as expected. Two observations are noteworthy. First, this result corresponds to the area under the constant function from \(x=0\) to \(L/2\) (the area of a square left of L /2). Second, this calculation requires an integration of the square of the wavefunction. A common mistake in performing such calculations is to forget to square the wavefunction before integration.
Example \(\PageIndex{1B}\): Where Is the Ball? (Part II)
A ball is again constrained to move along a line inside a tube of length L . This time, the ball is found preferentially in the middle of the tube. One way to represent its wavefunction is with a simple cosine function (Figure \(\PageIndex{4}\)). What is the probability of finding the ball in the last one-quarter of the tube?
Strategy
We use the same strategy as before. In this case, the wavefunction has two unknown constants: One is associated with the wavelength of the wave and the other is the amplitude of the wave. We determine the amplitude by using the boundary conditions of the problem, and we evaluate the wavelength by using the normalization condition. Integration of the square of the wavefunction over the last quarter of the tube yields the final answer. The calculation is simplified by centering our coordinate system on the peak of the wavefunction.
Solution
The wavefunction of the ball can be written
\[\Psi \, (x,t) = A \, \cos \, (kx) (-L/2 < x < L/2), \nonumber \]
where \(A\) is the amplitude of the wavefunction and \(k = 2\pi/\lambda\) is its wavenumber . Beyond this interval, the amplitude of the wavefunction is zero because the ball is confined to the tube. Requiring the wavefunction to terminate at the right end of the tube gives
\[\Psi \left(x = \dfrac{L}{2}, 0 \right) = 0. \nonumber \]
Evaluating the wavefunction at \(x = L/2\) gives
\[A \, \cos \, (kL/2) = 0. \nonumber \]
This equation is satisfied if the argument of the cosine is an integral multiple of \(π/2\), \(3π/2\), \(5π/2\), and so on. In this case, we have
\[\dfrac{kL}{2} = \dfrac{\pi}{2}, \nonumber \] or \[k = \dfrac{\pi}{L}. \nonumber \]
Applying the normalization condition gives \(A = \sqrt{2/L}\), so the wavefunction of the ball is
\[\Psi \, (x,0) = \sqrt{\dfrac{2}{L}} \, cos \, (\pi x/L), \, -L/2 < x < L/2. \nonumber \]
To determine the probability of finding the ball in the last quarter of the tube, we square the function and integrate:
\[P(x = L/4, L/2) = \int_{L/4}^{L/2} \left|\sqrt{\dfrac{2}{L}} \, \cos \, \left(\dfrac{\pi x}{L}\right) \right| ^2 dx = 0.091. \nonumber \]
Significance
The probability of finding the ball in the last quarter of the tube is 9.1%. The ball has a definite wavelength (\(\lambda = 2L\)). If the tube is of macroscopic length (\(L = 1 \, m\)), the momentum of the ball is
\[p = \dfrac{h}{\lambda} = \dfrac{h}{2L} \approx10^{-36} m/s. \nonumber \]
This momentum is much too small to be measured by any human instrument.
An Interpretation of the Wavefunction
We are now in position to begin to answer the questions posed at the beginning of this section. First, for a traveling particle described by \(\Psi \, (x,t) = A \, \sin \, (kx - \omega t)\), what is “waving?” Based on the above discussion, the answer is a mathematical function that can, among other things, be used to determine where the particle is likely to be when a position measurement is performed. Second, how is the wavefunction used to make predictions? If it is necessary to find the probability that a particle will be found in a certain interval, square the wavefunction and integrate over the interval of interest. Soon, you will learn soon that the wavefunction can be used to make many other kinds of predictions, as well.
Third, if a matter wave is given by the wavefunction \(\Psi \, (x,t)\), where exactly is the particle? Two answers exist: (1) when the observer is not looking (or the particle is not being otherwise detected), the particle is everywhere (\(x = -\infty, +\infty\)); and (2) when the observer is looking (the particle is being detected), the particle “jumps into” a particular position state (\(x,x + dx\)) with a probability given by
\[P(x,x + dx) = |\Psi \, (x,t)|^2 dx \nonumber \]
via a process called state reduction or wavefunction collapse . This answer is called the Copenhagen interpretation of the wavefunction, or of quantum mechanics.
To illustrate this interpretation, consider the simple case of a particle that can occupy a small container either at \(x_1\) or \(x_2\) (Figure \(\PageIndex{5}\)). In classical physics, we assume the particle is located either at \(x_1\) or \(x_2\) when the observer is not looking. However, in quantum mechanics, the particle may exist in a state of indefinite position—that is, it may be located at \(x_1\) and \(x_2\) when the observer is not looking. The assumption that a particle can only have one value of position (when the observer is not looking) is abandoned. Similar comments can be made of other measurable quantities, such as momentum and energy.
The bizarre consequences of the Copenhagen interpretation of quantum mechanics are illustrated by a creative thought experiment first articulated by Erwin Schrödinger ( National Geographic , 2013) (\(\PageIndex{6}\)):
“A cat is placed in a steel box along with a Geiger counter, a vial of poison, a hammer, and a radioactive substance. When the radioactive substance decays, the Geiger detects it and triggers the hammer to release the poison, which subsequently kills the cat. The radioactive decay is a random [probabilistic] process, and there is no way to predict when it will happen. Physicists say the atom exists in a state known as a superposition—both decayed and not decayed at the same time. Until the box is opened, an observer doesn’t know whether the cat is alive or dead—because the cat’s fate is intrinsically tied to whether or not the atom has decayed and the cat would [according to the Copenhagen interpretation] be “living and dead ... in equal parts” until it is observed.”
Schrödinger took the absurd implications of this thought experiment (a cat simultaneously dead and alive) as an argument against the Copenhagen interpretation. However, this interpretation remains the most commonly taught view of quantum mechanics.
Quantum Computing: qubits
Two-state systems (left and right, atom decays and does not decay, and so on) are often used to illustrate the principles of quantum mechanics. These systems find many applications in nature, including electron spin and mixed states of particles, atoms, and even molecules. Two-state systems are also finding application in the quantum computer, as mentioned in the introduction of this chapter. Unlike a digital computer, which encodes information in binary digits (zeroes and ones), a quantum computer stores and manipulates data in the form of quantum bits, or qubits. In general, a qubit is not in a state of zero or one, but rather in a mixed state of zero and one. If a large number of qubits are placed in the same quantum state, the measurement of an individual qubit would produce a zero with a probability p , and a one with a probability \(q = 1 - p\). Some scientists believe that quantum computers are the future of the computer industry.
Complex Conjugates
Later in this section, you will see how to use the wavefunction to describe particles that are “free” or bound by forces to other particles. The specific form of the wavefunction depends on the details of the physical system. A peculiarity of quantum theory is that these functions are usually complex functions . A complex function is one that contains one or more imaginary numbers (\(i = \sqrt{-1}\)). Experimental measurements produce real (nonimaginary) numbers only, so the above procedure to use the wavefunction must be slightly modified. In general, the probability that a particle is found in the narrow interval \((x, x + dx)\) at time \(t\) is given by
\[P (x,x + dx) = |\Psi \, (x,t)|^2 dx = \Psi^* (x,t) \, \Psi \, (x,t) \, dx, \label{7.5} \]
where \(\Psi^* (x,t)\) is the complex conjugate of the wavefunction. The complex conjugate of a function is obtaining by replacing every occurrence of \(i = \sqrt{-1}\) in that function with \(-i\). This procedure eliminates complex numbers in all predictions because the product \(\Psi^* (x,t) \, \Psi \, (x,t)\) is always a real number.
Exercise \(\PageIndex{1}\)
If \(a = 3 + 4i\), what is the product \(a^*a\)?
- Answer
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\[(3 + 4i)(3 - 4i) = 9 - 16i^2 = 25 \nonumber \]
Consider the motion of a free particle that moves along the x -direction. As the name suggests, a free particle experiences no forces and so moves with a constant velocity. As we will see in a later section of this chapter, a formal quantum mechanical treatment of a free particle indicates that its wavefunction has real and complex parts. In particular, the wavefunction is given by
\[\Psi \, (x,t) = A \, \cos \, (kx - \omega t) + i A \, \sin \, (kx - \omega t), \label{eq56} \]
where \(A\) is the amplitude , \(k\) is the wave number , and \(ω\) is the angular frequency . Euler’s formula
\[\underbrace{e^{i\phi} = \cos \, (\phi) + i \, \sin \, (\phi)}_{\text{Euler’s formula}} \nonumber \]
can be used to rewrite Equation \ref{eq56} in the form
\[\Psi \, (x,t) = Ae^{i(kx - \omega t)} = Ae^{i\phi}, \nonumber \]
where \(\phi\) is the phase angle. If the wavefunction varies slowly over the interval \(\Delta x\), the probability of finding the particle in that interval is
\[P (x,x + \Delta x) \approx \Psi^* (x,t) \, \Psi \, (x,t) \, \Delta x = (A^* e^{-i\phi})(Ae^{i\phi}) \, \Delta x = (A^*A) \Delta x. \nonumber \]
If \(A\) has real and complex parts (\(a+ib\), where \(a\) and \(b\) are real constants), then
\[A^*A = (a - ib)(a + ib) = a^2 + b^2. \nonumber \]
Notice that the complex numbers have vanished. Thus,
\[P(x,x + \Delta x) \approx |A|^2 \delta x \nonumber \]
is a real quantity. The interpretation of \(\Psi^* (x,t) \, \Psi \, (x,t)\) as a probability density ensures that the predictions of quantum mechanics can be checked in the “real world.”
Exercise \(\PageIndex{2}\)
Suppose that a particle with energy E is moving along the x -axis and is confined in the region between 0 and L . One possible wavefunction is
\[\psi (x,t) =\begin{cases}
Ae^{-iEt/\hbar} \sin \, \dfrac{\pi x}{L} & 0 \leq x \leq L \\
0 & x < 0 \text{ and } x > L \end{cases} \nonumber \]
Determine the normalization constant.
- Answer
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\(A = \sqrt{2/L}\)
Expectation Values
In classical mechanics, the solution to an equation of motion is a function of a measurable quantity, such as \(x(t)\), where \(x\) is the position and \(t\) is the time. Note that the particle has one value of position for any time \(t\). In quantum mechanics, however, the solution to an equation of motion is a wavefunction, \(\Psi \, (x,t)\). The particle has many values of position for any time \(t\), and only the probability density of finding the particle, \(|\Psi \, (x,t)|^2\), can be known. The average value of position for a large number of particles with the same wavefunction is expected to be
\[\langle x \rangle = \int_{-\infty}^{\infty} xP(x,t) \, dx = \int_{-\infty}^{\infty} x \Psi^* (x,t) \, \Psi \, (x,t) \, dx. \label{7.6} \]
This is called the expectation value of the position. It is usually written
\[\langle x \rangle = \int_{-\infty}^{\infty} \Psi^* (x,t) \, x \Psi \, (x,t) \, dx. \label{7.7} \]
where the \(x\) is sandwiched between the wavefunctions. The reason for this will become apparent soon. Formally, \(x\) is called the position operator .
At this point, it is important to stress that a wavefunction can be written in terms of other quantities as well, such as velocity (\(v\)), momentum (\(p\)), and kinetic energy (\(K\)). The expectation value of momentum, for example, can be written
\[\langle p \rangle = \int_{-\infty}^{\infty} \Psi^* (p,t) \, p\Psi \, (p,t) \, dp, \label{7.8} \]
where \(dp\) is used instead of \(dx\) to indicate an infinitesimal interval in momentum. In some cases, we know the wavefunction in position, \(\Psi \, (x,t)\), but seek the expectation of momentum. The procedure for doing this is
\[\langle p \rangle = \int_{-\infty}^{\infty} \Psi^* (x,t) \, \left(-i\hbar \dfrac{d}{dx}\right) \, \Psi \, (x,t) \, dx, \label{7.9} \]
where the quantity in parentheses, sandwiched between the wavefunctions, is called the momentum operator in the x -direction. [The momentum operator in Equation \ref{7.9} is said to be the position-space representation of the momentum operator.] The momentum operator must act (operate) on the wavefunction to the right, and then the result must be multiplied by the complex conjugate of the wavefunction on the left, before integration. The momentum operator in the x -direction is sometimes denoted
\[\langle p \rangle = - i\hbar \dfrac{d}{dx},\label{7.10} \]
Momentum operators for the y - and z -directions are defined similarly. This operator and many others are derived in a more advanced course in modern physics. In some cases, this derivation is relatively simple. For example, the kinetic energy operator is just
\[\begin{align} (K)_{op} &= \dfrac{1}{2}m(v_x)_{op}^2 \\[5pt] &= \dfrac{(p_x)^2_{op}}{2m} \\[5pt] &= \dfrac {\left(-i\hbar \dfrac{d}{dx}\right)^2}{2m} \\[5pt] &= \dfrac{-\hbar^2}{2m} \left(\dfrac{d}{dx}\right)\left(\dfrac{d}{dx}\right).\label{7.11} \end{align} \]
Thus, if we seek an expectation value of kinetic energy of a particle in one dimension, two successive ordinary derivatives of the wavefunction are required before integration.
Symmetry can simplify calculations
Expectation-value calculations are often simplified by exploiting the symmetry of wavefunctions. Symmetric wavefunctions can be even or odd. An even function is a function that satisfies
\[\psi(x) = \psi(-x). \label{7.12} \]
In contrast, an odd function is a function that satisfies
\[\psi(x) = -\psi(-x).\label{7.13} \]
An example of even and odd functions is shown in Figure \(\PageIndex{7}\). An even function is symmetric about the y -axis. This function is produced by reflecting \(\psi (x)\) for \(x > 0\) about the vertical y -axis. By comparison, an odd function is generated by reflecting the function about the y -axis and then about the x -axis. (An odd function is also referred to as an anti-symmetric function .)
In general, an even function times an even function produces an even function. A simple example of an even function is the product \(x^2e^{-x^2}\) (even times even is even). Similarly, an odd function times an odd function produces an even function, such as x sin x (odd times odd is even). However, an odd function times an even function produces an odd function, such as \(x^2e^{-x^2}\) (odd times even is odd). The integral over all space of an odd function is zero, because the total area of the function above the x -axis cancels the (negative) area below it. As the next example shows, this property of odd functions is very useful.
Example \(\PageIndex{2A}\): Expectation Value (Part I)
The normalized wavefunction of a particle is
\[\psi(x) = e^{-|x|/x_0} /\sqrt{x_0}. \nonumber \]
Find the expectation value of position.
Strategy
Substitute the wavefunction into Equation \ref{7.7} and evaluate. The position operator introduces a multiplicative factor only, so the position operator need not be “sandwiched.”
Solution
First multiply, then integrate:
\[\begin{align*} \langle x \rangle &= \int_{-\infty}^{\infty} dx\,x|\psi(x)|^2 \nonumber \\[4pt] &= \int_{-\infty}^{\infty} dx\, x|\dfrac{e^{-|x|/x_0}}{\sqrt{x_0}}|^2 \nonumber \\[4pt] &= \dfrac{1}{x_0} \int_{-\infty}^{\infty} dx\, xe^{-2|x|/x_0} \nonumber \\[4pt] &= 0. \nonumber \end{align*} \nonumber \]
Significance
The function in the integrand (\(xe^{-2|x|/x_0}\)) is odd since it is the product of an odd function ( x ) and an even function (\(e^{-2|x|/x_0}\)). The integral vanishes because the total area of the function about the x -axis cancels the (negative) area below it. The result (\(\langle x \rangle = 0\)) is not surprising since the probability density function is symmetric about \(x = 0\).
Example \(\PageIndex{2B}\): Expectation Value (Part II)
The time-dependent wavefunction of a particle confined to a region between 0 and L is
\[\psi(x,t) = A \, e^{-i\omega t} \sin \, (\pi x/L) \nonumber \]
where \(\omega\) is angular frequency and \(E\) is the energy of the particle. ( Note: The function varies as a sine because of the limits (0 to L ). When \(x = 0\), the sine factor is zero and the wavefunction is zero, consistent with the boundary conditions.) Calculate the expectation values of position, momentum, and kinetic energy.
Strategy
We must first normalize the wavefunction to find A . Then we use the operators to calculate the expectation values.
Solution
Computation of the normalization constant:
\[\begin{align*} 1 &= \int_0^L dx\, \psi^* (x) \psi(x) \nonumber \\[4pt] &= \int_0^L dx \, \left(A e^{+i\omega t} \sin \, \dfrac{\pi x}{L}\right) \left(A e^{-i\omega t} \sin \, \dfrac{\pi x}{L}\right) \nonumber \\[4pt] &= A^2 \int_0^L dx \, \sin^2 \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= A^2 \dfrac{L}{2} \nonumber \\[4pt] \Rightarrow A &= \sqrt{\dfrac{2}{L}}. \nonumber \end{align*} \nonumber \]
The expectation value of position is
\[\begin{align*}\langle x \rangle &= \int_0^L dx \, \psi^* (x) x \psi(x) \nonumber \\[4pt] &= \int_0^L dx \, \left(A e^{+i\omega t} \sin \, \dfrac{\pi x}{L}\right) x \left(A e^{-i\omega t} \sin \, \dfrac{\pi x}{L}\right) \nonumber \\[4pt] &= A^2 \int_0^L dx\,x \, \sin^2 \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= A^2 \dfrac{L^2}{4} \nonumber \\[4pt] \Rightarrow A &= \dfrac{L}{2}. \nonumber \end{align*} \nonumber \]
The expectation value of momentum in the x -direction also requires an integral. To set this integral up, the associated operator must— by rule—act to the right on the wavefunction \(\psi(x)\):
\[\begin{align*} -i\hbar\dfrac{d}{dx} \psi(x) &= -i\hbar \dfrac{d}{dx} Ae^{-i\omega t}\sin \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= - i\dfrac{Ah}{2L} e^{-i\omega t} \cos\, \dfrac{\pi x}{L}. \nonumber \end{align*} \nonumber \]
Therefore, the expectation value of momentum is
\[ \begin{align*} \langle p \rangle &= \int_0^L dx \left(Ae^{+i\omega t}sin \dfrac{\pi x}{L}\right)\left(-i \dfrac{Ah}{2L} e^{-i\omega t} cos \, \dfrac{\pi x}{L}\right) \nonumber \\[4pt] &= -i \dfrac{A^2h}{4L} \int_0^L dx \, \sin \, \dfrac{2\pi x}{L} \nonumber \\[4pt] &= 0. \nonumber \end{align*} \nonumber \]
The function in the integral is a sine function with a wavelength equal to the width of the well, L —an odd function about \(x = L/2\). As a result, the integral vanishes.
The expectation value of kinetic energy in the x -direction requires the associated operator to act on the wavefunction:
\[ \begin{align} -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} \psi (x) &= - \dfrac{\hbar^2}{2m} \dfrac{d^2}{dx^2} Ae^{-i\omega t} \, \sin \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= - \dfrac{\hbar^2}{2m} Ae^{-i\omega t} \dfrac{d^2}{dx^2} \, \sin \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= \dfrac{Ah^2}{8mL^2} e^{-i\omega t} \, \sin \, \dfrac{\pi x}{L}. \nonumber \end{align} \nonumber \]
Thus, the expectation value of the kinetic energy is
\[\begin{align*} \langle K \rangle &= \int_0^L dx \left( Ae^{+i\omega t} \, \sin \, \dfrac{\pi x}{L}\right) \left(\dfrac{Ah^2}{8mL^2} e^{-i\omega t} \, \sin \, \dfrac{\pi x}{L}\right) \nonumber \\[4pt] &= \dfrac{A^2h^2}{8mL^2} \int_0^L dx \, \sin^2 \, \dfrac{\pi x}{L} \nonumber \\[4pt] &= \dfrac{A^2h^2}{8mL^2} \dfrac{L}{2} \nonumber \\[4pt] &= \dfrac{h^2}{8mL^2}. \end{align*} \nonumber \]
Significance
The average position of a large number of particles in this state is \(L/2\). The average momentum of these particles is zero because a given particle is equally likely to be moving right or left. However, the particle is not at rest because its average kinetic energy is not zero. Finally, the probability density is
\[|\psi|^2 = (2/L) \, \sin^2 (\pi x/L). \nonumber \]
This probability density is largest at location \(L/2\) and is zero at \(x = 0\) and at \(x = L\). Note that these conclusions do not depend explicitly on time.
Exercise \(\PageIndex{3}\)
For the particle in the above example, find the probability of locating it between positions \(0\) and \(L/4\).
- Answer
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\((1/2 - 1/\pi) /2 = 9\%\)
Quantum mechanics makes many surprising predictions. However, in 1920, Niels Bohr (founder of the Niels Bohr Institute in Copenhagen, from which we get the term “Copenhagen interpretation”) asserted that the predictions of quantum mechanics and classical mechanics must agree for all macroscopic systems, such as orbiting planets, bouncing balls, rocking chairs, and springs. This correspondence principle is now generally accepted. It suggests the rules of classical mechanics are an approximation of the rules of quantum mechanics for systems with very large energies. Quantum mechanics describes both the microscopic and macroscopic world, but classical mechanics describes only the latter.