4.3: Second Quantization
In the usual tensor product notation, symmetric and antisymmetric states become quite cumbersome to deal with when the number of particles is large. We will now introduce a formalism called second quantization , which greatly simplifies manipulations of such multi-particle states. (The reason for the name “second quantization” will not be apparent until later; it is a bad name, but one we are stuck with for historical reasons.)
We start by defining a convenient way to specify states of multiple identical particles, called the occupation number representation . Let us enumerate a set of single-particle states, \(\{|1\rangle, |2\rangle, |3\rangle, \cdots\}\) , that form a complete orthonormal basis for the single-particle Hilbert space \(\mathscr{H}^{(1)}\) . Then, we build multi-particle states by specifying how many particles are in state \(|1\rangle\) , denoted \(n_1\) ; how many are in state \(|2\rangle\) , denoted \(n_2\) ; and so on. Thus,
\[|n_1,n_2,n_3,\dots\rangle\]
is defined as the appropriate symmetric or antisymmetric multi-particle state, constructed using Equation (4.2.6) if we’re dealing with bosons (Section 4.2), or using Equation (4.2.14) if we’re dealing with fermions (Section 4.2).
Let us run through a couple of examples:
Example \(\PageIndex{1}\)
The two-particle state \(|0,2,0,0,\dots\rangle\) has both particles in the single-particle state \(|2\rangle\) . This is only possible if the particles are bosons, since fermions cannot share the same state. Written out in tensor product form, the symmetric state is
\[|0,2,0,0,\dots\rangle \equiv |2\rangle|2\rangle.\]
Example \(\PageIndex{2}\)
The three-particle state \(|1,1,1,0,0,\dots\rangle\) has one particle each occupying \(|1\rangle\) , \(|2\rangle\) , and \(|3\rangle\) . If the particles are bosons, this corresponds to the symmetric state
\[\begin{align} \begin{aligned}|1,1,1,0,0,\dots\rangle &\equiv \frac{1}{\sqrt{6}}\Big(|1\rangle|2\rangle|3\rangle + |3\rangle|1\rangle|2\rangle + |2\rangle|3\rangle|1\rangle \\&\;\qquad+ |1\rangle|3\rangle|2\rangle + |2\rangle|1\rangle|3\rangle + |3\rangle|2\rangle|1\rangle \Big). \end{aligned}\end{align}\]
And if the particles are fermions, the appropriate antisymmetric state is
\[\begin{align} \begin{aligned}|1,1,1,0,0,\dots\rangle &\equiv \frac{1}{\sqrt{6}} \Big(|1\rangle|2\rangle|3\rangle + |3\rangle|1\rangle|2\rangle + |2\rangle|3\rangle|1\rangle \\&\;\qquad - |1\rangle|3\rangle|2\rangle - |2\rangle|1\rangle|3\rangle - |3\rangle|2\rangle|1\rangle\Big). \end{aligned}\end{align}\]
Fock space
There is a subtle point that we have glossed over: what Hilbert space do these state vectors reside in? The state \(|0,2,0,0,\dots\rangle\) is a bosonic two-particle state, which is a vector in the two-particle Hilbert space \(\mathscr{H}^{(2)} = \mathscr{H}^{(1)}\otimes \mathscr{H}^{(1)}\) . However, \(\mathscr{H}^{(2)}\) also contains two-particle states that are not symmetric under exchange, which is not allowed for bosons. Thus, it would be more rigorous for us to narrow the Hilbert space to the space of state vectors that are symmetric under exchange. We denote this reduced space by \(\mathscr{H}^{(2)}_S\) .
Likewise, \(|1,1,1,0,\dots\rangle\) is a three-particle state lying in \(\mathscr{H}^{(3)}\) . If the particles are bosons, we can narrow the space to \(\mathscr{H}^{(3)}_S\) . If the particles are fermions, we can narrow it to the space of three-particle states that are antisymmetric under exchange, denoted by \(\mathscr{H}^{(3)}_A\) . Thus, \(|1,1,1,0,\dots\rangle \in \mathscr{H}^{(3)}_{S/A}\) , where the subscript \(S/A\) depends on whether we are dealing with symmetric states ( \(S\) ) or antisymmetric states ( \(A\) ).
We can make the occupation number representation more convenient to work with by defining an “extended” Hilbert space, called the Fock space , that is the space of bosonic/fermionic states for arbitrary numbers of particles . In the formal language of linear algebra, the Fock space can be written as
\[\mathscr{H}_{S/A}^F = \mathscr{H}^{(0)} \oplus \mathscr{H}^{(1)} \oplus \mathscr{H}^{(2)}_{S/A} \oplus \mathscr{H}^{(3)}_{S/A} \oplus \mathscr{H}^{(4)}_{S/A} \oplus \cdots \label{fock}\]
Here, \(\oplus\) represents the direct sum operation, which combines vector spaces by directly grouping their basis vectors into a larger basis set; if \(\mathscr{H}_1\) has dimension \(d_1\) and \(\mathscr{H}_2\) has dimension \(d_2\) , then \(\mathscr{H}_1\oplus\mathscr{H}_2\) has dimension \(d_1+d_2\) . (By contrast, the space \(\mathscr{H}_1\otimes\mathscr{H}_2\) , defined via the tensor product, has dimension \(d_1d_2\) .) Once again, the subscript \(S/A\) depends on whether we are dealing with bosons ( \(S\) ) or fermions ( \(A\) ).
The upshot is that any multi-particle state that we can write down in the occupation number representation, \(|n_1,n_2,n_3,\dots\rangle\) , is guaranteed to lie in the Fock space \(\mathscr{H}^{F}_{S/A}\) . Moreover, these states form a complete basis for \(\mathscr{H}^{F}_{S/A}\) .
In Equation \(\eqref{fock}\), the first term of the direct sum is \(\mathscr{H}^{(0)}\) , the “Hilbert space of 0 particles”. This Hilbert space contains only one distinct state vector, denoted by
\[|\varnothing\rangle \equiv |0,0,0,0,\dots\rangle.\]
This refers to the vacuum state , a quantum state in which there are literally no particles. Note that \(|\varnothing\rangle\) is not the same thing as a zero vector; it has the standard normalization \(\langle\varnothing|\varnothing\rangle = 1\) . The concept of a “state of no particles” may seem silly, but we will see that there are very good reasons to include it in the formalism.
Another subtle consequence of introducing the Fock space concept is that it is now legitimate to write down quantum states that lack definite particle numbers. For example,
\[\frac{1}{\sqrt{2}} \Big(|1,0,0,0,0,\dots\rangle + |1,1,1,0,0,\dots\rangle\Big)\]
is a valid state vector describing the superposition of a one-particle state and a three-particle state. We will revisit the phenomenon of quantum states with indeterminate particle numbers in Section 4.3, and in the next chapter.
Second quantization for bosons
After this lengthy prelude, we are ready to introduce the formalism of second quantization. Let us concentrate on bosons first.
We define an operator called the boson creation operator , denoted by \(\hat{a}_\mu^\dagger\) and acting in the following way:
\[\hat{a}_\mu^\dagger \big|n_1, n_2, \dots, n_\mu, \dots\big\rangle = \sqrt{n_\mu+1} \; \big|n_1, n_2, \dots, n_\mu + 1, \dots\big\rangle. \label{amu}\]
In this definition, there is one particle creation operator for each state in the single-particle basis \(\{|\varphi_1\rangle,|\varphi_2\rangle,\dots\}\) . Each creation operator is defined as an operator acting on state vectors in the Fock space \(\mathscr{H}^F_S\) , and has the effect of incrementing the occupation number of its single-particle state by one. The prefactor of \(\sqrt{n_\mu+1}\) is defined for later convenience.
Applying a creation operator to the vacuum state yields a single-particle state:
\[\begin{align} \begin{aligned}\hat{a}_\mu^\dagger |\varnothing\rangle \, &= \, |0,\dots,0, 1, 0, 0, \dots\rangle. \\ &\qquad\qquad\quad\;\; \leftarrow \mu \end{aligned}\end{align}\]
The creation operator’s Hermitian conjugate, \(\hat{a}_\mu\) , is the boson annihilation operator . To characterize it, first take the Hermitian conjugate of Equation \(\eqref{amu}\):
\[\big\langle n_1, n_2, \dots\big| \hat{a}_\mu = \sqrt{n_\mu+1} \; \big\langle n_1, n_2, \dots, n_\mu + 1, \dots\big|.\]
Right-multiplying by another occupation number state \(|n_1',n_2',\dots\rangle\) results in
\[\begin{align} \begin{aligned}\big\langle n_1, n_2, \dots \big| \hat{a}_\mu \big|n_1',n_2',\dots\big\rangle &= \sqrt{n_\mu+1} \; \big\langle \dots, n_\mu + 1, \dots\big| \dots, n_\mu',\dots\big\rangle \\&= \sqrt{n_\mu+1}\; \delta^{n_1}_{n_1'}\; \delta^{n_2}_{n_2'} \cdots \delta^{n_\mu+1}_{n_\mu'} \dots \\ &= \sqrt{n_\mu'}\; \delta^{n_1}_{n_1'}\; \delta^{n_2}_{n_2'} \cdots \delta^{n_\mu+1}_{n_\mu'}\cdots\end{aligned} \end{align}\]
From this, we can deduce that
\[\hat{a}_\mu \big|n_1', n_2', \dots, n_\mu', \dots\big\rangle = \begin{cases} \sqrt{n_\mu'} \; \big|n_1', n_2', \dots, n_\mu' - 1, \dots\big\rangle, & \mathrm{if}\; n_\mu' > 0 \\ 0, & \mathrm{if}\; n_\mu' = 0.\end{cases}\]
In other words, the annihilation operator decrements the occupation number of a specific single-particle state by one (hence its name). As a special exception, if the given single-particle state is unoccupied ( \(n_\mu = 0\) ), applying \(\hat{a}_\mu\) results in a zero vector (note that this is not the same thing as the vacuum state \(|\varnothing\rangle\) ).
The boson creation/annihilation operators obey the following commutation relations:
\[\,[\hat{a}_\mu,\hat{a}_\nu] = [\hat{a}_\mu^\dagger,\hat{a}_\nu^\dagger] = 0, \qquad [\hat{a}_\mu,\hat{a}_\nu^\dagger] = \delta_{\mu\nu}. \label{commutations}\]
These can be derived by taking the matrix elements with respect to the occupation number basis. We will go through the derivation of the last commutation relation; the others are left as an exercise (Exercise 4.5.5).
To prove that \([\hat{a}_\mu,\hat{a}_\nu^\dagger] = \delta_{\mu\nu}\) , first consider the case where the creation/annihilation operators act on the same single-particle state:
\[\begin{align} \begin{aligned}\big\langle n_1, n_2, \dots \big| \hat{a}_\mu \hat{a}_\mu^\dagger \big| n_1', n_2'\dots\big\rangle &= \sqrt{(n_\mu+1)(n_\mu'+1)}\; \big\langle \dots, n_\mu+1, \dots \big| \dots, n_\mu'+1, \dots\big\rangle \\ &= \sqrt{(n_\mu+1)(n_\mu'+1)} \delta^{n_1}_{n_1'} \; \delta^{n_2}_{n_2'} \; \cdots \delta^{n_\mu+1}_{n_\mu'+1}\cdots \\ &= (n_\mu+1) \delta^{n_1}_{n_1'} \; \delta^{n_2}_{n_2'} \; \cdots \delta^{n_\mu}_{n_\mu'}\cdots \\ \big\langle n_1, n_2, \dots \big| \hat{a}_\mu^\dagger \hat{a}_\mu \big| n_1', n_2'\dots\big\rangle &= \sqrt{n_\mu n_\mu'}\; \big\langle \dots, n_\mu-1, \dots \big| \dots, n_\mu'-1, \dots\big\rangle \\&= \sqrt{n_\mu n_\mu'} \delta^{n_1}_{n_1'} \; \delta^{n_2}_{n_2'} \; \cdots \delta^{n_\mu-1}_{n_\mu'-1}\cdots \\ &= n_\mu \delta^{n_1}_{n_1'} \; \delta^{n_2}_{n_2'} \; \cdots \delta^{n_\mu}_{n_\mu'}\cdots \end{aligned}\end{align}\]
In the second equation, we were a bit sloppy in handling the \(n_\mu = 0\) and \(n_\mu' = 0\) cases, but you can check for yourself that the result on the last line remains correct. Upon taking the difference of the two equations, we get
\[\big\langle n_1, n_2, \dots \big| \left(\hat{a}_\mu \hat{a}_\mu^\dagger - \hat{a}_\mu^\dagger \hat{a}_\mu\right) \big| n_1', n_2'\dots\big\rangle = \delta^{n_1}_{n_1'} \; \delta^{n_2}_{n_2'} \; \cdots \delta^{n_\mu}_{n_\mu'}\cdots = \big\langle n_1, n_2, \dots \big| n_1', n_2'\dots\big\rangle.\]
Since the occupation number states form a basis for \(\mathscr{H}^F_S\) , we conclude that
\[\hat{a}_\mu \hat{a}_\mu^\dagger - \hat{a}_\mu^\dagger \hat{a}_\mu = \hat{I}.\]
Next, consider the case where \(\mu \ne \nu\) :
\[\begin{align} \begin{aligned}\big\langle n_1, \dots \big| \hat{a}_\mu \hat{a}_\nu^\dagger \big| n_1', \dots\big\rangle &= \sqrt{(n_\mu+1)(n_\nu'+1)}\, \langle \dots, n_\mu+1, \dots, n_\nu, \dots | \dots, n_\mu', \dots, n_\nu'+1, \dots\rangle \\ &= \sqrt{n_\mu' n_\nu} \;\, \delta^{n_1}_{n_1'} \; \cdots \delta^{n_\mu+1}_{n_\mu'} \cdots \delta^{n_\nu}_{n_\nu' + 1}\cdots \\ \big\langle n_1, \dots \big| \hat{a}_\nu^\dagger \hat{a}_\mu \big| n_1', \dots\big\rangle &= \sqrt{n_\mu' n_\nu}\, \langle \dots, n_\mu, \dots,n_\nu-1,\dots | \dots, n_\mu'-1, \dots, n_\nu'\dots\rangle \\&= \sqrt{n_\mu' n_\nu} \;\delta^{n_1}_{n_1'} \; \cdots \delta^{n_\mu}_{n_\mu'-1}\cdots \delta^{n_\nu-1}_{n_\nu'} \cdots \\ &= \sqrt{n_\mu' n_\nu} \; \delta^{n_1}_{n_1'} \; \cdots \delta^{n_\mu+1}_{n_\mu'}\cdots \delta^{n_\nu}_{n_\nu'+1} \cdots\end{aligned}\end{align}\]
Hence,
\[\hat{a}_\mu \hat{a}_\nu^\dagger - \hat{a}_\nu^\dagger \hat{a}_\mu = 0 \;\;\;\mathrm{for}\;\;\mu\ne\nu.\]
Combining these two results gives the desired commutation relation, \([\hat{a}_\mu, \hat{a}_\nu^\dagger] = \delta_{\mu\nu}\) .
Another useful result which emerges from the first part of this proof is that
\[\big\langle n_1, n_2, \dots \big| \hat{a}_\mu^\dagger \hat{a}_\mu \big| n_1', n_2'\dots\big\rangle = n_\mu \big\langle n_1, n_2, \dots \big| n_1', n_2'\dots\big\rangle.\]
Hence, we can define the Hermitian operator
\[\hat{n}_\mu \equiv \hat{a}_\mu^\dagger \hat{a}_\mu,\]
whose eigenvalue is the occupation number of single-particle state \(\mu\) .
If you are familiar with the method of creation/annihilation operators for solving the quantum harmonic oscillator, you will have noticed the striking similarity with the particle creation/annihilation operators for bosons. This is no mere coincidence. We will examine the relationship between harmonic oscillators and bosons in the next chapter.
Second quantization for fermions
For fermions, the multi-particle states are antisymmetric. The fermion creation operator can be defined as follows:
\[\begin{align} \begin{aligned}\hat{c}_\mu^\dagger |n_1, n_2, \dots, n_\mu,\dots\rangle &= \begin{cases}(-1)^{n_1 + n_2 + \cdots + n_{\mu-1}} |n_1, n_2, \dots, n_{\mu-1}, 1, \dots\rangle & \mathrm{if} \; n_\mu = 0 \\ \;\;0 & \mathrm{if}\; n_\mu = 1. \end{cases} \\&= (-1)^{n_1 + n_2 + \cdots + n_{\mu-1}}\; \delta^{n_\mu}_0\, \big|n_1, n_2, \dots, n_{\mu-1}, 1, \dots\big\rangle. \end{aligned}\end{align}\]
In other words, if state \(\mu\) is unoccupied, then \(\hat{c}_\mu^\dagger\) increments the occupation number to 1, and multiplies the state by an overall factor of \((-1)^{n_1 + n_2 + \cdots + n_{\mu-1}}\) (i.e, \(+1\) if there is an even number of occupied states preceding \(\mu\) , and \(-1\) if there is an odd number). The role of this factor will be apparent later. Note that this definition requires the single-particle states to be ordered in some way; otherwise, it would not make sense to speak of the states “preceding” \(\mu\) . It does not matter which ordering we choose, so long as we make some choice, and stick to it consistently.
If \(\mu\) is occupied, applying \(\hat{c}_\mu^\dagger\) gives the zero vector. The occupation numbers are therefore forbidden from being larger than 1, consistent with the Pauli exclusion principle.
The conjugate operator, \(\hat{c}_\mu\) , is the fermion annihilation operator. To see what it does, take the Hermitian conjugate of the definition of the creation operator:
\[\langle n_1, n_2, \dots, n_\mu,\dots| \hat{c}_\mu = (-1)^{n_1 + n_2 + \cdots + n_{\mu-1}}\; \delta^{n_\mu}_0\, \big\langle n_1, n_2, \dots, n_{\mu-1}, 1, \dots\big|.\]
Right-multiplying this by \(|n_1',n_2',\dots\rangle\) gives
\[\langle n_1, n_2, \dots, n_\mu,\dots| \hat{c}_\mu |n_1', n_2', \dots\rangle = (-1)^{n_1 + \cdots + n_{\mu-1}} \;\delta^{n_1}_{n_1'} \cdots \;\delta^{n_{\mu-1}}_{n_{\mu-1}'} \left(\delta^{n_\mu}_0 \delta^1_{n_\mu'} \right) \delta^{n_{\mu+1}}_{n_{\mu+1}'} \dots\]
Hence, we deduce that
\[\begin{align} \begin{aligned}\hat{c}_\mu |n_1', \dots, n_\mu',\dots\rangle &= \begin{cases} \;\;0 & \mathrm{if}\; n_\mu' = 0 \\ (-1)^{n_1' + \cdots + n_{\mu-1}'} |n_1', \dots, n_{\mu-1}', 0, \dots\rangle & \mathrm{if} \; n_\mu' = 1. \end{cases} \\ &= (-1)^{n_1' + \cdots + n_{\mu-1}'} \; \delta^1_{n_\mu'} \, \big|n_1', \dots, n_{\mu-1}', 0, \dots\big\rangle.\end{aligned}\end{align}\]
In other words, if state \(\mu\) is unoccupied, then applying \(\hat{c}_\mu\) gives the zero vector; if state \(\mu\) is occupied, applying \(\hat{c}_\mu\) decrements the occupation number to \(0\) , and multiplies the state by the aforementioned factor of \(\pm 1\) .
With these definitions, the fermion creation/annihilation operators can be shown to obey the following anticommutation relations:
Definition: Anticommutation Relations
\[\big\{\hat{c}_\mu,\hat{c}_\nu\big\} = \big\{\hat{c}_\mu^\dagger,\hat{c}_\nu^\dagger\big\} = 0, \qquad \big\{\hat{c}_\mu,\hat{c}_\nu^\dagger\big\} = \delta_{\mu\nu}. \label{anticommutation}\]
Here, \(\{\cdot,\cdot\}\) denotes an anticommutator, which is defined by
\[\left\{\hat{A},\hat{B}\right\} \equiv \hat{A}\hat{B} + \hat{B}\hat{A}.\]
Similar to the bosonic commutation relations \(\eqref{commutations}\), the anticommutation relations \(\eqref{anticommutation}\) can be derived by taking matrix elements with occupation number states. We will only go over the last one, \(\big\{\hat{c}_\mu,\hat{c}_\nu^\dagger\big\} = \delta_{\mu\nu}\) ; the others are left for the reader to verify.
First, consider creation/annihilation operators acting on the same single-particle state \(\mu\) :
\[\begin{align} \begin{aligned}\big\langle \dots,n_\mu,\dots \big| \hat{c}_\mu \hat{c}_\mu^\dagger \big|\dots, n_\mu', \dots\big\rangle &= (-1)^{n_1+\cdots+n_{\mu-1}} (-1)^{n_1'+\cdots+n_{\mu-1}'} \,\delta^{n_\mu}_0 \delta^0_{n_\mu'} \\ &\quad\times \big\langle n_1,\dots,n_{\mu-1}, 1, \dots\big| n_1',\dots,n_{\mu-1}', 1, \dots \big\rangle \\ &= \delta^0_{n_\mu'} \; \cdot \; \delta^{n_1}_{n_1'} \, \delta^{n_2}_{n_2'} \cdots \delta^{n_\mu}_{n_\mu'} \cdots\end{aligned}\end{align}\]
By a similar calculation,
\[\big\langle \dots,n_\mu,\dots \big| \hat{c}_\mu^\dagger \hat{c}_\mu \big|\dots, n_\mu', \dots\big\rangle = \delta^1_{n_\mu'} \; \cdot \; \delta^{n_1}_{n_1'} \, \delta^{n_2}_{n_2'} \cdots \delta^{n_\mu}_{n_\mu'} \cdots\]
By adding these two equations, and using the fact that \(\delta^0_{n_\mu'} + \delta^1_{n_\mu'} = 1\) , we get
\[\big\langle \dots,n_\mu,\dots \big| \left\{ \hat{c}_\mu, \hat{c}_\mu^\dagger\right\} \big|\dots, n_\mu', \dots\big\rangle = \big\langle \dots,n_\mu,\dots \big|\dots, n_\mu', \dots\big\rangle\]
And hence,
\[\left\{ \hat{c}_\mu, \hat{c}_\mu^\dagger\right\} = \hat{I}.\]
Next, we must prove that \(\left\{ \hat{c}_\mu, \hat{c}_\nu^\dagger\right\} = 0\) for \(\mu \ne \nu\) . We will show this for \(\mu < \nu\) (the \(\mu > \nu\) case follows by Hermitian conjugation). This is, once again, by taking matrix elements:
\[\begin{align} \begin{aligned}\big\langle \dots,n_\mu,\dots, n_\nu,\dots \big| \hat{c}_\mu \hat{c}_\nu^\dagger \big| & \dots, n_\mu', \dots, n_\nu',\dots\big\rangle = (-1)^{n_1+\cdots+n_{\mu-1}} (-1)^{n_1'+\cdots+n_{\nu-1}'} \,\delta^{n_\mu}_0 \delta^0_{n_\nu'} \\ &\qquad\qquad\qquad\qquad\qquad\times \big\langle \dots, 1, \dots, n_\nu,\dots\big| \dots,n_{\mu}', \dots,1, \dots \big\rangle \\ &\qquad\quad = (-1)^{n_\mu'+\cdots+n_{\nu-1}'}\, \delta^{n_1}_{n_1'} \, \delta^{n_2}_{n_2'} \cdots \left(\delta^{n_\mu}_0\delta^1_{n_\mu'}\right) \cdots \left(\delta^{n_\nu}_1 \delta^0_{n_\nu'}\right) \cdots \\ &\qquad\quad = (-1)^{1 + n_{\mu+1} + \cdots+n_{\nu-1}}\, \delta^{n_1}_{n_1'} \, \delta^{n_2}_{n_2'} \cdots \left(\delta^0_{n_\mu}\delta^1_{n_\mu'}\right) \cdots \left(\delta^0_{n_\nu'} \delta^1_{n_\nu}\right) \cdots \\ \big\langle \dots,n_\mu,\dots, n_\nu,\dots \big| \hat{c}_\nu^\dagger \hat{c}_\mu \big| &\dots, n_\mu', \dots, n_\nu',\dots\big\rangle = (-1)^{n_1+\cdots+n_{\nu-1}} (-1)^{n_1'+\cdots+n_{\mu-1}'} \,\delta^{n_\nu}_1 \delta^1_{n_\mu'} \\ &\qquad\qquad\qquad\qquad\qquad\times \big\langle \dots, n_\mu, \dots, 0,\dots\big| \dots,0, \dots,n_\nu', \dots \big\rangle \\ &\qquad\quad= (-1)^{n_{\mu}+\cdots+n_{\nu-1}}\, \delta^{n_1}_{n_1'} \, \delta^{n_2}_{n_2'} \cdots \left(\delta^{n_\mu}_0\delta^1_{n_\mu'}\right) \cdots \left(\delta^{n_\nu}_1 \delta^0_{n_\nu'} \right) \cdots \\ &\qquad\quad= (-1)^{0 + n_{\mu+1}+\cdots+n_{\nu-1}}\, \delta^{n_1}_{n_1'} \, \delta^{n_2}_{n_2'} \cdots \left(\delta^{n_\mu}_0\delta^1_{n_\mu'}\right) \cdots \left(\delta^{n_\nu}_1\delta^0_{n_\nu'}\right) \cdots \end{aligned}\end{align}\]
The two equations differ by a factor of \(-1\) , so adding them gives zero. Putting everything together, we conclude that \(\big\{c_\mu,c_\nu^\dagger\big\} = \delta_{\mu\nu}\) , as stated in \(\eqref{anticommutation}\).
As you can see, the derivation of the fermionic anticommutation relations is quite hairy, in large part due to the \((-1)^{(\cdots)}\) factors in the definitions of the creation and annihilation operators. But once these relations have been derived, we can deal entirely with the creation and annihilation operators, without worrying about the underlying occupation number representation and its \((-1)^{(\cdots)}\) factors. By the way, if we had chosen to omit the \((-1)^{(\cdots)}\) factors in the definitions, the creation and annihilation operators would still satisfy the anticommutation relation \(\{\hat{c}_\mu,\hat{c}_\nu^\dagger\}=\delta_{\mu\nu}\) , but two creation operators or two annihilation operators would commute rather than anticommute . During subsequent calculations, the “algebra” of creation and annihilation operators ends up being much harder to deal with.
Second-quantized operators
One of the key benefits of second quantization is that it allows us to express multi-particle quantum operators clearly and succinctly, using the creation and annihilation operators defined in Section 4.3 as “building blocks”.
Non-interacting particles
Consider a system of non-interacting particles. When there is just one particle ( \(N=1\) ), let the single-particle Hamiltonian be \(\hat{H}^{(1)}\) , which is a Hermitian operator acting on the single-particle Hilbert space \(\mathscr{H}^{(1)}\) . For general \(N\) , the multi-particle Hamiltonian \(\hat{H}\) is a Hermitian operator acting on the Fock space \(\mathscr{H}^F\) . How is \(\hat{H}\) related to \(\hat{H}^{(1)}\) ?
Let us take the bosonic case. Then the multi-particle Hamiltonian should be
\[\hat{H} = \sum_{\mu\nu} \hat{a}^\dagger_\mu H_{\mu\nu} \hat{a}_\nu, \;\;\; \mathrm{where}\;\;\; H_{\mu\nu} = \langle\mu|\hat{H}^{(1)}|\nu\rangle, \label{twobosonH}\]
where \(\hat{a}_\mu\) and \(\hat{a}_\mu^\dagger\) are the boson creation and annihilation operators, and \(|\mu\rangle\) , \(|\nu\rangle\) refer to single-particle state vectors drawn from some orthonormal basis for \(\mathscr{H}^{(1)}\) .
To understand why Equation \(\eqref{twobosonH}\) is right, consider its matrix elements with respect to various states. First, for the vacuum state \(|\varnothing\rangle\) ,
\[\langle\varnothing|\hat{H}|\varnothing\rangle = 0.\]
This makes sense. Second, consider the matrix elements between single-particle states:
\[\begin{align}\begin{aligned} \langle n_\mu = 1 |\hat{H}|n_\nu = 1\rangle &= \langle\varnothing|a_\mu \Big(\sum_{\mu'\nu'} \hat{a}^\dagger_{\mu'} H_{\mu'\nu'} \hat{a}_{\nu'}\Big) a_\nu^\dagger |\varnothing\rangle \\ &= \sum_{\mu'\nu'} H_{\mu'\nu'} \, \langle\varnothing|a_\mu \hat{a}^\dagger_{\mu'} \hat{a}_{\nu'} a_\nu^\dagger |\varnothing\rangle \\ &= \sum_{\mu'\nu'} H_{\mu'\nu'} \, \delta^{\mu}_{\mu'} \delta^{\nu}_{\nu'} \\&= H_{\mu\nu}.\end{aligned}\end{align}\]
This exactly matches the matrix element defined in Equation \(\eqref{twobosonH}\).
Finally, consider the case where \(\{|\mu\rangle\}\) forms an eigenbasis of \(\hat{H}_1\) . Then
\[\hat{H}^{(1)}|\mu\rangle = E_\mu |\mu\rangle \;\;\;\Rightarrow\;\;\; H_{\mu\nu} = E_\mu\,\delta_{\mu\nu} \;\;\;\Rightarrow \;\; \hat{H} = \sum_{\mu} E_\mu \,\hat{n}_\mu.\]
As previously noted, \(\hat{n}_\mu = \hat{a}^\dagger_\mu \hat{a}_\mu\) is the number operator, an observable corresponding to the occupation number of single-particle state \(\mu\) . Thus, the total energy is the sum of the single-particle energies, as expected for a system of non-interacting particles.
We can also think of the Hamiltonian \(\hat{H}\) as the generator of time evolution. Equation \(\eqref{twobosonH}\) describes an infinitesimal time step that consists of a superposition of alternative evolution processes. Each term in the superposition, \(\hat{a}^\dagger_\mu H_{\mu\nu} \hat{a}_\nu\) , describes a particle being annihilated in state \(\nu\) , and immediately re-created in state \(\mu\) , which is equivalent to “transferring” a particle from \(\nu\) to \(\mu\) . The quantum amplitude for this process is described by the matrix element \(H_{\mu\nu}\) . This description of time evolution is applicable not just to single-particle states, but also to multi-particle states containing any number of particles.
Note also that the number of particles does not change during time evolution. Whenever a particle is annihilated in a state \(\nu\) , it is immediately re-created in some state \(\mu\) . This implies that the Hamiltonian commutes with the total particle number operator:
\[ [\hat{H}, \hat{N}]\: = 0, \;\;\;\mathrm{where}\;\;\hat{N} \equiv \sum_\mu \hat{a}_\mu^\dagger a_\mu. \label{Hdiag}\]
The formal proof for this is left as an exercise (see Exercise 4.5.6). It follows directly from the creation and annihilation operators’ commutation relations (for bosons) or anticommuattion relations (for fermions).
Apart from the total energy, other kinds of observables—the total momentum, total angular momentum, etc.—can be expressed in a similar way. Let \(\hat{A}^{(1)}\) be a single-particle observable. For a multi-particle system, the operator corresponding to the “total \(A\) ” is
\[\hat{A} = \sum_{\mu\nu} \hat{a}^\dagger_\mu A_{\mu\nu} \hat{a}_\nu, \;\;\; \mathrm{where}\;\; A_{\mu\nu} = \langle\mu|\hat{A}^{(1)}|\nu\rangle. \label{Aop}\]
For fermions, everything from Equation \(\eqref{twobosonH}\)–\(\eqref{Aop}\) also holds, but with the \(a\) operators replaced by fermionic \(c\) operators.
Change of basis
A given set of creation and annihilation operators is defined using a basis of single-particle states \(\{|\mu\rangle\}\) , but such a choice is obviously not unique. Suppose we have a different single-particle basis \(\{|\alpha\rangle\}\) , such that
\[|\alpha\rangle = \sum_\mu U_{\alpha\mu} |\mu\rangle, \label{alphabasis}\]
where \(U_{\alpha\mu}\) are the elements of a unitary matrix. Let \(\hat{a}_\alpha^\dagger\) and \(\hat{a}_\mu^\dagger\) denote the creation operators defined using the two different basis (once again, we will use the notation for bosons, but the equations in this section are valid for fermions too). Writing Equation \(\eqref{alphabasis}\) in terms of the creation operators,
\[\begin{aligned} \hat{a}_\alpha^\dagger|\varnothing\rangle = \sum_\mu U_{\alpha\mu} \hat{a}_\mu^\dagger |\varnothing\rangle,\end{aligned}\]
We therefore deduce that
\[\hat{a}_\alpha^\dagger = \sum_\mu U_{\alpha\mu} \hat{a}_\mu^\dagger, \;\;\; \hat{a}_\alpha = \sum_\mu U_{\alpha\mu}^* \hat{a}_\mu. \label{atrans}\]
Using the unitarity of \(U_{\alpha\mu}\) , we can verify that \(\hat{a}_\alpha\) and \(\hat{a}_\alpha^\dagger\) satisfy bosonic commutation relations if and only if \(\hat{a}_\mu\) and \(\hat{a}_\mu^\dagger\) do so. For fermions, we put \(c\) operators in place of \(a\) operators in Equation \(\eqref{atrans}\), and use anticommutation rather than commutation relations.
To illustrate how a basis change affects a second quantized Hamiltonian, consider a system of non-interacting particles whose single-particle Hamiltonian is diagonal in the \(\alpha\) basis. The multi-particle Hamiltonian is
\[\hat{H} = \sum_\alpha E_\alpha \hat{a}_\alpha^\dagger \hat{a}_\alpha,\]
consistent with Equation \(\eqref{Hdiag}\). Applying Equation \(\eqref{atrans}\),
\[\hat{H} = \sum_{\mu\nu} \hat{a}_\mu^\dagger \left(\sum_\alpha E_\alpha U_{\alpha\mu} U_{\alpha\nu}^*\right) \hat{a}_\nu \label{Htransformed}\]
Compare this to single-particle bracket
\[\begin{align}\begin{aligned} H_{\mu\nu} \equiv \langle\mu | \hat{H}^{(1)} | \nu\rangle &= \sum_{\alpha\beta} \langle \alpha | U_{\alpha\mu} \hat{H}^{(1)} U_{\beta\nu}^* |\beta\rangle \\ &= \sum_{\alpha\beta} U_{\alpha\mu} U_{\beta\nu}^* \, E_\beta \, \delta_{\alpha\beta} \\ &= \sum_{\alpha} E_\alpha\, U_{\alpha\mu} U_{\alpha\nu}^*.\end{aligned}\end{align}\]
This precisely matches the term in parentheses in Equation \(\eqref{Htransformed}\). This is consistent with the general form of \(\hat{H}\) for non-interacting particles, Equation \(\eqref{twobosonH}\).
Particle interactions
Hermitian operators can also be constructed out of other kinds of groupings of creation and annihilation operators. For example, a pairwise (two-particle) potential can be described with a superposition of creation and annihilation operator pairs, of the form
\[\hat{V} = \frac{1}{2} \sum_{\mu\nu\lambda\sigma} \hat{a}^\dagger_\mu \hat{a}^\dagger_\nu \, V_{\mu\nu\lambda\sigma}\, \hat{a}_\sigma \hat{a}_\lambda. \label{Vop}\]
The prefactor of \(1/2\) is conventional. In terms of time evolution, \(\hat{V}\) “transfers” (annihilates and then re-creates) a pair of particles during each infinitesimal time step. Since the number of annihilated particles is always equal to the number of created particles, the interaction conserves the total particle number. We can ensure that \(\hat{V}\) is Hermitian by imposing a constraint on the coefficents:
\[\hat{V}^\dagger = \frac{1}{2} \sum_{\mu\nu\lambda\sigma} \hat{a}^\dagger_\lambda \hat{a}^\dagger_\sigma \, V_{\mu\nu\lambda\sigma}^* \, \hat{a}_\nu \hat{a}_\mu = \hat{V} \;\;\Leftarrow \;\; V_{\lambda\sigma\mu \nu}^* = V_{\mu\nu\lambda\sigma}.\]
Suppose we are given the two-particle potential as an operator \(\hat{V}^{(2)}\) acting on the two-particle Hilbert space \(\mathscr{H}^{(2)}\) . We should be able to express the second-quantized operator \(\hat{V}\) in terms of \(\hat{V}^{(2)}\) , by comparing their matrix elements. For example, consider the two-boson states
\[\begin{align} \begin{aligned} |n_\mu = 1, n_\nu = 1\rangle &= \frac{1}{\sqrt{2}} \Big(|\mu\rangle|\nu\rangle + |\nu\rangle|\mu\rangle \Big), \\ |n_\lambda = 1, n_\sigma = 1\rangle &= \frac{1}{\sqrt{2}} \Big(|\lambda\rangle|\sigma\rangle + |\sigma\rangle|\lambda\rangle \Big), \end{aligned}\end{align}\]
where \(\mu \ne \nu\) and \(\lambda \ne \sigma\) . The matrix elements of \(\hat{V}^{(2)}\) are
\[\begin{align}\begin{gathered} \langle n_\mu = 1, n_\nu = 1 | \hat{V}^{(2)} |n_\lambda = 1, n_\sigma = 1\rangle \\ = \frac{1}{2} \Big( \langle\mu|\langle\nu|\hat{V}^{(2)}|\lambda\rangle\sigma\rangle + \langle\nu|\langle\mu|\hat{V}^{(2)}|\lambda\rangle\sigma\rangle + \langle\mu|\langle\nu|\hat{V}^{(2)}|\sigma\rangle\lambda\rangle + \langle\nu|\langle\mu|\hat{V}^{(2)}|\sigma\rangle\lambda\rangle \Big). \end{gathered}\label{mat0}\end{align}\]
On the other hand, the matrix elements of the second-quantized operator \(\hat{V}\) are
\[\begin{align} \langle n_\mu = 1, n_\nu = 1 | \hat{V} |n_\lambda = 1, n_\sigma = 1\rangle &= \sum_{\mu'\nu'\lambda'\sigma'} V_{\mu'\nu'\lambda'\sigma'} \langle \varnothing | \hat{a}_\nu \hat{a}_\mu \hat{a}_{\mu'}^\dagger \hat{a}_{\nu'}^\dagger \hat{a}_{\sigma'} \hat{a}_{\lambda'} \hat{a}_\lambda^\dagger \hat{a}_\sigma^\dagger |\varnothing\rangle \label{mat1}\\ &= \frac{1}{2} \left(V_{\mu\nu\lambda\sigma} + V_{\mu\nu\sigma\lambda} + V_{\nu\mu\lambda\sigma} + V_{\nu\mu\sigma\lambda} \right).\label{mat2}\end{align}\]
In going from Equation \(\eqref{mat1}\) to \(\eqref{mat2}\), we use the bosonic commutation relations repeatedly to “pushing” the annihilation operators to the right (so that they can act upon \(|\varnothing\rangle\) ) and the creation operators to the left (so that they can act upon \(\langle\varnothing|\) ). Comparing Equation \(\eqref{mat0}\) to Equation \(\eqref{mat2}\), we see that the matrix elements match if we take
\[\begin{align} V_{\mu\nu\lambda\sigma} = \langle\mu|\langle\nu|\hat{V}^{(2)} |\lambda\rangle|\sigma\rangle.\end{align}\]
For instance, if the bosons have a position representation, we would have something like
\[\begin{align} V_{\mu\nu\lambda\sigma} = \int d^d r_1 \int d^d r_2\; \varphi_\mu^*(r_1) \,\varphi_\nu^*(r_2) \, V(r_1, r_2) \, \varphi_\lambda(r_1) \, \varphi_\sigma(r_2).\end{align}\]
The appropriate coefficients for \(\mu = \nu\) and/or \(\lambda = \sigma\) , as well as for the fermionic case, are left for the reader to work out.
Other observables?
Another way to build a Hermitian operator from creation and annihilation operators is
\[\hat{A} = \sum_\mu \left(\alpha_\mu \hat{a}_\mu^\dagger + \alpha^*_\mu \hat{a}_\mu\right).\]
If such a term is added to a Hamiltonian, it breaks the conservation of total particle number. Each infinitesimal time step will include processes that decrement the particle number (due to \(\hat{a}_\mu\) ), as well as processes that increment the particle number (due to \(\hat{a}_\mu^\dagger\) ). Even if the system starts out with a fixed number of particles, such as the vacuum state \(|\varnothing\rangle\) , it subsequently evolves into a superposition of states with different particle numbers. In the theory of quantum electrodynamics, this type of operator is used to describe the emission and absorption of photons caused by moving charges.
Incidentally, the name “second quantization” comes from this process of using creation and annihilation operators to define Hamiltonians. The idea is that single-particle quantum mechanics is derived by “quantizing” classical Hamiltonians via the imposition of commutation relations like \([x,p] = i\hbar\) . Then, we extend the theory to multi-particle systems by using the single-particle states to define creation/annihilation operators obeying commutation or anticommutation relations. This can be viewed as a “second” quantization step.


1.14.34: Internal Pressure
According to the Thermodynamic Equation of State,
\[\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}=\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}-\mathrm{p} \nonumber \]
The partial differential \((\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}\) (with units, \(\mathrm{N m}^{-2}\)) is the internal pressure \(\pi_{\mathrm{int}}\).
\[\pi_{\mathrm{int}}=\mathrm{T} \, \beta_{\mathrm{V}}-\mathrm{p} \nonumber \]
\(\pi_{\mathrm{int}\) describes the sensitivity of energy \(\mathrm{U}\) to a change in volume. A high \(\phi_{\mathrm{int}\) implies strong inter-molecular cohesion [1-8]. For many liquids, \(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}>>\mathrm{p}\) such that
\[(\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}} \cong \mathrm{T} \, \beta_{\mathrm{V}} \nonumber \]
\(\mathrm{T} \, \boldsymbol{\beta}_{\mathrm{V}}\) is sometimes called the thermal pressure. By definition, for \(\mathrm{n}\) moles of a perfect gas,
\[\mathrm{p} \, \mathrm{V}=\mathrm{n} \, \mathrm{R} \, \mathrm{T} \nonumber \]
Then
\[\mathrm{V} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R} \nonumber \]
Or,
\[\mathrm{T} \,\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\mathrm{n} \, \mathrm{R} \, \mathrm{T} / \mathrm{V}=\mathrm{p} \nonumber \]
From equation (a), for a perfect gas, \(\left(\frac{\partial \mathrm{U}}{\partial \mathrm{V}}\right)_{\mathrm{T}}\) is zero.
The internal pressure for water(\(\ell\)) presents an interesting puzzle [9]. From equations (a) and (c), it follows that [1]
\[\pi_{\mathrm{int}}=\mathrm{T} \,\left(\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}}\right)-\mathrm{p} \nonumber \]
But at the temperature of maximum density (TMD), \(\alpha_{p}\) is zero. So near the TMD, \(\pi_{\mathrm{int}\) is zero. We understand this pattern if we think about hydrogen bonding. In order to form a strong hydrogen bond between two neighboring water molecules the O-H---O link has to be close to if not actually linear. In other words the molar volume for water(\(\ell\)) is larger than the molar volume of a system comprising close-packed water molecules. Consequently hydrogen bonding has a strong ‘repulsive’ component to intermolecular interaction. However once formed hydrogen bonding has a strong cohesive contribution to intermolecular forces. Hence for water between \(273\) and \(298 \mathrm{~K}\) cohesive and repulsive components of hydrogen bonding play almost competitive roles.
Footnotes
[1] Using a calculus operation, \(\left(\frac{\partial p}{\partial T}\right)_{V}=-\left(\frac{\partial V}{\partial T}\right)_{p} \,\left(\frac{\partial p}{\partial V}\right)_{T}\) For equilibrium properties,
\[\left(\frac{\partial \mathrm{p}}{\partial \mathrm{T}}\right)_{\mathrm{V}}=\frac{\alpha_{\mathrm{p}}}{\kappa_{\mathrm{T}}} \nonumber \]
[2] Some authors use the term ‘isochoric thermal pressure coefficient’ for the property, \(\left(\frac{\partial p}{\partial T}\right)_{V}\)
[3] For details of original proposals concerning internal pressures see the following references.
- W. Westwater, H. W. Frantz and J. H. Hildebrand, Phys.Rev.,1928, 31,135.
- J. H. Hildebrand, Phys.Rev.,1929,34,649 and 984.
- See also S. E. Wood, J.Phys.Chem.,1962,66, 600.
[4] Internal pressures are quoted in the literature using many units. Here we use \(\mathrm{N m}^{-2}\). We list some internal pressures and relative permitivities at \(298.15 \mathrm{~K}\).
| liquid | \(\varepsilon_{\mathrm{r}}\( | \(\pi_{\mathrm{int}} / 10^{5} \mathrm{~N m}^{-2}\) |
| water | 78.5 | 1715 |
| methanol | 32.63 | 2849 |
| ethanol | 24.30 | 2908 |
| propanone | 20.7 | 3368 |
| diethyl ether | 4.3 | 2635 |
| tetrachloromethane | 2.24 | 3447 |
| dioxan | 2.2 | 4991 |
The above details are taken from M. R. J. Dack, J.Chem.Educ.,1974, 51 ,231;see also
- Aust. J. Chem.,1976, 29 ,771 and 779.
- D. D. MacDonald and J. B. Hyne, Can.J.Chem.,1971, 49 ,2636
[5] For a discussion of effects of solvents on rates of chemical reactions with reference to internal pressures, see
- K. Colter and M. L. Clemens, J.Phys.Chem.,1964, 68 ,651.
- A. P. Stefani, J. Am. Chem.Soc.,1968, 90 ,1694.
[6] For comments on solvent polarity and internal pressures see J. E. Gordon, J. Phys. Chem.,1966, 70 ,2413.
[7] For comments on internal pressures of binary aqueous mixtures see D. D. Macdonald, Can. J Chem.,1976, 54 ,3559; and references therein.
[8] For comments on effect of internal pressure on conformational equilibria see R. J. Ouellette and S. H. Williams, J. Am. Chem.Soc.,1971, 93 ,466.
[9] For details concerning the dependence of internal pressure of water and \(\mathrm{D}_{2}\mathrm{O}\), see M. J. Blandamer, J. Burgess and A.W.Hakin, J. Chem. Soc. Faraday Trans. 1, 1987, 83 , 1783.
[10] For comments on the calculation of excess internal pressures for binary liquid mixtures using equation (h) see W. Marczak, Phys.Chem.Chem.Phys.,2002, 4 ,1889.


4.5: The Fauna of California
The fauna of the State of California may be the most diverse in the United States of America. Of the lower 48 conterminous states. California also has the greatest diversity in climate, terrain, and geology in general. The state's six life zones are the lower Sonoran (desert); upper Sonoran (foothill regions and some coastal lands); transition (coastal areas and moist northeastern counties); and the Canadian, Hudsonian, and Arctic zones, comprising California's highest elevations. California’s diverse geography gives rise to dozens of different ecosystems, each of which has its own unique native plants and animals. California is a huge state, the 3rd largest in the U.S., and can range broadly in habitat type.
Earth scientists typically divide California into eleven distinct geomorphic provinces with clearly defined boundaries. They are, from north to south, the Klamath Mountains, the Cascade Range, the Modoc Plateau, the Basin and Range, the Coast Ranges, the Central Valley, the Sierra Nevada, the Transverse Ranges, the Mojave Desert, the Peninsular Ranges, and the Colorado Desert. Here, the Los Angeles Basin, the Channel Islands, and the Pacific Ocean are treated as distinct regions.
Common animals that live throughout all the state, and its coasts, include raccoons, weasels, otters, beavers, hawks, lizards, owls, coyotes, skunks, snakes, cougars, black bears, deer, squirrels, and whales. As of 2024, there are 685 bird species on the official California checklist but the California Birds Records Committee, ten of which are introduced species which are not native to the state. The California quail, the official state bird, has a breeding habit of mainly shrubby areas and open woodland. Another bird which winters in California is the American white pelican, which is a large seabird, with a wingspan reaching up to 110 inches (280 cm).
Venomous spiders in California include Arizona recluse, Baja recluse, Chilean recluse, desert recluse, Martha’s recluse, Russell’s recluse, brown widow, and western black widow.
Northern California
The forests in northern parts of California have an abundant fauna, which includes for instance the black-tailed deer, black bear, gray fox, North American cougar, bobcat, and Roosevelt elk. Garter snakes and rattlesnakes are common, as are such amphibians as the mudpuppy and redwood salamander. The kingfisher, chickadee, towhee, and hummingbird represent the bird life of this region. There are an estimated 3,487 wild horses in Northern California as of 2024, according to the Bureau of Land Management. Gray wolves began repopulating California in 2011 as they entered Lassen, Siskiyou, and Plumas Counties from the Cascade Range of Oregon.
Case Study - Sierra Nevada
Mammals of the Canadian zone include the snowshoe hare, mountain chickadee, and several species of chipmunk. Conspicuous birds include the blue-fronted jay, hermit thrush, American dipper, and Townsend's solitaire. Birds become scarcer as one ascends to the Hudsonian zone, and the wolverine is now regarded as rare. The only bird native to the high Arctic region is the gray-crowned rosy finch, but others often visit, including the Anna's hummingbird and Clark's nutcracker.
Principal mammals of this region are also visitors from other zones; the Sierra coney and white-tailed jackrabbit make their homes here. The bighorn sheep also live in this mountainous terrain; the bighorn sheep was listed as endangered by the US Fish and Wildlife Service. Some animals in the Yosemite Valley include bobcats, mountain lions, ring-tailed cats, the Steller's jay, California ground squirrels and the American black bear.
Southern California
Southern California constitutes one of the more varied collections of geologic, topographic, and natural ecosystem landscapes in a diversity outnumbering other major regions in the country. The region spans from Pacific Channel Islands, shorelines, beaches, and coastal plains, through the Transverse and Peninsular Ranges with their peaks, into the large and small interior valleys, to the vast deserts of California.
Several varieties of rattlesnakes are indigenous to the region. While only the Pacific Northwest rattler makes its home in Northern California, almost a dozen rattlesnakes make their home in the deserts of Southern California, including the western diamondback and the Mojave rattlesnake. Birds in the region include the Anna's hummingbird, acorn woodpecker, northern flicker, California towhee, California vulture, red-tailed hawk and many more.
Case Study - Mojave Desert
The Mojave Desert appears to have little in the way of wildlife but has large, diverse populations. The extremely warm desert environment has animals that have adapted to their environment with each filling an important niche in the desert ecosystem.  One of the more iconic ‘trees’ of the region are the Joshua trees. Joshua trees are succulents, a type of plant known for storing water in their leaves or stems. Despite their tree-like appearance with a single trunk and branches, they belong to the yucca family. The story behind their name is quite interesting. Legend has it that 19 th -century Mormon settlers, traversing the harsh desert, saw the tree's outstretched limbs resembling a biblical figure, Joshua, raising his arms in prayer. This association led to the name "Joshua tree."
Coastal California
Along the coast of California is the California sea lion, which can grow up to seven feet long and can be found in shallow ocean water, near beaches, and among rocks. In the open ocean is the northern elephant seal, which grows up to a massive 14 feet (4.3 meters) and has a population of just over 150,000. The California ocean is home to six species of seals: Guadalupe fur seal, northern fur seal, northern sea lion, California sea lion, northern elephant seal and harbor seal.
California waters are also home to eleven species of dolphins, including the short-beaked common dolphin and the Pacific white-sided dolphin. A dozen species of whales live in California, including the killer whale and the gray whale. At least 34 species of shark have been recorded off the California Coast, including the great white shark and tiger shark.
Case Study - Channel Islands
More than 2000 species of plants and animals can be found within the Channel Islands National Park, which consists of five out of the eight islands that compromise the California Channel Islands. Three mammals are endemic to the archipelago: The Channel Islands fox, the deer mouse and the Channel Islands spotted skunk. Introduced mammal species include feral pigs, cats, rats, deer, cattle, the Santa Cruz sheep, and the Catalina Island bison herd.
Other mammals include the harvest mouse, the ground squirrel, and the ornate shrew. Other animals in the islands include island fence lizard, island scrub jay, harbor seal, California sea lion, island night lizard, barn owl, bald eagles, American kestrel, horned lark and meadowlark and California brown pelican. One hundred and forty-five of these species are unique to the islands and found nowhere else in the world. Marine life ranges from microscopic plankton to the endangered blue whale, the largest animal ever to live on earth. The oceans surrounding the islands have a rich marine life; species include orcas, swellshark, bat ray, California moray, great white shark, and sea lions.


8.4: Pythagoras
Musical Investigations and Theories
According to legend, Pythagoras discovered the foundations of musical tuning by listening to the sounds of four blacksmith’s hammers, which produced consonance and dissonance when they were struck simultaneously. According to Nicomachus in his 2nd century CE Enchiridion harmonices, Pythagoras noticed that hammer A produced consonance with hammer B when they were struck together, and hammer C produced consonance with hammer A, but hammers B and C produced dissonance with each other. Hammer D produced such perfect consonance with hammer A that they seemed to be “singing” the same note. Pythagoras rushed into the blacksmith shop to discover why, and found that the explanation was in the weight ratios. The hammers weighed 12, 9, 8, and 6 pounds respectively. Hammers A and D were in a ratio of 2:1, which is the ratio of the octave. Hammers B and C weighed 9 and 8 pounds. Their ratios with hammer A were (12:9 = 4:3 = perfect fourth) and (12:8 = 3:2 = perfect fifth). The space between B and C is a ratio of 9:8, which is equal to the musical whole tone, or whole step interval.
The legend is, at least with respect to the hammers, demonstrably false. These proportions are indeed relevant to string length (e.g. that of a monochord) — using these founding intervals, it is possible to construct the chromatic scale and the basic seven-tone diatonic scale used in modern music, and Pythagoras might well have been influential in the discovery of these proportions — but the proportions do not have the same relationship to hammer weight and the tones produced by them.
Earlier sources mention Pythagoras’ interest in harmony and ratio. Xenocrates (4th century BCE), while not as far as we know mentioning the blacksmith story, described Pythagoras’ interest in general terms: “Pythagoras discovered also that the intervals in music do not come into being apart from number; for they are an interrelation of quantity with quantity. So he set out to investigate under what conditions concordant intervals come about, and discordant ones, and everything well-attuned and ill-tuned.” Whatever the details of the discovery of the relationship between music and ratio, it is regarded as historically the first empirically secure mathematical description of a physical fact. As such, it is symbolic of, and perhaps leads to, the Pythagorean conception of mathematics as nature’s modus operandi . As Aristotle was later to write, “the Pythagoreans construct the whole universe out of numbers.”
Pythagoreans elaborated on a theory of numbers, the exact meaning of which is still debated among scholars. Another belief attributed to Pythagoras was that of the “harmony of the spheres.” Thus the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.
In 1894, E. Cobham Brewer (1894), wrote:
The music or harmony of the spheres. Pythagoras, having ascertained that the pitch of notes depends on the rapidity of vibrations, and also that the planets move at different rates of motion, concluded that the sounds made by their motion must vary according to their different rates of motion. As all things in nature are harmoniously made, the different sounds must harmonize, and the combination he called the “harmony of the spheres.” Kepler has a treatise on the subject. [1]
- E. Cobham Brewer (1894). Dictionary of Phrase and Fable, p. 2614. ↵
Contributors and Attributions
- Pythagoras. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Pythagoras . License : CC BY-SA: Attribution-ShareAlike
- Pythagorean hammers. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Pythagorean_hammers . License : CC BY-SA: Attribution-ShareAlike
- Gaffurio Pythagoras . Authored by : Franchino Gaffurio. Provided by : Wikimedia Commons. Located at : https://commons.wikimedia.org/wiki/File:Gaffurio_Pythagoras.png#/media/File:Gaffurio_Pythagoras.png . License : Public Domain: No Known Copyright


10.7: How can teachers continue to learn throughout life?
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by Christa Portlock
“Give a man a fish, feed him for a day. Teach a man to fish; feed him for a lifetime.” The founder of Taoism, Lao Tzu, spoke these words over 2000 years ago. (Smith, xi) Education is similar to this principle. For centuries students were “fed” information and taught to memorize facts, in the hope they would retain a portion of what was taught. Today it isn’t enough just to feed information to students. Our world is changing so fast that new information becomes available faster than it can be taught. If a nation wants to keep pace with this ever-changing world, then learning can’t stop at graduation. Students as well as teachers must learn “how to fish” for knowledge by becoming life long learners. Teachers can become life-long learners by realizing and acknowledging that life long-learning is a necessity; by learning to keep up with changes through personal and professional development; and by teaching with passion, inspiring young minds to see learning as something wonderful.
Life-long Learning is a Necessity
The last century has brought about an explosion of inventions, new technologies and new information about our world and universe. Teachers have had to keep up with all the new information and knowledge. What would it be like if they hadn’t? Picture a classroom somewhere in America. The teacher has been teaching for many years; she might be in her 50’s. That means she started teaching during the 1970s. Think back about 30 years. What were American schools like then? What kind of things would you have found in her first classroom? Classrooms in those days had chalkboards and pull-down maps. Some schools had TVs, film-and overhead projectors that were shared between classes and could be rolled into the room. Calculators existed but were generally not used in the classroom. Now imagine for a minute what it would be like if this teacher had not adapted to change. What would it be like if she had buried herself in her classroom, not furthered her learning, not embraced all the new technologies, events and discoveries? If she had not been a life-long learner, what would she have missed? What would she not have learned, embraced, and passed on to students? Think of the things that have happened in the past 35 years. She would have missed, among other things, the hostage crisis in the late 70’s, Presidents coming and going, this country at war more than once, the attack of 9/11. She would not have seen the Soviet Union collapse, would not have heard of Aids or global warming. She would not have known much, if anything, about DNA. She wouldn’t know how to use a personal computer, the internet or cell phones. It is easy to see that if we stand still the world will pass us by. We live in a global society that is changing at breathtaking speed. We need to adapt and change. Life-long learning is a necessity for everyone.
Personal and Professional Development
In order to keep up with our changing environment and to stay informed, teachers make a conscious effort to continue learning. They will take part in professional training through the school district or through classes taken at universities. Most teachers read books and professional journals to keep up with the newest information about child development. They read magazines, newspapers, watch the news and surf the internet to keep up with changes that happen around the world. Teachers attend workshops to keep up with technology, the newest teaching strategies, and classroom management techniques. They are eager to take in new information about our world and look for ways to improve their teaching practices in the classroom. Another way a teacher can improve and move forward is to learn more about him or herself. In the book Those Who can, Teach by Ryan Cooper, a teacher is likened to a sculptor who “begins with a vision of what he or she wants to be and then sets to work transforming the vision into a reality… The process requires an understanding of … self … it takes long hours of chipping away and then smoothing the surfaces… To be a teacher, particularly a teacher who is continuously moving forward, is a lifelong commitment …” (Cooper, 473) A sculptor needs time, patience and the ability to see beyond what is easily visible on the surface to create his work of art. A teacher uses those same qualities, just like a sculptor, to patiently reflect and look deep within him- or herself to see where changes could be made. At the end of a day it is helpful for a teacher to think deeply about the events of that day, to analyze him- or herself and therefore better understand and react to situations in the future. It is good to take time to remember happy events, a chance to feel content about good things that may have happened. Reflection can help a person to learn and adjust to his or her environment. As previously discussed, teachers have to become life-long learners and have to adjust to keep up with our ever-changing world through personal and professional development. They are however, not the only ones who need to keep pace with new information and technology. One of the reasons it is so important for teachers to stay informed, is so they can pass this new knowledge and information on to students.
Teaching with Passion
Note
If we can have “students see our passions – it gives them proof that we enjoy learning.” (Thomas, 2007)
Children have a sense of wonder when they are young. “Keeping the joy in learning should be a priority of every school and every teacher.” (Warner, 2006)If children can keep their sense of excitement about the world, they will love to learn and that original excitement and love of learning will most likely stay with them for life. To help keep the excitement alive, teachers aim to inspire young minds. They want their students to see how much fun learning can be. In the article “A Return to Community: Inquiry in Action” Ann H. MacKenzie wrote: “Life-long learning implies a buy-in curiosity about life... an insatiable desire to be alive in ‘the now, the future,’ and immersed in the teeming life surrounding us every moment.” (Thomas, 2007) If we can have “students see our passions – it gives them proof that we enjoy learning.” (Thomas, 2007) It is wonderful to see the sparks when students understand a concept and take off excited with their new-found knowledge. “We must model that inquisitiveness in the classrooms. Our students must see us involved in our own life-long learning… Watch us passionately immerse ourselves in our own curiosities.” (MacKenzie)
When teachers model excitement about learning, students will see learning as something positive to be imitated. In turn they will hopefully become life-long learners themselves. Rafe Esquinth wrote an inspiring book entitled Teach Like Your Hair’s on Fire . He teaches in a “rough” Los Angeles neighborhood and found a way to inspire his students to learn, imagine, and eventually succeed in school. Esquinth relates a story about a day that didn’t start out well. In the course of that day he became so involved in teaching that unbeknownst to him, his hair caught on fire during an experiment. He was so involved in teaching, he didn’t even notice. That day he realized that the key to teaching was to become absorbed and excited about learning. He was determined from then on to make his goal to “teach like his hair is on fire”. (Esquinth, xii) If we can convey to students excitement about a subject, about learning, and about life we will teach as if our “hair is on fire”. That kind of excitement will inspire children to become life-long learners.
Note
“We must model that inquisitiveness in the classrooms. Our students must see us involved in our own life-long learning… Watch us passionately immerse ourselves in our own curiosities.” (MacKenzie)
Conclusion
As individuals and as citizens of a global economy we have to keep pace with our world. We must continue to learn and then pass our knowledge and love of learning on to the next generation. We have to inspire the next generation to find joy in continuous learning so that, even after leaving school, students will be prepared for this ever-changing world, ready to embrace a world filled with new, exciting things, ready to investigate and ready to keep learning, never giving up on their quest for knowledge. A fish is only one meal… we want to teach students to be excited about learning, excited about fishing for information; we want to give them the tools to become life-long fishermen for knowledge.
Exercise \(\PageIndex{1}\)
1)Does learning stop at graduation? Why?
a)Yes, because there are too many other things to do.
b)Yes, because learning makes hair catch on fire.
c)No, because it’s important to keep up with our ever-changing world.
d)Of course we stop learning at graduation! We’re done with school!
2)What is an example of professional development for teachers?
a)Reading articles about child development
b)Attending workshops
c)Keeping up with technology
d)All of the above
3)What is an important part of personal development for teachers?
a)Developing a seventh sense
b)Getting a pedicure
c)Practicing reflection
d)Joining a teacher’s organization
4)How does the practice of reflection help a teacher?
a)Reflection helps a teacher learn about him or herself.
b)Reflection can help a teacher to learn about and adjust to his or her environment.
c)It’s important for a teacher to see her own reflection to make sure she looks proper before entering the classroom.
d)Both a and b.
5)Why is it important to teach with passion?
a)It gives children proof that we enjoy learning.
b)It shows children that learning is something positive to be imitated.
c)It keeps excitement about learning alive.
d)All of the above.
- Answer
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1. c
2. d
3. c
4. d
5. d
Reaction Paper
My Thoughts About Life-long Learning
by Christa Portlock
Investigating life-long learning and what it means made me think about what it means to me as a student and as a prospective future teacher. To me the world has always been a very interesting, exciting, and beautiful place to investigate and to learn about. I can’t imagine what my life would be like without that wonder and appreciation about learning new things. Life is so much more stimulating, rich and pleasurable when we engage our minds, use our imagination and fill our hearts with the beauty that surrounds us. I would love to pass that excitement on to someone else. It would be exciting to spark the imagination of children and watch them find the same joy that I have always felt when learning something new. I know I have much to learn to be able to know how to best pass on knowledge, as well as excitement about learning. I am on a learning quest right now. I want to learn how to become that teacher who can create those sparks that light up a child’s eyes and light his or her path through life with inquisitiveness and excitement. I will spend the next few years in school learning all I can. I have spent much time in classrooms in the past, but from this time on I will be spending time in the classroom watching with different eyes. I will have chances to observe experienced teachers.  I know, however, that this is just the beginning of my journey. Most of my learning about teaching will come through the time spent in the classroom. I want to venture and say that many of my most profound lessons will be taught to me by my future students without them even knowing they’ve been teaching me. I am looking forward to those lessons and am excited about what is to come. As a teacher I will continue to learn. I will take advantage of the training opportunities offered to me and learn individually as well. I hope I will have the opportunity to work with a mentor, learning through observation. I am excited about all the new things to learn and embrace. Life-long learning goes beyond the time spent at school; it goes beyond the classroom and beyond the teaching years. Life-long learning is an adventurous journey through life. Keeping the mind active has enormous health benefits, physically, mentally and spiritually. Life-long learning has the ability to shape, influence, and determine the future. It has the ability to influence mankind’s achievements, technologies, and ultimately our destination through individuals who never stand still and make life-long learning their quest that fuels their investigations. Who knows what sparks will fly when the imagination soars in our future classrooms? I am determined to find out!
Essay Question
How can a teacher be likened to a sculptor?
Bibliography
Cooper, Ryan (2007). Those Who Can, Teach. Boston: Houghton Mifflin Company
Esquith, Rafe (2007). Teach like your hair is on fire. New York, NY: Penguin Group
MacKenzie, A. (2006, November). A Return to the Community: Inquiry in Action. American Biology Teacher, pp. 518, 519. Retrieved February 2, 2008, from Education Research Complete Database. search.ebscohost.com/login.aspc?direct=true&db=ehh&AN=23771678&site=ehost-live
Smith, Rick (2004). Conscious classroom management. San Rafael, CA: Conscious Teaching Publications.
Thomas, J. (2007, November). Teaching with Passion. Education Digest, 73(3), 63-65. Retrieved February 2, 2008, from Education Research Complete database. http://search.ebscohost.com/login.aspx?direct=true&db=ehh&AN=27518663&site=ehost-live
Warner, S. (2006, April 1). Keeping Joy in Technology Education. Technology Teacher, 65(7), 6. (ERIC Document Reproduction Service No. EJ747900) Retrieved February 2, 2008,fromERICdatabase . http://search.ebscohost.com.ezproxy.vccs.edu:2048/login.aspx?direct=true&db=eric&AN=EJ747900&site=ehost-live


5.5: Theories of Cognitive Development, Learning, and Memory
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Pavlov
Ivan Pavlov (1880-1937) was a Russian physiologist interested in studying digestion. As he recorded the amount of salivation his laboratory dogs produced as they ate, he noticed that they actually began to salivate before the food arrived as the researcher walked down the hall and toward the cage. The dogs knew that the food was coming because they had learned to associate the footsteps with the food. The keyword here is “learned”. A learned response is called a “conditioned” response.
Pavlov began to experiment with this “psychic” reflex. He began to ring a bell, for instance, prior to introducing the food. Sure enough, after making this connection several times, the dogs could be made to salivate to the sound of a bell. Once the bell had become an event to which the dogs had learned to salivate, it was called a conditioned stimulus. The act of salivating to a bell was a response that had also been learned, now termed in Pavlov’s jargon, a conditioned response.
Notice that the response, salivation, is the same whether it is conditioned or unconditioned (unlearned or natural). What changed is the stimulus to which the dog salivates. One is natural (unconditioned) and one is learned (conditioned).
Let’s think about how classical conditioning is used on us. One of the most widespread applications of classical conditioning principles was brought to us by the psychologist, John B. Watson. 15
Classical Conditioning
Classical conditioning is a form of learning whereby a conditioned stimulus (CS) becomes associated with an unrelated unconditioned stimulus (US), in order to produce a behavioral response known as a conditioned response (CR). The conditioned response is the learned response to the previously neutral stimulus. The unconditioned stimulus is usually a biologically significant stimulus such as food or pain that elicits an unconditioned response (UR) from the start. The conditioned stimulus is usually neutral and produces no particular response at first, but after conditioning, it elicits the conditioned response.
If we look at Pavlov's experiment, we can identify these four factors at work:
- The unconditioned response was the salivation of dogs in response to seeing or smelling their food.
- The unconditioned stimulus was the sight or smell of the food itself.
- The conditioned stimulus was the ringing of the bell. During conditioning, every time the animal was given food, the bell was rung. This was repeated during several trials. After some time, the dog learned to associate the ringing of the bell with food and to respond by salivating. After the conditioning period was finished, the dog would respond by salivating when the bell was rung, even when the unconditioned stimulus (the food) was absent.
- The conditioned response, therefore, was the salivation of the dogs in response to the conditioned stimulus (the ringing of the bell). 16
Neurological Response to Conditioning
Consider how the conditioned response occurs in the brain. When a dog sees food, the visual and olfactory stimuli send information to the brain through their respective neural pathways, ultimately activating the salivary glands to secrete saliva. This reaction is a natural biological process as saliva aids in the digestion of food. When a dog hears a buzzer and at the same time sees food, the auditory stimuli activates the associated neural pathways. However, since these pathways are being activated at the same time as the other neural pathways, there are weak synapse reactions that occur between the auditory stimuli and the behavioral response. Over time, these synapses are strengthened so that it only takes the sound of a buzzer to activate the pathway leading to salivation.
Operant Conditioning
Operant conditioning is a theory of behaviorism, a learning perspective that focuses on changes in an individual's observable behaviors. In operant conditioning theory , new or continued behaviors are impacted by new or continued consequences. Research regarding this principle of learning was first studied by Edward L. Thorndike in the late 1800's, then brought to popularity by B.F. Skinner in the mid-1900's. Much of this research informs current practices in human behavior and interaction.
Skinner's Research
Thorndike's initial research was highly influential on another psychologist, B.F. Skinner. Almost half a century after Thorndike's first publication of the principles of operant conditioning, Skinner attempted to prove an extension to this theory—that all behaviors were in some way a result of operant conditioning. Skinner theorized that if a behavior is followed by reinforcement, that behavior is more likely to be repeated, but if it is followed by punishment, it is less likely to be repeated. He also believed that this learned association could end, or become extinct if the reinforcement or punishment was removed.
To prove this, he placed rats in a box with a lever that when tapped would release a pellet of food. Over time, the amount of time it took for the rat to find the lever and press it became shorter and shorter until finally, the rat would spend most of its time near the lever eating. This behavior became less consistent when the relationship between the lever and the food was compromised. This basic theory of operant conditioning is still used by psychologists, scientists, and educators today.
Shaping, Reinforcement Principles, and Schedules of Reinforcement
Operant conditioning can be viewed as a process of action and consequence. Skinner used this basic principle to study the possible scope and scale of the influence of operant conditioning on animal behavior. His experiments used shaping, reinforcement, and reinforcement schedules in order to prove the importance of the relationship that animals form between behaviors and results.
All of these practices concern the setup of an experiment. Shaping is the conditioning paradigm of an experiment. The form of the experiment in successive trials is gradually changed to elicit a desired target behavior. This is accomplished through reinforcement, or reward, of the segments of the target behavior, and can be tested using a large variety of actions and rewards. The experiments were taken a step further to include different schedules of reinforcement that become more complicated as the trials continued. By testing different reinforcement schedules, Skinner learned valuable information about the best ways to encourage a specific behavior, or the most effective ways to create a long-lasting behavior. Much of this research has been replicated on humans, and now informs practices in various environments of human behavior. 17
Positive and Negative Reinforcement
Sometimes, adding something to the situation is reinforcing as in the cases we described above with cookies, praise and money. Positive reinforcement involves adding something to the situation in order to encourage a behavior. Other times, taking something away from a situation can be reinforcing. For example, the loud, annoying buzzer on your alarm clock encourages you to get up so that you can turn it off and get rid of the noise. Children whine in order to get their parents to do something and often, parents give in just to stop the whining. In these instances, negative reinforcement has been used.
Operant conditioning tends to work best if you focus on trying to encourage a behavior or move a person into the direction you want them to go rather than telling them what not to do. Reinforcers are used to encourage a behavior; punishers are used to stop behavior. A punisher is anything that follows an act and decreases the chance it will reoccur. But often a punished behavior doesn’t really go away. It is just suppressed and may reoccur whenever the threat of punishment is removed. For example, a child may not cuss around you because you’ve washed his mouth out with soap, but he may cuss around his friends. Or a motorist may only slow down when the trooper is on the side of the freeway. Another problem with punishment is that when a person focuses on punishment, they may find it hard to see what the other does right or well. And punishment is stigmatizing; when punished, some start to see themselves as bad and give up trying to change.
Reinforcement can occur in a predictable way, such as after every desired action is performed, or intermittently, after the behavior is performed a number of times or the first time it is performed after a certain amount of time. The schedule of reinforcement has an impact on how long a behavior continues after reinforcement is discontinued. So a parent who has rewarded a child’s actions each time may find that the child gives up very quickly if a reward is not immediately forthcoming. Think about the kinds of behaviors that may be learned through classical and operant conditioning. But sometimes very complex behaviors are learned quickly and without direct reinforcement. Bandura’s Social Learning covered later in the chapter explains how. 19
Watson and Behaviorism
Another theorist who added to the spectrum of the behavioral movement was John B. Watson. Watson believed that most of our fears and other emotional responses are classically conditioned. He had gained a good deal of popularity in the 1920s with his expert advice on parenting offered to the public. He believed that parents could be taught to help shape their children’s behavior and tried to demonstrate the power of classical conditioning with his famous experiment with an 18 month old boy named “Little Albert”. Watson sat Albert down and introduced a variety of seemingly scary objects to him: a burning piece of newspaper, a white rat, etc. But Albert remained curious and reached for all of these things. Watson knew that one of our only inborn fears is the fear of loud noises so he proceeded to make a loud noise each time he introduced one of Albert’s favorites, a white rat. After hearing the loud noise several times paired with the rat, Albert soon came to fear the rat and began to cry when it was introduced.
Watson filmed this experiment for posterity and used it to demonstrate that he could help parents achieve any outcomes they desired, if they would only follow his advice. Watson wrote columns in newspapers and in magazines and gained a lot of popularity among parents eager to apply science to household order. Parenting advice was not the legacy Watson left us, however. Where he really made his impact was in advertising. After Watson left academia, he went into the world of business and showed companies how to tie something that brings about a natural positive feeling to their products to enhance sales. Thus the union of sex and advertising! 20 Sometimes we do things because we’ve seen it pay off for someone else. They were operantly conditioned, but we engage in the behavior because we hope it will pay off for us as well. This is referred to as vicarious reinforcement (Bandura, Ross and Ross, 1963).
Do parents socialize children or do children socialize parents?
Bandura (1986) suggests that there is interplay between the environment and the individual. We are not just the product of our surroundings, rather we influence our surroundings. There is interplay between our personality and the way we interpret events and how they influence us. This concept is called reciprocal determinism. An example of this might be the interplay between parents and children. Parents not only influence their child’s environment, perhaps intentionally through the use of reinforcement, etc., but children influence parents as well. Parents may respond differently with their first child than with their fourth. Perhaps they try to be the perfect parents with their firstborn, but by the time their last child comes along they have very different expectations both of themselves and their child. Our environment creates us and we create our environment.
Social Learning Theory
Albert Bandura is a leading contributor to social learning theory . He calls our attention to the ways in which many of our actions are not learned through conditioning; rather, they are learned by watching others (1977). Young children frequently learn behaviors through imitation. Sometimes, particularly when we do not know what else to do, we learn by modeling or copying the behavior of others. A new employee, on his or her first day of a new job might eagerly look at how others are acting and try to act the same way to fit in more quickly. Adolescents struggling with their identity rely heavily on their peers to act as role-models. Newly married couples often rely on roles they may have learned from their parents and begin to act in ways they did not while dating and then wonder why their relationship has changed.
Contributors and Attributions
15. Lifespan Development - Module 4: Infancy by Lumen Learning references Psyc 200 Lifespan Psychology by Laura Overstreet, licensed under CC BY 4.0
16. Children’s Development by Ana R. Leon is licensed under CC BY 4.0
17. Children’s Development by Ana R. Leon is licensed under CC BY 4.0
19. Lifespan Development - Module 4: Infancy by Lumen Learning references Psyc 200 Lifespan Psychology by Laura Overstreet, licensed under CC BY 4.0
20. Lifespan Development - Module 4: Infancy by Lumen Learning references Psyc 200 Lifespan Psychology by Laura Overstreet, licensed under CC BY 4.0