## Problem 9.14 from Levine

Specifically, simplifying the final integral from the problem into Eqn 9.53.


```python
from sympy import integrate, oo, symbols, exp, pi
```


```python
Z, a0, r1, r2, epsilon0, e = symbols('Z a0 r1 r2 epsilon0 e', real=True, positive=True)
```


```python
lower = integrate(exp(-2*Z*r1/a0)*exp(-2*Z*r2/a0)*1/r2*r1**2*r2**2, (r1, 0, r2))
lower
```




$\displaystyle \frac{a_{0}^{3} r_{2} e^{- \frac{2 Z r_{2}}{a_{0}}}}{4 Z^{3}} + \frac{\left(- 2 Z^{5} a_{0} r_{2}^{3} e^{\frac{4 Z r_{2}}{a_{0}}} - 2 Z^{4} a_{0}^{2} r_{2}^{2} e^{\frac{4 Z r_{2}}{a_{0}}} - Z^{3} a_{0}^{3} r_{2} e^{\frac{4 Z r_{2}}{a_{0}}}\right) e^{- \frac{8 Z r_{2}}{a_{0}}}}{4 Z^{6}}$




```python
upper = integrate(exp(-2*Z*r1/a0)*exp(-2*Z*r2/a0)*1/r1*r1**2*r2**2, (r1, r2, oo))
upper
```




$\displaystyle \frac{a_{0}^{2} r_{2}^{2} \left(\frac{Z r_{2}}{2 a_{0}} + \frac{1}{4}\right) e^{- \frac{4 Z r_{2}}{a_{0}}}}{Z^{2}}$




```python
lower + upper
```




$\displaystyle \frac{a_{0}^{2} r_{2}^{2} \left(\frac{Z r_{2}}{2 a_{0}} + \frac{1}{4}\right) e^{- \frac{4 Z r_{2}}{a_{0}}}}{Z^{2}} + \frac{a_{0}^{3} r_{2} e^{- \frac{2 Z r_{2}}{a_{0}}}}{4 Z^{3}} + \frac{\left(- 2 Z^{5} a_{0} r_{2}^{3} e^{\frac{4 Z r_{2}}{a_{0}}} - 2 Z^{4} a_{0}^{2} r_{2}^{2} e^{\frac{4 Z r_{2}}{a_{0}}} - Z^{3} a_{0}^{3} r_{2} e^{\frac{4 Z r_{2}}{a_{0}}}\right) e^{- \frac{8 Z r_{2}}{a_{0}}}}{4 Z^{6}}$




```python
final = integrate(lower + upper, (r2, 0, oo))
final
```




$\displaystyle \frac{5 a_{0}^{5}}{128 Z^{5}}$




```python
((16*Z**6*e**2) / (4*pi*epsilon0*a0**6)) * final
```




$\displaystyle \frac{5 Z e^{2}}{32 \pi a_{0} \epsilon_{0}}$



The above is the result from Eqn 9.53, just factored a bit differently.


```python

```

c1f12d3ce32c7d10ca151ef0aae0516520abe81e
ipynb
Jupyter Notebook
notebooks/problem_9_14.ipynb
akey7/basic-quantum-models
b2f7edd2fdd15b6f85ee3708f8c2dae24686fd07
BSD-2-Clause
notebooks/problem_9_14.ipynb
akey7/basic-quantum-models
b2f7edd2fdd15b6f85ee3708f8c2dae24686fd07
BSD-2-Clause
2021-07-12T15:49:00.000Z
2021-07-12T15:54:55.000Z
notebooks/problem_9_14.ipynb
akey7/basic-quantum-models
b2f7edd2fdd15b6f85ee3708f8c2dae24686fd07
BSD-2-Clause
Qwen/Qwen-72B
1. YES
2. YES