## Problem 9.3


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


```python
v, nu, m, h_bar, omega, h, alpha, c, d = symbols('v nu m h_bar omega h alpha c d', real=True, positive=True)
x = symbols('x', real=True)
```

Define the unperturbed first excited state wavefunction from Eqn 4.55


```python
psi_1_0 = (4*alpha**3/pi)**0.25*x*exp(-alpha*x**2/2)
psi_1_0
```




$\displaystyle \frac{1.4142135623731 \alpha^{0.75} x e^{- \frac{\alpha x^{2}}{2}}}{\pi^{0.25}}$



As a sanity check, make sure \\(\psi_{1}^{(0)}\\) is normalized


```python
integrate(psi_1_0**2, (x, -oo, oo))
```




$\displaystyle 1.0$



Calculate the first order perturbation correction.


```python
E_1_1 = integrate(psi_1_0*(c*x**3 + d*x**4)*psi_1_0, (x, -oo, oo))
E_1_1
```




$\displaystyle \frac{3.75 d}{\alpha^{2.0}}$




```python
E_1_1.subs({alpha: 4*pi**2*nu*m/h})
```




$\displaystyle \frac{0.234375 d h^{2.0}}{\pi^{4.0} m^{2.0} \nu^{2.0}}$



This matches the answer in the book!


```python

```

33c317df6c39c86ed2ff04473650ee46b90d0c6e
ipynb
Jupyter Notebook
notebooks/problem_9_3.ipynb
akey7/basic-quantum-models
b2f7edd2fdd15b6f85ee3708f8c2dae24686fd07
BSD-2-Clause
notebooks/problem_9_3.ipynb
akey7/basic-quantum-models
b2f7edd2fdd15b6f85ee3708f8c2dae24686fd07
BSD-2-Clause
2021-07-12T15:49:00.000Z
2021-07-12T15:54:55.000Z
notebooks/problem_9_3.ipynb
akey7/basic-quantum-models
b2f7edd2fdd15b6f85ee3708f8c2dae24686fd07
BSD-2-Clause
Qwen/Qwen-72B
1. YES
2. YES