```python
from sympy import init_session
init_session()
```

    IPython console for SymPy 1.5.1 (Python 3.6.10-64-bit) (ground types: gmpy)
    
    These commands were executed:
    >>> from __future__ import division
    >>> from sympy import *
    >>> x, y, z, t = symbols('x y z t')
    >>> k, m, n = symbols('k m n', integer=True)
    >>> f, g, h = symbols('f g h', cls=Function)
    >>> init_printing()
    
    Documentation can be found at https://docs.sympy.org/1.5.1/
    



```python
#init_printing?
```

### Parameters and two Gaussians


```python
a, b, c, a1, a2 = symbols('a b c a1 a2', positive=True, real=True)
```


```python
g1=x*exp(-a1*x**2)
g2=x*exp(-a2*x**2)
g1, g2
```

### Normalization constant


```python
N=integrate(g1*g1, (x, -oo, oo))
N
```


```python
1/sqrt(N)
```


```python
printing.sstrrepr(1/sqrt(N))
```




    '2*2**(1/4)*a1**(3/4)/pi**(1/4)'



### Overlap integral S


```python
S=integrate(g1*g2, (x, -oo, oo))
S
```


```python
S.simplify()
```


```python
printing.sstrrepr(S.simplify())
```




    'sqrt(pi)/(2*(a1 + a2)**(3/2))'



### Kinetic energy $T = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} = \frac{1}{2m}\left(\frac{\hbar}{i}\frac{d}{dx} \right)^2$


```python
d1=diff(g1,x)
d2=diff(g2,x)
d1, d2
```


```python
T = 1/2 * integrate(d1*d2, (x, -oo, oo))
#T=T.simplify()
#T=T.factor()
T.factor()
```


```python
printing.sstrrepr(T.factor())
```




    '1.5*sqrt(pi)*a1*a2/(a1 + a2)**(5/2)'



### Potential $V(x) = (ax^2 - b)e^{-cx^2}$


```python
v=(a*x**2-b)*exp(-c*x**2)
v
```


```python
V = integrate(g1*v*g2, (x, -oo, oo))
V
```


```python
V.factor()
```


```python
printing.sstrrepr(V.factor())
```




    'sqrt(pi)*(3*a - 2*a1*b - 2*a2*b - 2*b*c)/(4*(a1 + a2 + c)**(5/2))'




```python

```

3e650140d17fc8dd6b288156c730455860125bd0
ipynb
Jupyter Notebook
notebooks/GTO_integrals/GTO_1D_P.ipynb
tsommerfeld/L2-methods_for_resonances
acba48bfede415afd99c89ff2859346e1eb4f96c
MIT
notebooks/GTO_integrals/GTO_1D_P.ipynb
tsommerfeld/L2-methods_for_resonances
acba48bfede415afd99c89ff2859346e1eb4f96c
MIT
notebooks/GTO_integrals/GTO_1D_P.ipynb
tsommerfeld/L2-methods_for_resonances
acba48bfede415afd99c89ff2859346e1eb4f96c
MIT
Qwen/Qwen-72B
1. YES
2. YES