I solved the Schrödinger equation for a given potential and then found the energy equation to substitute into the thermodynamic equations. However, I struggled to write a program code using Python with the help of GPT. I came up with this, but it doesn't work or solve the integral.

import sympy as sp

#variables 
beta, Q1, Q2, Q3, delta, lambda_var, rho, k_B = sp.symbols('beta Q1 \
Q2 Q3 lambda_var delta rho k_B', real=True, positive=True)

lambda_var = -delta + sp.sqrt(Q3) 

# partition function of energy
Z_beta_integral = sp.integrate(sp.exp(beta * (Q2 * rho**2 + Q2 * Q3 / rho**2)), (rho, 
delta, delta + lambda_var))


Z_beta_solution = sp.simplify(Z_beta_integral)


print(f'Z_beta_integral = {Z_beta_integral}')
print(f'Z_beta_solution = {Z_beta_solution}')