Schrödinger Equation

Time-Dependent

$$ i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle $$

Time-Independent

For stationary states:

$$ \hat{H}|\psi\rangle = E|\psi\rangle $$

1D Position Space

$$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi $$

Python (Numerical Solution)

1import numpy as np
2from scipy.integrate import odeint
3
4def schrodinger_1d(psi, x, E, V, m=1, hbar=1):
5    """Solve 1D time-independent Schrödinger equation"""
6    psi_real, psi_imag = psi
7    dpsi_real = psi_imag
8    dpsi_imag = (2*m/hbar**2) * (V - E) * psi_real
9    return [dpsi_real, dpsi_imag]

Further Reading

• Schrödinger Equation - Wikipedia

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