Question

Schrodinger's equation for a 3D cube

Answer 1

The Schrödinger equation in 3D is

$$-\frac{\hbar^2}{2m}\bigg(\frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}{\partial y^2}+\frac{\partial^2\psi}{\partial z^2}\bigg)=E\psi(x,y,z).$$

The wavefunction for a cubic box with side L is

$$\psi(x,y,z)=A\text{ sin}\bigg(\frac{n_1\pi x}{L}\bigg)\text{ sin}\bigg(\frac{n_2\pi y}{L}\bigg)\text{ sin}\bigg(\frac{n_3\pi z}{L}\bigg).$$

The normalization constant is

$$A=\bigg(\frac{2}{L}\bigg)^{3/2}.$$