Question #111167
Schrodinger's equation for a 3D cube
1
Expert's answer
2020-04-26T18:49:19-0400

The Schrödinger equation in 3D is


22m(2ψx2+2ψy2+2ψz2)=Eψ(x,y,z).-\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


ψ(x,y,z)=A sin(n1πxL) sin(n2πyL) sin(n3πzL).\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=(2L)3/2.A=\bigg(\frac{2}{L}\bigg)^{3/2}.

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