10.1: Correspondence between time-dependent and time-independent solutions

The time dependent Schrödinger equation is

\[-\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} \psi(x, t)+V(x) \psi(x, t)=\frac{\hbar i \partial}{\partial t} \psi(x, t) .\]

As we remember, a solution of the form

\[\psi(x, t)=\phi(x) e^{-i E t / \hbar}\]

leads to a solution of the time-independent Schrödinger equation of the form

\[-\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial x^2} \phi(x)+V(x) \phi(x)=E \phi(x)\]
 

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