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10.1 correspondence between time-dependent and time-independent solutions

The time dependent Schrödinger equation is

- $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{\partial^2}{\partial x^2}}$$\displaystyle \psi$(x, t) + V(x)$\displaystyle \psi$(x, t) = $\displaystyle {\frac{\hbar i\partial}{\partial t}}$$\displaystyle \psi$(x, t). (10.1)

As we remember, a solution of the form

$\displaystyle \psi$(x, t) = $\displaystyle \phi$(x)e-iEt/$\scriptstyle \hbar$ (10.2)

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

- $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{\partial^2}{\partial x^2}}$$\displaystyle \phi$(x) + V(x)$\displaystyle \phi$(x) = E$\displaystyle \phi$(x). (10.3)



© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk