Schrodinger Equation

Time-Dependent

The time-dependent Schrodinger equation is

\[i\hbar \frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}\]

Where the form of $\Psi$ is

\[\Psi(x,t) = Ae^{i(kx-\omega t)}\]

which covers all our bases for some oscillatory motion.

General Time-Dependent

The general time-dependent Schrodinger equation is

\[-\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + U(x)\Psi = i\hbar \frac{\partial\Psi}{\partial t}\]

Which is equivalent to saying

\[K + U = E\]

The form of $\Psi$ is

\[\Psi(x,t) = \psi(x)e^{-i\omega t}\]

Time-Independent (TISE)

Stationary states are where it is time-independent. We can combine stationary states to describe any general state.

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