Stationary States

Stationary states are the energy eigenstates of the Hamiltonian operator. These states are called "stationary" because their probability distributions are independent of time.

For a conservative system with a time independent potential, $V({\textbf {r}})$, the Schrödinger equation takes the form:

$i\hbar {\frac {\partial \psi ({\textbf {r}},t)}{\partial t}}=\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\textbf {r}})\right]\psi ({\textbf {r}},t)$

Since the potential and the Hamiltonian do not depend on time, solutions to this equation can be written as

$\psi ({\textbf {r}},t)=e^{-iEt/\hbar }\psi ({\textbf {r}})$.

Obviously, for such state the probability density is

$|\psi ({\textbf {r}},t)|^{2}=|\psi ({\textbf {r}})|^{2}$

which is independent of time. Hence, the name is "stationary state".

The same thing happens in calculating the expectation value of any dynamical variable.

For some operator $\langle Q(x,p,t)\rangle \!$

$\langle Q(x,p)\rangle =\int \psi ^{*}(x)Q\left(x,{\frac {\hbar }{i}}{\frac {d}{dx}}\right)\psi (x)dx$

The Schrödinger equation now becomes

$\left[-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\textbf {r}})\right]\psi ({\textbf {r}},t)=E\psi ({\textbf {r}})$

which is an eigenvalue equation with eigenfunction $\psi ({\textbf {r}})$ and eigenvalue $E\!$. This equation is known as the time-independent Schrödinger equation.

A sample problem: A free particle Schrödinger equation.