Stationary state

A stationary state is described by a wavefunction that is associated with a probability density and an expectation value that are both independent of time.

For the time-independent Schrodinger equation, $\hat{H}\psi =E\psi$ , its solutions are stationary states. This implies that stationary states of the time-independent $\hat{H}$ can be represented by basis wavefunctions but not linear combinations of basis wavefunctions with non-zero coefficients. For example, the wavefunction $\psi\left ( x,t \right )=\phi\left ( x \right )e^{-iEt/\hbar}$ describes a stationary state because:

$\left | \psi\left ( x,t \right ) \right |^{2}=\psi^{*}\psi=\phi^{*}e^{iEt/\hbar}\phi e^{-iEt/\hbar}=\left | \psi\left ( x \right ) \right |^{2}$

and

$\left \langle H \right \rangle=\int \psi^{*}\hat{H}\psi dx=\int \phi^{*}e^{iEt/\hbar}\hat{H}\phi e^{-iEt/\hbar}dx=E\int \phi^{*}\phi dx=E$

whereas $\psi\left ( x,t \right )=\sum_{n=1}^{2}c_n\phi_n\left ( x \right )e^{-iE_nt/\hbar}=c_1\phi_1\left ( x \right )e^{-iE_1t/\hbar}+c_2\phi_2\left ( x \right )e^{-iE_2t/\hbar}$ does not describe a stationary state because:

$\left | \psi\left ( x,t \right ) \right |^{2}=c_{1}^{*}c_1\phi_{1}^{*}(x)\phi_{1}(x)+c_{2}^{*}c_2\phi_{2}^{*}(x)\phi_{2}(x)+c_{1}^{*}c_2\phi_{1}^{*}(x)\phi_{2}(x)e^{i(E_1-E_2)t/\hbar}+c_{2}^{*}c_1\phi_{2}^{*}(x)\phi_{1}(x)e^{i(E_2-E_1)t/\hbar}$

where the last two terms of the RHS of the last equality are time-dependent.

If $\phi_!$ and $\phi_2$ describe a degenerate state, where $E_1=E_2=E$ , then $\psi\left ( x,t \right )$ describes a stationary state. Since an observable of a stationary state, e.g. $H$ , is independent of time, every measurement of $H$ of systems in such a state results in the same value $E$ .

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