3.10: Stationary States

An eigenstate of the energy operator \(H\equiv {\rm i}\,\hbar\,\partial/\partial t\) corresponding to the eigenvalue \(E_i\) satisfies

\[{\rm i}\,\hbar\,\frac{\partial \psi_E(x,t,E_i)}{\partial t} = E_i\,\psi_E(x,t,E_i).\] It is evident that this equation can be solved by writing \[\psi_E(x,t,E_i) = \psi_i(x)\,{\rm e}^{-{\rm i}\,E_i\,t/\hbar},\] where \(\psi_i(x)\) is a properly normalized stationary (i.e., non-time-varying) wavefunction. The wavefunction \(\psi_E(x,t,E_i)\) corresponds to a so-called stationary state, because the probability density \(|\psi_E|^{\,2}\) is non-time-varying. Note that a stationary state is associated with a unique value for the energy. Substitution of the previous expression into Schrödinger’s equation ([e3.1]) yields the equation satisfied by the stationary wavefunction:

\[\label{etimeii} \frac{\hbar^{\,2}}{2\,m}\,\frac{d^{\,2} \psi_i}{d x^{\,2}} = \left[V(x)-E_i\right]\psi_i.\] This is known as the time-independent Schrödinger equation. More generally, this equation takes the form \[\label{etimei} H\,\psi_i = E_i\,\psi_i,\] where \(H\) is assumed not to be an explicit function of \(t\). Of course, the \(\psi_i\) satisfy the usual orthonormality condition:

\[\label{e4.157} \int_{-\infty}^\infty \psi_i^\ast\,\psi_j\,dx = \delta_{ij}.\]

Moreover, we can express a general wavefunction as a linear combination of energy eigenstates:

\[\label{e4.158} \psi(x,t) = \sum_i c_i\,\psi_i(x)\,{\rm e}^{-{\rm i}\,E_i\,t/\hbar},\] where \[c_i = \int_{-\infty}^{\infty} \psi_i^\ast(x)\,\psi(x,0)\,dx.\] Here, \(|c_i|^{\,2}\) is the probability that a measurement of the energy will yield the eigenvalue \(E_i\). Furthermore, immediately after such a measurement, the system is left in the corresponding energy eigenstate. The generalization of the previous results to the case where \(H\) has continuous eigenvalues is straightforward.

If a dynamical variable is represented by some Hermitian operator \(A\) that commutes with \(H\) (so that it has simultaneous eigenstates with \(H\)), and contains no specific time dependence, then it is evident from Equations ([e4.157]) and ([e4.158]) that the expectation value and variance of \(A\) are time independent. In this sense, the dynamical variable in question is a constant of the motion.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

  \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)