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4.12: Stationary States

  • Page ID
    1169
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    An eigenstate of the energy operator \(\begin{equation}H \equiv \mathrm{i} \hbar \partial / \partial t\end{equation}\) corresponding to the eigenvalue \(\begin{equation}E_{i}\end{equation}\) satisfies 

    \begin{equation}i \hbar \frac{\partial \psi_{E}\left(x, t, E_{i}\right)}{\partial t}=E_{i} \psi_{E}\left(x, t, E_{i}\right)\end{equation}

     

    It is evident that this equation can be solved by writing
    \begin{equation}\psi_{E}\left(x, t, E_{i}\right)=\psi_{i}(x) \mathrm{e}^{-i E_{i} t / \hbar}\end{equation}

    where \(\begin{equation}\psi_{i}(x)\end{equation}\) is a properly normalized stationary (i.e., non-time-varying) wavefunction. The wavefunction \(\begin{equation}\psi_{E}\left(x, t, E_{i}\right)\end{equation}\) corresponds to a so-called stationary state, since the probability density \(\begin{equation}\left|\psi_{E}\right|^{2}\end{equation}\) is non-time-varying. Note that a stationary state is associated with a unique value for the energy. Substitution of the above expression into Schrödinger's equation (137) yields the equation satisfied by the stationary wavefunction:

    \begin{equation}\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi_{i}}{d x^{2}}=\left[V(x)-E_{i}\right] \psi_{i}\end{equation}

     

    This is known as the time-independent Schrödinger equation. More generally, this equation takes the form
    \begin{equation}H \psi_{i}=E_{i} \psi_{i}\end{equation}

    where $H$ is assumed not to be an explicit function of $t$. Of course, the \(\begin{equation}\psi_{i}\end{equation}\) satisfy the usual orthonormality condition:

    \begin{equation}\int_{-\infty}^{\infty} \psi_{i}^{*} \psi_{j} d x=\delta_{i j}\end{equation}

     

    Moreover, we can express a general wavefunction as a linear combination of energy eigenstates:
    \begin{equation}\psi(x, t)=\sum_{i} c_{i} \psi_{i}(x) \mathrm{e}^{-i E_{i} t / \hbar}\end{equation}

    where

    \begin{equation}c_{i}=\int_{-\infty}^{\infty} \psi_{i}^{*}(x) \psi(x, 0) d x\end{equation}

    Here, \(\begin{equation}\left|c_{i}\right|^{2}\end{equation}\) is the probability that a measurement of the energy will yield the eigenvalue \(\begin{equation}E_{i}\end{equation}\). Furthermore, immediately after such a measurement, the system is left in the corresponding energy eigenstate. The generalization of the above results to the case where $H$ has continuous eigenvalues is straightforward.

    If a dynamical variable is represented by some Hermitian operator $A$ which commutes with $H$ (so that it has simultaneous eigenstates with $H$), and contains no specific time dependence, then it is evident from Eqs. (297) and (298) 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

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

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

    This page titled 4.12: Stationary States is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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