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4.2: Schrödinger's Equation

  • Page ID
    1159
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    Consider a dynamical system consisting of a single non-relativistic particle of mass \(\begin{equation}m\end{equation}\) moving along the x-axis in some real potential \(\begin{equation}V(x)\end{equation}\). In quantum mechanics, the instantaneous state of the system is represented by a complex wave function \(\begin{equation}\psi(x, t)\end{equation}\). This wavefunction evolves in time according to Schrödinger's equation: 

    \begin{equation}\mathrm{i} \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \psi}{\partial x^{2}}+V(x) \psi\end{equation}

    The wavefunction is interpreted as follows: \(\begin{equation}|\psi(x, t)|^{2}\end{equation}\)  is the probability density of a measurement of the particle's displacement yielding the value x. Thus, the probability of a measurement of the displacement giving a result between \(\begin{equation}a \text { and } b \text { (where } a<b \text { ) is }\end{equation}\) 

    \begin{equation}P_{x \in a: b}(t)=\int_{a}^{b}|\psi(x, t)|^{2} d x\end{equation}

    Note that this quantity is real and positive definite.

    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 4.2: Schrödinger's Equation is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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