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4.5: Motion in Central Field

  • Page ID
    1201
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    Consider a particle of mass \(M\) moving in a spherically symmetric potential. The Hamiltonian takes the form

    \[H = \frac{{p}^2}{2\,M} + V(r). \label{387}\]

    Adopting Schrödinger's representation, we can write $ {\bf p} = -({\rm i}/\hbar)
\nabla$ . Hence,

    \( H = -\frac{\hbar^2}{2\,M}\, \nabla^2 + V(r).\) \ref{388}

    When written in spherical polar coordinates, the above equation becomes

    $ H= -\frac{\hbar^2}{2\,M}\left[ \frac{1}{r^2}\frac{\partial}{\part...
...+ \frac{1}{r^2\sin^2\theta} \frac{\partial^2}{\partial\varphi^2}\right] + V(r).$ \ref{389}

    Comparing this equation with Equation \ref{374}, we find that

    \( L^2\) (see Section 4.1). We also know, from Equations \ref{369}-\ref{371}, that \( L_y\), and \( [{\bf L}, H] = 0.\) \ref{391}

    It is also easily seen that \( L^2\) (which can be expressed as a purely angular differential operator) commutes with the Hamiltonian:

    \( {\bf L}\) and its magnitude squared \( L^2\) are both constants of the motion. This is as expected for a spherically symmetric potential.

    Consider the energy eigenvalue problem

    \( E\) is a number. Since \( L^2\) and \( L^2\), \( H\). But, we already know that the most general form for the wavefunction of a simultaneous eigenstate of \( L^2\) and \( \psi(r, \theta, \varphi) = R(r) \,Y_{l\,m}(\theta, \varphi).\) \ref{394}

    Substituting Equation \ref{394} into Equation \ref{390}, and making use of Equation \ref{382}, we obtain

    \( R(r)\). We know, from the general properties of this type of equation, that if \( r=0\) and as $ r\rightarrow
\infty$ then solutions only exist for a discrete set of values of \( E\). These are the energy eigenvalues. In general, the energy eigenvalues depend on the quantum number \( m\).

    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 4.5: Motion in Central Field is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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