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Physics LibreTexts

4.5: Motion in Central Field

( \newcommand{\kernel}{\mathrm{null}\,}\)

Consider a particle of mass M moving in a spherically symmetric potential. The Hamiltonian takes the form

H=p22M+V(r).

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

H=22M2+V(r). ???

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).$ ???

Comparing this equation with Equation ???, we find that

L2 (see Section 4.1). We also know, from Equations ???-???, that Ly, and [L,H]=0. ???

It is also easily seen that L2 (which can be expressed as a purely angular differential operator) commutes with the Hamiltonian:

L and its magnitude squared L2 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 L2 and L2, H. But, we already know that the most general form for the wavefunction of a simultaneous eigenstate of L2 and ψ(r,θ,φ)=R(r)Ylm(θ,φ). ???

Substituting Equation ??? into Equation ???, and making use of Equation ???, 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)


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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