4.5: Motion in Central Field
- Page ID
- 1201
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 . Hence,
When written in spherical polar coordinates, the above equation becomes
Comparing this equation with Equation \ref{374}, we find that
It is also easily seen that \( L^2\) (which can be expressed as a purely angular differential operator) commutes with the Hamiltonian:
Consider the energy eigenvalue problem
Substituting Equation \ref{394} into Equation \ref{390}, and making use of Equation \ref{382}, we obtain
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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