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 . Hence,
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).$](http://farside.ph.utexas.edu/teaching/qm/lectures/img927.png)
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
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)