Let us consider the isotropic three-dimensional quantum harmonic oscillator: we have (where the particle mass is now *m*_{0} to prevent confusion later). In cartesian coordinates, this becomes:

where *H _{x}*,

*H*, and

_{y}*H*are each the hamiltonian for a one-dimensional harmonic oscillator, in the x, y, and z directions respectively. Thus, the energy eigenstates will be products of eigenstates of these three; we have energy levels

_{z}. Thus, we have energies , with degeneracy equal to the number of solutions of , which, via combinatorics, is . (Only the ground state

*n*=0 is non-degenerate.)

Now, suppose instead we consider spherical coordinates. As depends on

*r*only (spherical symmetry), we see that under separation of variables, the angular components will be given by the spherical harmonics (see here and here). We will then have . With this in place, our Schrödinger equation becomes, for the radial component:

Defining , we see that this differential equation simplifies to

.

Now, we rescale the radial coordinate by defining dimensionless . Then the above equation becomes

, where is the rescaled

*u*(

*r*). We can expect that far from the origin, we should have something like a Gaussian. If we attempt the substitution (where

*k*is a constant for which we will find later a value convenient for solving the equation), then this becomes, after eliminating a factor of ,

.

We see that this simplifies greatly if , so that , and then

.

Lastly, defining , and , we get

,

which is the associated Laguerre differential equation

with parameters and .

The solutions are associated Laguerre functions , , and to be physically valid, we require that be a non-negative integer. Thus

*l*≤

*n*, and they are both odd or both even integers. Thus, for even

*n*,

*l*can take values 0,2,4,…,

*n*-2,

*n*; and for odd

*n*,

*l*can take values 1,3,5,…,

*n*-2,

*n*. Adding the fact that

*m*is an integer with possible values –

*l*,-

*l*+1,…,

*l*-1,

*l*; we have 2

*l*+1 different possibilities for

*m*. So, for even

*n*, we have

different possible angular momentum states; and for odd n, there are

angular momentum states; the same formula as for even

*n*. Looking back at the start of this post, we note that this formula matches the result for the degeneracies of the energy states we found using cartesian coordinates.

Tags: Angular Momentum, Degeneracy, Friday Physics, Harmonic Oscillator, Laguerre Differential Equation, physics, Quantum Harmonic Oscillator, Quantum Mechanics, Schrödinger Equation, Spherical Harmonics

February 20, 2009 at 2:24 am |

[…] coordinates, with the angular components being the spherical harmonics (see here and here). As in here, when we perform the spherical coordinate separation , we obtain radial equation Making the […]