Set The Schrodinger equation is

in Cartesian coordinate, it is,

We can set the wave function to be

we can see, there are three repeated terms, we can set

We decoupled the X, Y, Z. Each equation is a quadratic equation with energy

and

The number of states for each energy level is

The first few numbers of states are 1, 3, 6, 10, 15, 21, 28, … The accumulated numbers of states are 1, 4, 10, 20, 35, 56, 84, … Due to the spin-state, the accumulated numbers of particles are 2, 8, 20, 40, 70, 112, 168, … The few magic numbers are reproduced.

The wave function is the product of the Hermite functions and exponential function

If we simply replace , we can see the ground state consists of s-orbit, the 1st excited state consists of p-orbit, and the 2nd excited state consists of d-orbit.

To see more clearly, we can project the function onto spherical harmonic, which is fixed angular momentum, i.e.

.

where is coefficients and is radial function.

To have a better understanding, the radial function has to be solved. The procedure is very similar to solving Coulomb potential.

separate the radial part and angular part.

Set

Set

Set

as usual, the short range behaviour is , long range behaviour is , as stated in the Cartesian coordinate. Thus, we set

with change of variable , the equation becomes

This is our friend, the Laguerre polynomial! In the Laguerre polynomial, must be non-negative integer. Now we set , than the energy is

In order to have are integer, when

The overall solution without a normalization factor is

The normalization constant can be calculate easily using the integral formula of Laguerre polynomial.

change of variable

The integration is

we can use

Thus,

replace . The total wave function is

Here are some drawing of the square of the wave functions. From the below is , from left to right, are s-orbit, p-orbit, d-orbit, f-orbit.

With the LS coupling, the spatial function does not affected, unless the coupling has spatial dependence. With the LS coupling, the good quantum numbers are