It is obvious that our solution in Cartesian coordinates is simply,

(3.1) |

Less simple, but more edifying is the case in spherical coordinates. With the conversions,

(3.2) |

we have,

(3.3) |

We invert the matrix,

Thus,

(3.4) |

Of course,

(3.5) |

Here we begin an aside on the angular momentum operators. This will be useful because it will bring us half the solution. We have,

(3.6) |

We convert this,

We let

(3.7) |

(3.8) |

Properly, none of the terms involves . Throwing in the proper we find,

(3.9) |

With some algebra we can show that

(3.10) |

We know that . We also have that the Laplacian in spherical coordinates is,

Thus, in three dimensions and spherical coordinates, the Schrödinger equation is,

(3.11) |

By separation of variables, the radial term and the angular term can be divorced. The solution to the angular equation are hydrogeometrics. The reader is referred to the supplement on the basic hydrogen atom for a detailed and self-contained derivation of these solutions. Our resulting radial equation is, with the Harmonic potential specified,

(3.12) |

We can quickly solve this equation by applying the SAP method (Simplify, Asymptote, Power Series). We set the stage by first rewriting the above equation in a form which will later simplify our process,

(3.13) |

Write , then our first asymptote is towards infinity where the terms dominate; our second is towards zero where dominates and we find that

(3.14) |

The next term is,

(3.15) |

To solve such equations, we suppose there exist a function which will bring this equation into a more conventionally solvable form. We will determine what z is by the consequences of this demand. With

(3.16) |

This equation is solvable if the coefficients are proportional, or even better, equal:

(3.17) |

This equation has solutions easily found as,

(3.18) |

But we can solve for z and find that , whereby,

(3.19) |

Our first derivative is,

(3.20) |

Our second derivative is,

Recalling that,

(3.21) |

our net equation thus requires that

(3.22) |

We seek to match the coefficients of r, since they must vanish independently, whereby,

(3.23) |

This gives us the recursion relation,

(3.24) |

Requiring this series to terminate to prevent non-physical behavior is our quantization condition, whereby we must have,

(3.25) |

This recursion relationship and eigenvalue formula thus define a three dimensional harmonic oscillator.