Required math: calculus
Required physics: 3-d Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 4.38.
The 3-d harmonic oscillator can be solved in rectangular coordinates by separation of variables. The Schrödinger equation to be solved for the 3-d harmonic oscillator is
To use separation of variables we define
Dividing 1 through by this product we get
where the double prime notation indicates the second derivative of a function with respect to its independent variable, so , etc.
We now have three groups of two terms each of which depends on only one of the variables and , and the sum of all these terms is the constant . We can therefore use the usual argument that each group of two terms must be a constant on its own, so the 3-d equation reduces to the sum of three 1-d harmonic oscillators. From the analysis of the 1-d harmonic oscillator, we know that each of these will contribute to the total energy, with the ground state at . Thus the ground state for the 3-d oscillator will have energy , and the general energy level will increase in steps of so the energy levels are given by
Unlike the 1-d case, the energies of the 3-d oscillator are degenerate. A given value of is composed of the sum of 3 quantum numbers: where all numbers are non-negative integers. Suppose we choose a value for so that . The number of pairs of integers that can be used for is (since can be anything between 0 and ). Since itself can range between 0 and , the total number of combinations of quantum states that can make up state is