We begin our study of Quantum Mechanics in 3 dimensions with a few simple cases of
problems that can be separated in Cartesian coordinates.
This is possible when the Hamiltonian can be written
One nice example of separation of variable in Cartesian coordinates is the
3D harmonic oscillator
Another problem that separates is the particle in a 3D box.
Again, energies depend on three quantum numbers
In classical physics, it takes three coordinates to give the location of a particle in 3D. In quantum mechanics, we are finding that it takes three quantum numbers to label and energy eigenstate (not including spin).
Jim Branson 2013-04-22