Question: Explain to your friend. who has just learned about simple one-dimensional standing waves on a string fixed at its ends, why hydrogen's electron has only certain energies, and why, for some of those energies, the electron can still be in different states?

One-dimensional standing waves can be observed when a medium is having its opposite ends fixed, and nodes are located at the endpoints. The simplest example of the one-dimensional standing wave will be the one having only one antinode in the middle. This will be half of the wavelength.

The electron of hydrogen behaves as a wave of the bound state, that is why they have only certain energy or certain allowed standing waves. But when multiple dimensions are introduced in space, it is possible to have different standing waves but still have the same frequency/energy.

Assume a square wave of two dimensions for instance, the energy/frequency of the wave will be the same for both the following conditions,

(i) when the wave contains two waves along –axis and one wave along -axis

(ii) when the wave contains two waves along -axis and one wave along –axis.

Thus, if the energy levels are limited, still waves can have different orientations and may result in a different state but of the same energy.