Question

The quantized energy levels in the infinite well get further apart as n increases, but in the harmonic oscillator they are equally spaced.

1. Explain the difference by considering the distance “between the walls” in each case and how it depends on the particles energy
2. A very important bound system, the hydrogen atom, has energy levels that actually get closer together as n increases. How do you think the separation between the potential energy “walls” in this system varies relative to the other two? Explain.

Short answer

As energy increases, the energy levels move closer and hence wavelength becomes smaller.

Step-by-step solution

 Energy of harmonic oscillator

1. The energy spacing is equal to $\left(n+\frac{1}{2}\right)\sout{h}\omega$. The ground state energy is larger than zero in a harmonic oscillator as n increases, the walls become further apart. But the energy of the oscillator is limited to certain values and hence allowed quantized energy levels are equally spaced.
   Higher energy states have higher total energies and hence the classical limits to amplitude of the displacement will be larger for these states. Thus, they have shorter wavelengths.

Explanation

(b) Since energy varies as n increases, the walls move apart and so the energy levels come closer to each other.Thus, the energy gap will be smaller and energy increases faster than the harmonic oscillator