Questions for 1-D harmonic oscillators

[ray.deng83]

In my textbook, it says "For a system of one-dimensional oscillators, the energy levels are equally spaced and non-degenerate, so the number of quantum states in an interval dE is proportional to dE so long as dE is much larger than the spacing h(h-bar)w between levels. In fact, we may conclude from this that g(E)dE must have the value dE/h(h-bar)w."

1. Why are the energy levels are equally spaced? According to the Bohr Model of hydrogen, as the energy level is getting higher, the distance between two levels are getting closer.

2. Why should it be non-degenerate? What's the difference between degenerate and non-degenerate energy levels?

3. Why is h(h-bar)w the spacing and as long as dE is much larger than it, the number of quantum states in an interval dE is proportional to dE? Also, dE is just the differential of energy, it should have no size and thus can't be measured to be compared with the spacing.

Can someone help to explain a bit on these?

 

[blkqi]

The harmonic oscillator models vibrational motion in a quantum mechanical system, where energy appears quantized and the wave nature of matter cannot be neglected. It is not a model of atomic structure like the Bohr model.

Degenerate states are different quantum states which lead to the same calculated value of total energy. The state of a harmonic oscillator is defined only by the principle number n, so there is no degeneracy.

Total energy is
$$E=(n+\frac{1}{2})\hbar \omega$$
so the energy levels are linear. Any interval dE larger than $$\hbar \omega$$ must contain at least one allowed state.

 

[Bob_for_short]

Here dE means not an infinitesimal differential but a finite difference ∆E. If there are many energy levels within this difference, then the textbook is right.

 

[jtbell]

1. Why are the energy levels are equally spaced? According to the Bohr Model of hydrogen, as the energy level is getting higher, the distance between two levels are getting closer.

The hydrogen atom and the harmonic oscillator have different potential energy functions, so they have different energy levels. In both cases you find the energy levels by solving the Schrödinger equation to find the states with definite energy. For the harmonic oscillator, the energies turn out to be equally spaced.