Quantum oscillators -- discrete states


Harmonic oscillation as exemplified by the mass on a spring serves as a prototype for describing all kinds of oscillations -- any time we have a stable situation with a restoring force (one that pushes back towards the stable point) with overshoot. (This doesn't have to look anything like a mass and a spring. One example are oscillators in electric circuits.) Things change a bit for sub-molecular systems when Newtonian mechanics has to be replaced by quantum mechanics. How this happens is a long story that we won't tell here (though see The Wave Model of Matter for a brief discussion).

But when we do describe oscillations in a quantum system, something interesting happens. The potential energy description is very similar -- the same language and terms are used to describe the interaction energies of atoms, nuclei, and electrons as for classical systems. But for systems whose potential energies are strong enough (negative enough) to keep them together (bound), something unexpected happens.

In a classical bound system, any total energy is allowed. In an analogous quantum bound system, only specific energies are allowed. For the toy model of a simple harmonic oscillator, the classical PE is shown in the figure below at the left.

 As with all quantum systems, the lowest energy state is not at 0. There is always some minimal quantum vibration. For the harmonic oscillator, the excited states (the higher levels) are equally spaced. In fact the spacing between the levels is given by the quantum constant (Planck's constant, $h = 6.63 x 10^{-24} \mathrm{J⋅s}$) times the classical frequency of the oscillator:

$$\Delta E = hf = h\bigg(\frac{\omega_0}{2\pi} \bigg) =\bigg(\frac{h}{2\pi} \bigg) \omega_0 = \hbar \omega_0 $$

where $ω_0$ is our old friend, the square root of $k/m$. (For working with omega instead of $f$, a new quantum constant, "h bar = $\hbar$", is defined to be $h$ divided by $2π$.)

The ground state — the lowest allowed level — is 1/2 this energy above the bottom of the well.  As a result, the energies of all of the harmonic oscillator states (measured from the bottom of the well) can be written

$$E_n = (n + ½) \hbar \omega_0 $$

where $n$ is any positive integer. With this convention, the ground state is the state $E_0$, with $n = 0$.

When we have a more general potential energy well — like the Lennard-Jones potential between two atoms —  we will still have discrete levels as shown in the figure at the right, but their spacing can be a bit different from the spacing of a pure oscillator. But usually near the bottom of the well, it can be reasonably well approximated by a parabola and the corresponding equally space energy levels.

Of course the parabola is a fairly unrealistic approximation. Any spring will bend and break if you pull it hard enough. This is better represented by the LJ potential that shows that you can get out of the well (separate to large distances and break the spring) if you put in enough energy.

One interesting fact about quantum levels is that they tell you information about the structure of the system. If a quantum system has energy levels spaced like a harmonic oscillator, you know you can think about its excitations as a vibration in some degree of freedom. This is particularly important since you can directly obtain information about the excited states of molecules and atoms by looking at the frequency of the light that they absorb since this frequency determines the energy of the light (see The photon model of light).

Joe Redish 4/16/13

Article 673
Last Modified: May 22, 2019