Conceptual understanding of quantum harmonic oscillators

The way I understand it is that we have the time-independent Schrödinger equation for a particle described by a wave function $$\psi$$ in a potential V(x)

$$-\frac{\hbar}{2m}\frac{d^2}{dx^2} \psi + V(x)\psi(x) = E\psi(x)\tag{1} \, .$$

If I were to approximate the potential $$V(x)$$ to be quadratic in $$x$$ (for example $$V(x) = \frac{1}{2} m \omega x^2$$) we know that this describes a harmonic oscillator.

If we then solve equation (1) for $$\psi$$ and $$E(x)$$ we will find that

$$E_n = \hbar\omega \left(n+\frac{1}{2} \right)$$ and the corresponding $$\psi_n$$.

Now it is mostly the sentences I can't really formulate. Like for example which of these statements is correct:

• Our particle described by $$\psi$$ is a harmonic oscillator.

• Our particle described by $$\psi$$ is modeled by a harmonic oscillator.

• Let us consider a particle described as a harmonic oscillator...

As you might see, I have the basic mathematical framework, but I can't really grasp the words/concepts surrounding it.

• If we know all eigenfunctions and eigenvalues for a 1D Schrodinger equation then from that information we should be able to reproduce the potential energy V(x) and find that it is quadratic in x, so we'd know that our particle is a harmonic oscillator indeed. – Maxim Umansky Jan 12 at 0:38
• This is a deep question and I recommend 1. Hartle, James B. "Quantum mechanics of individual systems." American Journal of Physics 36.8 (1968): 704-712, 2. Peres, Asher. "What is a state vector?." American Journal of Physics 52.7 (1984): 644-650. – ZeroTheHero Jan 12 at 2:16

There is no prior difference about naming the concept. IT depends on how you call things, this is just one more.

• A quantum free particle = a wavefunction that obeys the Schrödinger's equation with no potential energy.

• A particle enclosed in an infinite-well = a wavefunction that obeys the Schrödinger's equation with such infinite-well potential.

• A quantum Harmonic Oscillator = a wavefunction that obeys the Schrödinger's equation with the harmonic potential.

But there are two "problems" here. The first one is general: talking about "particle " at this level is "risky". How do you define "particle"? If you want to be clear, talk in terms of "wavefunctions" or "systems".

And now, here comes the problem. The quantum harmonic oscillator behaves completely differently from the classical one. In fact, the states with defined energies are stationary, so they do not change in time, so it cannot oscillate. Yep, states with a single energy do not change in time. So we've got here an oscillator which does not oscillate!

This means that, if your wavefunction if, for example, the one with $$n=0$$, it has only one possible energy, $$E_0=½\hbar\omega$$. Since it has only one possible energy, that wavefunction stays the same all the time. So it is not oscillating.

That's why saying "it's an harmonic oscillator" can be a little confusing. When we say "a quantum harmonic oscillator", we mean " a particle under the influence of a potential that classically would yield a harmonic oscillator", but we are lazy and we just say "harmonic oscillator". The thing is that quantum particles behave really differently from classical mechanics.

Your sentences are right, because everybody understands that it means "a wavefunction that obeys Schrödinger's equation with a quadratic potential". That quadratic potential would give a classical harmonic oscillator if we used classical mechanics and the scalar equation for energy, but this is quantum mechanics and it works differently. The name has been kept the same.

• Oh that was so helpful thank you! I was thinking about what you've said and came to the conclusion that if it's the potential V(x) (in our case the harmonic oscillator) which allows us to calculate $E_n$, there should be always one unique Hamiltonian to each V(x), no? I am asking because usually in most examples I am given the Hamilitonian is kind of 'god given' so this would be the first time for me to see how a Hamiltonian emerges out of the potential V(x) and the schrödinger equation. – Caito Jan 12 at 10:19
• Kind of, yes. Most problems just give you the potential, because the kinetic term is always there. – FGSUZ Jan 12 at 10:54
• Perfect! Finally things start to make sense. One last question. I assume that in quantum mechanics the Hamiltonian usually 'generates' the unitaries (which are allowed?) for a given system. I am currently working on a system in a harmonic potential, hence a quantum harmonic oscillator. The unitaries we are using there are 2-level rotations between the different eigenstates. My question: Are 2-level rotations THE unitary one can use given a harmonic potential? Or am I missing something here? – Caito Jan 12 at 11:55
• I have some comment to this answer. "particle = a wavefunction" The wave function is not the particle. The wave function describes it. For example an electron is for all we know a point particle, but its wavefunction can have any size. Then, a quantum harmonic oscillator behaves exactly like the classical one, except that it's energy, hence its amplitude, is quantized. And, if its ground state is not oscillating, why does it have energy and a varying, though on average zero momentum? We have to be careful when interpreting quantum mechanics. – my2cts Jan 12 at 12:00
• You're right, that's why I said "talking about particles is 'risky' here". Hm, maybe what I said can lead to some confussion if that aspect. Thank you. – FGSUZ Jan 12 at 12:19

The same problem arises classically. If you have a mass on a spring we have a harmonic oscillator. But do we say the mass is a harmonic oscillator? Perhaps. If so then we would say for your question that the particle is a harmonic oscillator

But the spring is an important part of the system too. If we really want to be picky I would say that that the spring mass system constitutes a harmonic oscillator system. In that case we would say the particle in the given potential can be treated as a harmonic oscillator system.

In that sense anything which has a Hamiltonian that looks like

$$H = ax^2+bp^2$$

can be called a harmonic oscillator.

In any case, all of the sentences you wrote down are reasonable and would be understood by anyone.