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As its name suggests, Quantum Mechanics is underpinned by the idea of quantisation: physical systems on the quantum scale can be split into discrete states one of which will always be observed when a measurement is made. Part IB Quantum Mechanics includes a section on the harmonic oscillator, possibly the simplest analytic nonzero potential, and then proves that states are quantised by the fairly unrevealing method of ‘spotting’ a Gaussian groundstate and then deriving series solutions and only taking those which won’t misbehave at infinity, giving a discrete spectrum of energy values.

In part II, a much more novel and surprising (at least from my current perspective) method is taken – dream up some operators that allow you to hop between energy levels by a constant amount, and then place bounds on where we can travel, giving a fixed lattice of hop locations which yield the discrete energy spectrum. This seems curious, and I’m wondering if there is something more fundamental going on or if this is just a trick. Maybe writing out this post will help (or responses from more physically inclined friends).

The (undimensionalised) harmonic oscillator is that with Hamiltonian

$H = \frac{p^2}{\hbar^2} + x^2 = (x-i\frac{p}{\hbar})(x+i\frac{p}{\hbar}) + [i\frac{p}{\hbar},x] = a^\dagger a + 1$

Where we have let $a=x+i\frac{p}{\hbar}$.

Our first nontrivial observation is that the form of the Hamiltonian leads to the relation

$\langle \psi |H-1|\psi \rangle = |a |\psi \rangle|^2 \geq 0.$

I.e. every state has energy at least 1, and $a|\psi\rangle$ is the zero state if and only if $\langle \psi |H|\psi \rangle =1$.

Our second nontrivial observation is that $a$ does not commute with the Hamiltonian. Indeed, let’s bash out some commutation relations (deriving everything from $[x,p] = i\hbar$) and see what we get.

$[a^\dagger, a] = [x-i\frac{p}{\hbar},x+i\frac{p}{\hbar}] = [x,i\frac{p}{\hbar}]-[i\frac{p}{\hbar},x]=-2$

Hence $[H,a] = [a^\dagger a, a] = [a^\dagger, a]a = -2a$.

What does this mean for our energy eigenstates (if it’s got energy $E$ we’ll write it $|E\rangle$? If we measure the energy of a state after the $a$ operator has been applied we get

$Ha |E\rangle = (aH + [H,a]) |E\rangle = (E-2)a|E\rangle$

In other words, provided $E\not=1$ (where both sides of the equation vanish by what we already proved about the modulus of $a|\psi\rangle$) applying the operator $a$ reduces the energy of any eigenstate by 2.

Hence the only possible energy eigenvalues are $E=1,3,5,.....$. Indeed, if there were any more, we could apply $a$ some finite number of times to obtain an eigenstate with energy value less than $1$, which we proved was impossible, so the energy spectrum of the harmonic oscillator is indeed discrete.

I still find that this method exists and works wholly mysterious and wonderful (given quite how much neater it is than just bashing out the differential equations directly). If anyone can give me some deeper insight into what is going on I’d be appreciative.