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

Where we have let .

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

I.e. every state has energy at least 1, and is the zero state if and only if .

Our second nontrivial observation is that does not commute with the Hamiltonian. Indeed, let’s bash out some commutation relations (deriving everything from ) and see what we get.

Hence .

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

In other words, provided (where both sides of the equation vanish by what we already proved about the modulus of ) applying the operator reduces the energy of any eigenstate by 2.

Hence the only possible energy eigenvalues are . Indeed, if there were any more, we could apply some finite number of times to obtain an eigenstate with energy value less than , 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.

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July 17, 2010 at 9:30 pm

YoniHey man,

Just came across this and thought I’d offer my two cents if you’re still curious. This “coincidence” that you can solve a horrid differential equation by operator methods is essentially the foundation for the entire field of Lie algebras. You’ve probably seen the quantization of angular momentum by this point, which is identical to the harmonic oscillator example except for the fact that you only have a finite number of states per representation, as opposed to an infinite number. In mathematical terms, you have an algorithm for constructing all finite-dimensional representations of the Lie algebra $su(2)$ based on repeated application of the lowering operator $L_-$ and the existence of a highest weight state killed by $L__+$. Now, it just so happens that all semisimple Lie algebras contain $su(2)$ subalgebras…in fact, many of them…and so you can construct a representation theory of ALL finite-dimensional semisimple Lie algebras based on the properties of $su(2)$. Neat.

So I think you should think of the harmonic oscillator “coincidence” in the same way. But the extension comes from the fact that the harmonic oscillator Hamiltonian is the prototype for ANY physical system whatsoever, near its equilibrium point – the dependence on $p^2$ holds for any non-relativistic particle, and $x^2$ is the first nonzero term in a Taylor expansion of an arbitrary potential $V(x)$ about an equilibrium. So, if all you care about is first-order approximations, you can take advantage of this wonderful coincidence to describe a whole bunch of physical systems. But if you want a more accurate result, you start with the harmonic oscillator, work in the basis defined by the creation and annihilation operators, add higher-order terms in the Hamiltonian, describe a systematic expansion in powers of these terms of the energy basis…and you’ve got yourself quantum field theory!

May 19, 2011 at 9:48 am

ChristianHello,

I would like to note that this operators are also called “Ladder operators” or “Operators of creation and annihilation”, respectively. Physicists (and mathematicians) have an exact formula for each of them and a matrix representation for calculations. For the reason that, these operators obey the rules which described above, lead someone to approach the Quantum Harmonic Oscillator in a different way (the other way is via Hermite polynomials and as follows via differential equations).

P.s. The first, who used these operators to solve the Quantum Harmonic Oscillator, was P.A.M. Dirac…