Friday, July 17, 2009

(|wood> + |trees>)/sqrt(2)

I think that SM358, the Open University's quantum mechanics course, is solid but perhaps slightly conservative (perhaps cut-down and lean for distance-learning is more accurate).

One of the challenges in teaching and learning QM is that the central organising concepts of the theory can't be comprehended until quite a bit of the machinery has been taken on board. So there's a lot which has to be taken on faith and is therefore mysterious to the student for a while - the wood and the trees problem.

Later in the course, it helps to try to set what has been learned into some kind of structured context, and here I think SM358 falters a bit. Here are some of the things which have mystified me, and my own views on their resolution.

Q. What's so special about the concepts of eigenfunction and eigenvalue?

A. These are foundational concepts in QM but the reason why is initially not very clear. The real explanation is that in QM, unlike classical mechanics, the problem-solving act is in two steps: (i) find the correct wave function or state vector; (ii) apply the boundary conditions of the specific problem to find the probabilities of the possible observable values to be measured. In classical mechanics one simply solves the 'well-known' equations in the presence of the boundary conditions.

Finding the correct wave function often comes down to solving Schrodinger's time-independent equation,

Hψ = Eψ where E is a constant (eigenvalue), for unknown functions ψ.

Solutions to this equation are indeed eigenfunctions - due to the form of the equation - and that's where the utility of the concept comes from.

Q. What is the significance of a quantum mechanical operator?

A. I was puzzled by this for a long time. Were operators in QM something to do with the act of observation (it is said that operators 'represent' observables)? Perhaps an operator corresponds to a piece of apparatus?

No, none of this is true. The operator appears at the earlier step, where the correct wavefunction for the problem has to be determined. The operator is a constituent of the ψ-equation which determines the correct wavefunction (or state-vector or wavepacket) for the problem under consideration (free particle, harmonic oscillator, Coulomb model of the hydrogen atom, ...).

The second stage, working out the probability of different observables being measured, is a calculation of amplitudes using the wavefunction/state-vector already found - it's this stage which is relevant to the apparatus configuration and the measurement process.