Friday, May 08, 2009

Of operators and observables in QM

I'm probably about 40% through the OU's first course on quantum mechanics and here summarise my present understanding of observables and operators.

Apparently, all observables such as position, momentum, energy, etc are represented in quantum mechanics by an operator which acts upon the wave function Ψ(r,t) representing any configuration of objects and fields.

A wavefunction itself may be written as the product of a space-dependent stationary state and a time-dependent coefficient (the probability amplitude).

When we apply the observable-operator to a stationary-state wavefunction, we find there are eigenfunction/eigenvalue solutions. So if A is the operator and ψn is a (stationary-state) eigenfunction, then

-- Aψn = αnψn where αn is the eigenvalue (may be different for various values of n).

So what the observable operator has done for us is to get us a set of eigenfunctions (the solutions to the eigenfunction equation above) together with the corresponding set of eigenvalues. None of this is yet connected with an actual act of observation.

The next stage is that we express the wave function corresponding to a particular experimental set-up as a sum of the eigenfunctions of the observable of interest. (One dimensional case below).

-- Ψ(x,t) = c1(t)ψ1(x) + c2(t)ψ2(x) + ...

The coefficient ci(t) of each eigenfunction ψi is the probability amplitude for that eigenfunction (or in vector language, the projection of Ψ onto that basis vector "axis") which will give - via the modulus squared - the probability that the observation measures the eigenvalue αi of that eigenfunction.

So to summarise. Given a wavefunction of a definite type (e.g. describing a particle of mass m bound within a harmonic well of a certain potential energy), an observable operator sets up the menagerie of a particular set of basis eigenfunctions together with their associated eigenvalues.

The modelling of the experimental situation requires the further step of defining its specific wavefunction and then projecting it onto that basis to read-off the probability amplitudes. These are then squared (well, the moduli are) to get the probabilities of the various eigenvalues which might be observed.

Anyway, that's what I figure for now.

Note: the above describes the case of discrete eigenfunctions/values. We also covered, albeit more briefly, the case of continuous observables where the same general approach holds. Sums go to integrals; probability amplitudes go to density functions; we're into Fourier transform territory.