Predictions in the Stern-Gerlach experiment

If one knows the state of the universe completely, one can, according to the thermal interpretation, predict the results of all single measurements.

In particular, one should therefore be able to predict the impact positions of particles in a Stern-Gerlach experiment.

Since we do not, in fact, know the state, this is of course an idle speculation, but it is the principle that is important. We work in the Schrödinger picture, and denote the state of the universe at time t by ρ(t). This is a density matrix, an operator on the Hilbert space of the universe. It determines the universal ensemble by the prescription

<f>_t := tr(ρ(t)f)

for the objective value of every (sufficiently slowly-varying) variable f.

In the classical Stern-Gerlach experiment silver atoms pass by a magnet and are then captured by a detector. The distribution of the silver on the detector is given by a field S(x) which can in principle be written down in terms of quantum field operators.

Since this field is macroscopic at the resolution relevant to the experiment, one can assume local equilibrium, and obtains (in accordance with statistical mechanics in local equilibrium, as described in many books, and up to a certain precision) the observable silver distribution

S(x,t) := <S(x)>_t = tr(ρ(t)S(x)).

What can be measured is therefore clearly determined by the state of the universe. If God
1. knows this state at the time t=0,
2. knows the Hamiltonian of the universe, and
3. can solve the von Neumann equation,
then he can calculate ρ(t) and thereby predict the distribution of silver at any given time.

If one models this situation in more detail, and includes the quantum source together with the magnets, one finds that S(x,t) undergoes a jump diffusion process, which is in good agreement with both individual observations and the predictable distribution (namely two spots, corresponding to the two spin eigenvalues).

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Arnold Neumaier (Arnold.Neumaier@univie.ac.at)

A theoretical physics FAQ