Interpretations of

Quantum Mechanics

Quantum Mechanics

Implications of

Quantum Mechanics

Quantum Mechanics

21. Bohm’s Hidden Variable Interpretation.

Summary

There are severe problems with hidden variable interpretations of quantum mechanics. Bohm’s model is described and its shortcomings pointed out.

It has been conjectured that even though one cannot show there are particles, there may still be ‘hidden’ (not detectable by experiment) variables that determine the unique outcome of an experiment, the outcome we perceive. At a given time, there is only one, unique set of these hidden variables (as opposed to the many versions of reality of the wave function), so they are objective.

The first observation is thatSecond, there is no definitive proof that there can

[P21] There is no experimental evidence for hidden variables.

*not*be a hidden variable theory underlying quantum mechanics. And third, there is no acceptable hidden variable theory at this time. Further, one encounters severe difficulties in attempting to construct a satisfactory underlying theory.

Bohm’s model. The most nearly successful hidden variable theory is that of Bohm [Ref. 4, Ref. 6]. Although it falls short, there are several reasons for presenting it. First, it

*is*successful in achieving its goal if one ignores certain shortcomings. Second, it is pretty much the only even minimally acceptable hidden variable model. Third, any acceptable hidden variable theory would presumably have to reduce to Bohm’s model in many situations. And fourth, it serves as a testing ground for conjectured ‘no-go’ theorems—arguments which purport to prove that hidden variables are impossible.

In this model, Bohm derived a set of trajectories through space from the Schrödinger equation (the velocity of the trajectory through point x is essentially the gradient of the phase angle, , of the wave function). He then assumed that for each ‘particle-like’ wave function, a ‘particle’ was put on one of the trajectories. This particle then followed a very complex trajectory. The ‘hidden variables’ in this case, not accessible to experiment, are the position and velocity of the particle.

To explain the success of the theory, we will use the example of a Stern-Gerlach experiment on a spin 1 particle (particle-like wave function). The z-component of spin of this wave function can take on the three values +1, 0, -1. When this wave function is shot through a magnetic field, the three parts of the wave function each follow separate trajectories, so the wave function, schematically, is

(21-1)

where the different wave functions correspond to the different ‘trajectories.’ For each run of the experiment, the probability law tells us that the probability of the particle following trajectory i is |a(i)|^{2}.

We now suppose the spin 1 particle consists of two spin 0 particles in a bound state (like the electron and proton in a hydrogen atom, ignoring their spins) and apply the Bohm model to this situation. Before the compound particle reaches the magnetic field, the two particles which are bound together will follow trajectories that

*move very rapidly*through the three possible states. And then at the magnetic field, the trajectory of the bound state will follow just one of the three possible paths. The success of Bohm’s model is that it predicts that path i will be followed a fraction |a(i)|

^{2}of the time, so that the probability law is satisfied.

Now for the problems with the model. First, it is not relativistic, and because of the way in which time is used, it is difficult to generalize it to a relativistic formulation. Second, there are as-yet-unsolved problems in handling the creation and annihilation of particles. Third, one must assume a specialized initial density of trajectories in order to obtain the probability law, and there is no clear reason why nature should choose that density.

Fourth, it is an arbitrary feature of the model that a particle is put on just one of the trajectories. There is nothing

*in the mathematics*that prevents there being

*two*particles, on two different trajectories, associated with a ‘single particle’ wave function. And it is extremely difficult to reformulate the Bohm model with a ‘source’ equation which

*mathematically forces*the assigning of one and only one particle to a single particle wave function.

Finally, there is the problem discussed in the Particle Interpretation. Because there is no collapse of the wave function, there is a valid version of the brain wave function of the observer in each version of reality. But there is nothing in hidden variable theories which would show that only the version singled out by the hidden variables is conscious. From the Particle Interpretation,

[P20]* It appears to be very difficult, and probably impossible, to explain in any non-collapse hidden variable model why the quantum versions of the brainnotassociated with the hidden variables cannot be consciously aware.