Individual particles pass through an entrance slit into an inhomogeneous magnetic field ). The initial spin state is an eigenstate of the Sy operator with eigenvalue 1/2 and so their spin vectors lie along the y direction (parallel to the plane of the slits.) The magnetic field is oriented along the z-direction and increases in magnitude along the y direction. The field direction and magnitude are represented by the red arrows.

The yellow/blue surface represents the probability density, where its magnitude is large it is more likely that a particle would be found in that region - should a measurement be carried out.

The yellow line represents the particle trajectory whilst the red dot represents the current particle position. As the particle moves along its trajectory its spin vector, represented by the red arrow, precesses. At first it precesses in the xy plane, as would be expected from the classical theory of Larmor precession. But as the beam splits into two under the influence of the magnetic field, the spin begins to behave in a non classical way under the influence of the quantum torque. It starts to precess out of the xy plane and to align either with or against the field, which alternative happens depends on where the particle starts off in the beam.

The green lines, under the probability density surface, display a representative sample of possible trajectories.

By observing where the particle ends up we can determine its z-component of spin, so in a sense the particle position is the measurement apparatus coordinate, or pointer. The inhomogeneous magnetic field introduces a correlation between the particle position (the apparatus coordinate) and its spin (which is being "measured" in the apparatus). Initially the z-component of spin is zero and this is not an eigenstate of the operator representing the observable being measured. In Bohm's theory, unless the quantum state is already an eigenstate of the operator representing the observable, the value it assigns will not be one of the eigenvalues of the operator. But we see clearly how, in Bohm's theory, the assigned value evolves to become one of the eigenvalues as the dynamical measurement process takes place in a deterministic way.

So, the outcome of the measurement is not actually a random event (although it appears that way to the experimenter), instead the outcome is determined from the start by the actual position of the particle. The actual particle position can neither be controlled, nor observed, by the experimenter without ruining the experiment. Its actual value is hidden. It is a hidden variable - until such time as the position is observed on the screen

The Stern-Gerlach measurement of a spin one half particle is the archetypical quantum measurement. But a similar story will be told for the measurement of any observable in Bohm's theory. The outcomes are not random, they are determined by the hidden variable - the particle position.

Although we refer to the particle position as a hidden variable, it is exactly what is observed if the position is measured. The black spot on a photographic plate occurs where the Bohm particle actually is. The position of the particle is uncontrollable, but nonetheless observable and so not so much hidden.

The Stern-Gerlach measurement process in Bohm's theory was first described in the paper:

What happens in a spin measurement? (article)
Author
C. Dewdney and P.R. Holland and A. Kyprianidis
Journal
Physics Letters A
Year
1986
Volume
119
Number
6

Doi
[ Ссылка ]90144-1
Issn
0375-9601
Url
[ Ссылка ]
Local Files
Remote URLs
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[ Ссылка ]90144-1

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