THE PROBLEM
When you make a measurement of a quantity, it would be natural to assume that you are simply becoming privy to a predetermined value. Values of position (x) at time (t) would assume identical values even if not observed. The tree falls in the absence of observers, and still makes a noise. However, quantum mechanics tells us the best approximation we can make is a probability distribution, describing the likelihood that a measurement takes a given value. The quantity in question assumes a definite state once observed, but not a moment sooner.
This problem has plagued quantum mechanics since its conception. Some scientists choose to accept the intrinsically undetermined nature of the theory, while others suggest we are missing a crucial piece of the puzzle. What if there is information about a quantum system we do not know, parameters we have not considered, or variables we have not taken into account? Possibly, by the addition of these unknowns, we could eliminate the need for probability distributions and restore a deterministic nature to quantum mechanics. If this is to be done, we need more information; we need “hidden” variables.
For ease of explanation we will be exploring the nature of hidden variables as they relate to the Bell inequality and the Stern-Gerlach experiment and device; considering the implications of spin measurements oriented along different axis; ultimately determining how hidden variables aid in narrowing the probabilistic nature of the quantum mechanical theory.
STERN-GERLACH AND CORRELATION
The Stern-Gerlach experiment (hereafter referred to as SG) was first conducted in 1992 by Otto Stern and Walther Gerlach who split a beam of silver atoms into 2 divergent beams using a magnetic field. The most profound consequence of their findings was “experimental proof of directional quantization”, later attributed to the directional quantization of spin (Friedrich). To see how this directional quantization relates to our search for hidden variables we will consider an extension of the Stern-Gerlach experiment (figure 2) in which a single particle is passed through 2 SG devices of relative angle varying from 0 to 180 degrees.
When the measuring devices are oriented along the same axis the second measurement of spin will yield the same result as the first measurement, consistent with quantum mechanics’ theory of non-local wave function collapse. The curious result emerges when we examine what effect the relative angle between measuring devices has on the result of probability correlation values. By varying the angle between the measuring devices one introduces a statistical variation in probability. As the angle between the two SG devices varies we find the probability of measuring the particles spin to be opposite of the first value varies as well. A probability of measurement correlation described by
(1)Where is the relative angle between axis 1 and 2; an expression that is valid through any relative angle, up to and including 2. And produces a graph of correlation vs. relative angle detailed in figure 1.
Figure1. How the probability of obtaining contradicting results of 2 successive Stern-Gerlach measurements varies with the relative angle between the devices.
It is this varying correlation with relative angle that will aid in our understanding of the Bell inequality.
JOHN BELL AND HIS INEQUALITY
John Bell famously summarized the bizarre results of correlation experiments with his inequality describing the statistical probability of particle spin measurements (Wigner). Bell proposed a case of two particles, whose spins were equal and opposite in every direction, traveling away from each other, observed when they reach SG devices oriented at relative angles to each other. The idea was to determine the spins of both particles along the same axis, and Bell anticipated results in which his measurement of the first particle would always be opposite to that of the second.
Bell found that when varying the angle between his SG devices he uncovered a statistical model of correlation between his results that perfectly agrees with Eq. (1). (Depending on the angle he chose between devices he found different probabilities of the particles spins being opposite.) He formulated his famous inequality by considering measurements of spin in three directions and applying the relationship concerning correlation explained previously. Using commuting characteristics of directional measurement Bell hypothesized that local hidden variables can account for quantum probability predictions only if the three directions in which the spin is measured are oriented such that they satisfy;
(2)Where the subscripts 1, 2, and 3 are used to denote the axis of measurement, and hence define the angles between them. This is a condition that can be violated by a large number of choices for the three directions (Wigner). Since many reasonable choices for our three directions of measurement do not allow hidden variables to fully account for observables it would seem they cannot be the means behind the workings of quantum mechanics. In order to solve this problem let’s examine what is so counterintuitive about our results thus far.
EXTENSION OF THE STERN GERLACH EXPERIMENT
As previously discussed, sending a particle through two successive SG devices yields a statistical distribution of correlation (resulting in Eq. (1)), a result which is expected when measuring non-commuting observables. Let’s now investigate a case where three SG devices successively divide a beam of particles according to their spin, and only the negative spin particles from the first device are sent through the second, and only the negative spin particles from the second are sent through the third. The second and third devices are oriented at 90 and 0 degrees relative to the first, respectively. (Figure 2)
Figure 2. three successive Stern-Gerlach measurements with device #2 oriented 90 degrees relative to device #1. and device #3 oriented 0 degrees relative to device #1. The particles with positive spin are blocked from entering the next device after each measurement.
In this example of SG measurements it is quite easy to see why these results do not agree with classical expectations. When our beam passes through the third device we would assume to find no particles with positive spin, although our third measurement of the beam yields an equal division of positive and negative spins. This result would seem to negate the “sorting” of the beam performed by the first device. Why is this so? Opponents of a hidden variable theory would argue that by making a second non-commuting measurement we have restored the wave function from its collapsed form. Conversely, the result of this particular experiment can be made clearer if we incorporate non-local hidden variables.
NON-LOCAL INFLUENCE
When our beam reaches the third device, it yields our final result by means of “knowing” the second measurement was made. Yet how can a particle know it has been observed? Bell’s inequality served to illustrate a local hidden variable theory’s maximum ability to reproduce the results of experiment in quantum mechanics but it leaves room for non-local hidden variables to influence the particle along its path. It is at this stage that our discussion concerning hidden variables takes a horribly self-destructive turn. The introduction of non-local hidden variables (also referred to as “action at a distance”) is primarily an attempt to finish the work of local hidden variable theories and restore the last piece of deterministic nature to the quantum theory. Non-local hidden variables would solve the Einstein-Podolsky-Rosen paradox for just about any interpretation, and explain the results of our 3 SG device experiment. Unfortunately, no theory proposed thus far mathematically accounts for this “spooky action” in a manner that is consistent with all experimental outcomes of the classical quantum mechanical theory.
For example, Bohm postulated that the Schrödinger potential could be separated into a classical potential and a quantum super-potential. At the same time Bohm introduced point like particles to the quantum picture, and concluded that the quantum super-potential was the non-local effect influencing the particles trajectory through space. If we could solve for the super-potential we would have a solution, and a restoration of determinism. While this sounds like a viable solution, Bohm’s theory falls short of describing the happenings of the real world for one simple reason. In his theory, Bohm attributes characteristics to his point like particles (ie. charge, mass) that we know are carried on the wave function. This is an example of a viable solution to the non-local hidden variable problem that upon closer examination has not solved the problem we set out to solve.
There continue to be searches for these non-local hidden variables; the scientists who ignore the Copenhagen interpretation are not ready to give up just yet. However the problem is approached with tentative vigor, one scientist expressing that his experiment has the capability of “probabilities [being] enhanced by a factor of the order of 400 or more over the average” while also acknowledging “The experiment may also reveal a breakdown in quantum theory” (Eberhard). Science is treading a fine line between modifying the current quantum theory, and finding reasons why we need a new theory entirely.
WHAT HAVE WE LEARNED?
We started our discussion concerning hidden variables to solve one fundamental anomaly concerning quantum theory. Position (x) at time (t) [or rather spin(z) at time (t)] has a value regardless of measurement, how can we find it without measuring? We have shown how a local theory of hidden variables can narrow the window of uncertainty, but does not account for the entire distribution of values our parameter can take. We then hypothesized a non-local influence to account for this discrepancy and found the scientific community scrambling for a solution which does not unravel the core of our quantum understanding. Maybe, according to the Copenhagen interpretation, we are asking one too many questions.
The Copenhagen interpretation of quantum mechanics was formulated soon after the creation of quantum mechanics itself, and was introduced to solve this very problem. When asked information concerning two non-commuting operators simultaneously the Copenhagen interpretation simply says this cannot be done. Information gathered at the same time concerning two non-commuting operators is simply not an allowed measurement and therefore is not a part of our physical world we should be concerned with. Quantum mechanics is intrinsically probabilistic; therefore asking questions concerning determinism is futile.
The statements laid forth by the Copenhagen interpretation, however damaging they may seem to the search for hidden variables, do not negate the scientific findings of maximum correlation and the possibility of hidden variables explaining the workings of the natural world. They simply provide an answer if one is not concerned with restoring determinism to our theory. The search for these variables will no doubt continue, fueled by the research of those who refuse to accept a probabilistic view of the world. Until a complete theory outlining hidden variables is found it is the non-deterministic theory that will be given credit for the workings of the universe. All the while scientists work furiously to find a theory of hidden-variables which might give us a greater insight to the mechanics of the quantum world; thinking that maybe, just maybe, things aren’t as random as they seem.
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