David Bohm

David Bohm is perhaps best known for new experimental methods to test Einstein’s supposed suggestion of “hidden variables” that would explain the EPR paradox by providing the information needed at the distant “entangled” particle, so it can coordinate its properties perfectly with the “local” particle.

Bohm wrote in 1952,

The usual interpretation of the quantum theory is based on an assumption having very far-reaching implications, ~is., that the physical state of an individual system is completely specified by a wave function that determines only the probabilities of actual results that can be obtained in a statistical ensemble of similar experiments. This assumption has been the object of severe criticisms, notably on the part of Einstein, who has always believed that, even at the quantum level, there must exist precisely definable elements or dynamical variables determining (as in classical physics) the actual behavior of each individual system, and not merely its probable behavior. Since these elements or variables are not now included in the quantum theory and have not yet been detected experimentally, Einstein has always regarded the present form of the quantum theory as incomplete, although he admits its internal consistency.

Five years later, Bohm and his Israeli student Yakir Aharonov reformulated the original EPR argument in terms of electron spin. They said experimental tests with continuous variables would be much more difficult than tests with discrete quantities, such as the spin of electrons or polarization of photons. They wrote: Bohmian mechanics provides a straightforward physical explanation.

First, close slit 1 and open slit 2.
The particle goes through slit 2.
It arrives at x on the plate with probability |ψ2(x)|2,
where ψ2 is the wave function which passed through slit 2.

Second, close slit 2 and open slit 1.
The particle goes through slit 1.
It arrives at x on the plate with probability |ψ1(x)|2,
where ψ1 is the wave function which passed through slit 1.

Third, open both slits.
The particle goes through slit 1 or slit 2.
It arrives at x with probability |ψ1(x)+ψ2(x)|2.

Now observe that in general,
1(x)+ψ2(x)|2 = |ψ1(x)+ψ2(x)|2= |ψ1(x)|2+|ψ2(x)|2 + 2ℜψ1(x) ψ2(x).

The last term comes from the interference of the wave packets ψ1 and ψ2 which passed through slit 1 and slit 2.

The probabilities of finding particles when both slits are open are different from the sum of slit 1 open and slit 2 open separately. The wave function determines the probabilities of finding particles, as Einstein first proposed..

We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore

ψ = (1/√2) [ ψ+ (1) ψ- (2) - ψ- (1) ψ+ (2) ]

where ψ+ (1) refers to the wave function of the atomic state in which one particle (A) has spin +ℏ/2, etc. The two atoms are then separated by a method that does not influence the total spin. After they have separated enough so that they cease to interact, any desired component of the spin of the first particle (A) is measured. Then, because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (8) is opposite to that of A.

Einstein may have pressed Bohm to develop hidden variables as the source of nonlocal behavior. Einstein had heartily approved of Bohm’s textbook and was initially supportive of Bohm’s new mechanics. Einstein thought Bohm was young enough and smart enough to produce the mathematical arguments that the older generation of “determinist” physicists like Erwin Schrödinger, Max Planck, and others had not been able to accomplish.

But when Bohm finished the work, based on Louis de Broglie’s 1923 “pilot-wave” idea (which Einstein had supported), Einstein rejected it as inconsistent with his theory of relativity.

Einstein wrote to Max Born on May 15, 1952,

Have you noticed that Bohm believes (as de Broglie did, by the way, 25 years ago) that he is able to interpret the quantum theory in deterministic terms? That way seems too cheap to me. But you, of course, can judge this better than I.

Now Richard Feynman's path integral formulation of quantum mechanics describes supraluminal paths and even some things moving backwards in time, so we must take a careful look at Bohm's work.

Bohm's search for "hidden variables" inspired John Bell to develop a theorem on "inequalities" that would need to be satisfied by hidden variables. To this date, every test of Bell's theorem has violated his inequalities and shown that the quantum theory cannot be replaced by one with "local" hidden variables. If they exist at all, "hidden variables" must also be "nonlocal."

Bohm's pilot-wave goes through both slits in the two-slit experiment, whereas the particle goes through only one, thus explaining what Richard Feynman called the "only mystery" in quantum mechanics.

The Measurement Process
David Bohm was particularly clear on the process of measurement. He said it involves macroscopic irreversibility, which was a sign and a consequence of treating the measuring apparatus as a macroscopic system that could not itself be treated quantum mechanically. The macroscopic system could, in principle, be treated quantum mechanically, but Bohm said its many degrees of internal freedom would destroy any interference effects. This is the modern theory of quantum decoherence.

Bohm's view is consistent with the information-philosophy solution to the measurement problem. A measurement has only been made when new information has come into the world and adequate entropy has been carried away to insure the stability of the new information, long enough for it to be observed by the "conscious" observer.

In his 1950 textbook Quantum Theory, Bohm discusses measurement in chapter 22, section 12.

12. Irreversibility of Process of Measurement and Its Fundamental Role in Quantum Theory.
From the previous work it follows that a measurement process is irreversible in the sense that, after it has occurred, re-establishment of definite phase relations between the eigenfunctions of the measured variable is overwhelmingly unlikely. This irreversibility greatly resembles that which appears in thermodynamic processes, where a decrease of entropy is also an overwhelmingly unlikely possibility.*

* There is, in fact, a close connection between entropy and the process of measurement. See L. Szilard, , 53, 840, 1929. The necessity for such a connection can be seen by considering a box divided by a partition into two equal parts, containing an equal number of gas molecules in each part. Suppose that in this box is placed a device that can provide a rough measurement of the position of each atom as it approaches the partition. This device is coupled automatically to a gate in the partition in such a way that the gate will be opened if a molecule approaches the gate from the right, but closed if it approaches from the left. Thus, in time, all the molecules can be made to accumulate on the left-hand side. In this way, the entropy of the gas decreases. If there were no compensating increase of entropy of the mechanism, then the second law of thermodynamics would be violated. We have seen, however, that in practice, every process which can provide a definite measurement disclosing in which side of the box the molecule actually is, must also be attended by irreversible changes in the measuring apparatus. In fact, it can be shown that these changes must be at least large enough to compensate for the decrease in entropy of the gas. Thus, the second law of thermodynamics cannot actually be violated in this way. This means, of course, that Maxwell's famous "sorting demon " cannot operate, if he is made of matter obeying all of the laws of physics. (See L. Brillouin, American Scientist, 38, 594, 1950.)

Because the irreversible behavior of the measuring apparatus is essential for the destruction of definite phase relations and because, in turn, the destruction of definite phase relation's is essential for the consistency of the quantum theory as a whole, it follows that thermodynamic irreversibility enters into the quantum theory in an integral way. This is in remarkable contrast to classical theory, where the concept of thermodynamic irreversibility plays no fundamental role in the basic sciences of mechanics and electrodynamics. Thus, whereas in classical theory fundamental variables (such as position or momentum of an elementary particle) are regarded as having definite values independently of whether the measuring apparatus is reversible or not, in quantum theory we find that such a quantity can take on a well defined value only when the system is coupled indivisibly to a classically describable system undergoing irreversible processes. The very definition of the state of any one system at the microscopic level therefore requires that matter in the large shall undergo irreversible processes. There is a strong analogy here to the behavior of biological systems, where, likewise, the very existence of the fundamental elements (for example, the cells) depends on the maintenance of irreversible processes involving the oxidation of food throughout an organism as a whole. (A stoppage of these processes would result in the dissolution of the cell.)

But Bohm changed his mind about irreversibility when he developed his more realistic and deterministic theory. Now he became concerned with the classic "problem" of microscopic irreversibility, namely how can the increase of entropy involve macroscopic irreversibility if microscopic collisions of particles are reversible?

References
A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I
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