The Projection Postulate

Dirac emphasized in his book on Quantum Mechanics that a measurement will result in the wave function being in an eigenstate. On the other hand, some authors do not focus on this, which raises the question of whether it is a fundamental assumption of quantum mechanics, or perhaps just a heuristic device to enable people to think in terms of ideal states.

In one sense, the proposition that the result of a measurement will be an eigenvalue of the measurement operator is indeed one of the fundamental postulates of formal quantum mechanics. See, for example, "Quantum Mechanics" by Davies and Betts, which summarizes the postulates of formal quantum mechanics as four statements, the second of which is

Every observable can be represented by a Hermitian operator, the eigenvalues of which are the various possible values that would be obtained on measurement. Immediately after a measurement the state of the system is the corresponding eigenstate associated with that eigenvalue.

Similarly, in his book "Quantum Mechanics", Rae presents five basic postulates, the second of which is virtually identical to the above:

Every dynamical variable may be represented by a Hermitian operator whose eigenvalues represent the possible results of carrying out a measurement of the value of the dynamical variable. Immediately after such a measurement, the wave function of the system will be identical with the eigenfunction corresponding to the eigenvalue obtained as a result of the measurement.

And of course von Neumann stressed this view of things in his book “Mathematical Foundations of Quantum Mechanics", as at the beginning of Chapter 5 where he says "After the measurement, the state of affairs is...", and goes on to say the value measured is one of the eigenvalues of the operator, and the state of the system is the corresponding eigenstate. But in Section 6 he acknowledges

In the course of this we found a peculiar dual nature of the quantum mechanical procedure, which could not be satisfactorily explained.  Namely, we found that on the one hand a state is transformed … under the action of an energy operator… which is purely causal. On the other hand, the state undergoes in a measurement a non-causal change…  The difference between these two processes is a very fundamental one: aside from the different behaviors in regard to the principle of causality, they are also different in that the former is (thermodynamically) reversible, while the latter is not.

In Section 5 he used the terms “process 1” for measurement and “process 2” for unitary evolution under the Schrodinger equation. Referring to Bohr’s 1929 explication in terms of “psycho-physical parallelism”, von Nuemann continued

We must always divide the world into two parts, the one being the observed system, and the other the observer… The boundary between these two is arbitrary to a very large extent.. but in each method of description the boundary must be put somewhere, if the method is not to proceed vacuously, i.e., if comparison with experiment is to be possible.

He procedes to consider a system to be observed, the measuring instrument, and the actual observer, which he denotes by I, II, and III respectively, and he shows that the results for I are the same, regardless of whether we apply process 2 (unitary evolution) to I and process 1 (measurement) to the interaction between I and (II+III), or if we apply process 2 to (I+II) and process 1 to the interaction between (I+II) and III. (Von Nuemann credits essential elements of this analysis to conservations with Leo Szilard.) It’s interesting that von Neumann seems to have been most concerned with what happens if we make the “boundary” smaller around conscious awareness.

That this boundary can be pushed arbitrarily deeply into the interior of the body of the actual observer is the content of the principle of the psycho-physical parallism…

In contrast, others have been more concerned about pushing the boundary outward, even to the point of considering the wave function of the entire universe, which is necessarily unobserved. As noted above, von Neumann regarded pure unitary evolution (process 2) with no measurement (process 1) as vacuous. But, although von Neumann’s three-partition analysis describes the effect of moving the boundary between observer and observed, it overlooks the connundrums of having more than one boundary, since we can have “observers” in both II and III. The idea of decoherence was not clear at that time, but even this does not fully resolve the issues. As Landau stressed, quantum mechanics is a theory that describes the interaction between a quantum system and a classical system.  To apply this to a pure quantum systems leads to conceptual difficulties, since the connection with empiricism requires (at least arguably) some other principle to play the role of von Neumann’s process 1.

In the less formal "The Quantum World" by Polkinghorne we find that “the eigenvectors must obviously correspond to special states… in which the observable definitely takes that particular value...", which, he says, occurs when a measurement takes place.

However, even though this "standard formulation" of the postulates of quantum mechanics is nearly universal in basic text books, it is much less universal in works whose purpose is to consider the philosophical aspects of quantum mechanics, and the measurement problem in particular. For example, in his book "Particles and Paradoxes", Peter Gibbins includes a whole chapter discussing various "Projection Postulates". The difficulty with the basic proposition as stated in the "standard formulation" is that it refers to "what happens" when a measurement is made, but we are given no clear idea of what constitutes a "measurement", i.e., how to know when we should stop propagating the wave function in accord with Schrodinger's equation and abruptly invoke the "2nd postulate". It seems to depend on where we draw the line between what is being measured and what is doing the measuring, but that seems to be a subjective line, and it's difficult to attribute to it several different objective significances simultaneously.

Not surprisingly, there are even interpretations of quantum mechanics that deny the occurrence of "measurements" altogether, but in the absence of a corresponding theory of consciousness it isn't clear how we are to understand the apparent definiteness of events, as well as the apparent "law-like" behavior of events and their probabilities. (The latter is related to lack of a well-behaved probabilistic measure on the various "branches" of the no-collapse interpretation.)

So, although the Projection Postulate is generally understood to be part of the fundamental basis of quantum mechanics, and certainly is invoked whenever we actually apply quantum mechanics in a practical situation, we can’t automatically say that any treatment of quantum mechanics must list this as a basic postulate, especially if the work is trying to address the subtleties of the measurement problem.

By the way, the word “eigen” was taken from the German word for "self", which is fitting, because an eigenvector is transformed to itself (up to scale factor). According to Eisberg and Resnick (in their book "Quantum Physics") regarding the use of the term eigen, "it is conventional not to translate it into English, perhaps in honor of the dominant role played by German speaking physicists in the development of quantum mechanics." This may be true, although the term eigenvalue is probably just as prominent in pure mathematics. In any case, many English books now use the word “characteristic”.

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