The Projection Postulate in Quantum Mechanics

 

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. Even 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. (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”.

 

Return to MathPages Main Menu