## 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" 2nd ed, 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 measurment 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 of the system is the
corresponding eigenstate.

Even in the less formal "The Quantum World" by Polkinghorne
we find [p 27] that

"The eigenvectors must obviously correspond to special
states.  They are taken to correspond to those states
in which the observable definitely takes that particular
value..."

which, he says, occurs when a measurment takes place.

However, even though this "standard formulation" of the postulates
of QM is nearly universal in basic "no-nonsense" practical treatments
of the subject, it is much less universal in works whose purpose
is to consider the philosophical aspects of QM, 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 QM 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 "lawlike" 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, my answer to the question would be that "postulate 2" is generally
understood to be part of the fundamental basis of quantum mechanics,
and certainly is invoked whenever we actually apply QM in a practical
situation, but I don't think we can automatically say that any paper
or book on QM *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
"characteristic", and many English books have reverted back to
the literal version.  However, according to Eisberg and Resnick
in their book "Quantum Physics", the use of the term eigen "is
conventional... it is also 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.]
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