----------------------------------------------------------
S1r. Quantum measurement theory for continuous observables
----------------------------------------------------------
The standard textbook measurement theory says that the possible
measurement results in measuring an observable given by a Hermitian
operator A are its possible eigenvalues, with a probability density
depending on the state of the system. This is part of the content of
Born's rule, and counts as one of the cornerstones of the
interpretation of quantum mechanics.
But Born's rule gives only a very idealized account of measurement
theory, and gives no sufficient explanation for what is going on in
many nontrivial measurements.
The spectrum of the Hamiltonian of the electron of a hydrogen atom
has a discrete part, catering for its bound states. According to the
idealized textbook measurement theory, a measurement of the energy
of a bound state should produce an infinitely accurate value agreeing
with one of the values in the (QED-corrected) Balmer (etc.) series.
But this is ridiculous. Repeated preparation and measurement of the
position of the ``same'' spectral lines (which provide these energy
measurements, relative to an appropriate zero of the energy) yields
different results, from which the energies themselves can be obtained
only to a certain accuracy.
Thus Born's rule does not account for the interpretation of a
measurement of the energy of an electron. For similar reasons,
measurements of particle masses or resonance energies do not reveal
the exact values (which they should according to Born's rule) but only
approximations whose quality depends a lot on the way the measurement
is done (an aspect that does not figure at all in Born's rule).
Measurements such as that of a particle lifetime, a reaction rate, or
the integral cross section of a particular reaction do not even have a
natural associated operator of which the measurement result would be
an eigenvalue.
The idealized textbook measurement theory based on Born's rule is
appropriate only for the measurement of spin and related variables
that result in recording decisions between a small number of cases.
Thus the measurement process as described by von Neumann (and copied
from there to numerous textbooks) is an unrealistic idealization
compared with many (and probably most) real measurements.
The latter are usually much better described by suitable POVMs
(positive operator valued measures) rather than by Born's rule,
which corresponds to PVMs (projection-valued measures), a special case
of POVMs in which the positive operators are in fact projections.
See Sections 7.3-7.5 of the book
A. Neumaier and D. Westra,
Classical and Quantum Mechanics via Lie algebras,
arXiv:0810.1019
for a realistic account of measurement theory not dependent on
Born's rule. The latter is derived there as a special case, together
with giving the condition in which it is applicable.