framework (requiring for instance the solution of a Schr ̈odinger equation in 10^23
variables).
Instead of trying to resolve in general this problem of how macroscopic clas-
sical physics behavior emerges in a measurement process, one can adopt the
following two principles as providing a phenomenological description of what
will happen, and these allow one to make precise statistical predictions using
quantum theory:
Principle(Observables).States for which the value of an observable can be
characterized by a well-defined number are the states that are eigenvectors for
the corresponding self-adjoint operator. The value of the observable in such a
state will be a real number, the eigenvalue of the operator.
This principle identifies the states we have some hope of sensibly associating
a label to (the eigenvalue), a label which in some contexts corresponds to an
observable quantity characterizing states in classical mechanics. The observ-
ables with important physical significance (for instance the energy, momentum,
angular momentum, or charge) will turn out to correspond to some group action
on the physical system.
Principle(The Born rule).Given an observableOand two unit-norm states
|ψ 1 〉and|ψ 2 〉that are eigenvectors ofOwith distinct eigenvaluesλ 1 andλ 2
O|ψ 1 〉=λ 1 |ψ 1 〉, O|ψ 2 〉=λ 2 |ψ 2 〉the complex linear combination state
c 1 |ψ 1 〉+c 2 |ψ 2 〉will not have a well-defined value for the observableO. If one attempts to
measure this observable, one will get eitherλ 1 orλ 2 , with probabilities
|c^21 |
|c^21 |+|c^22 |and
|c^22 |
|c^21 |+|c^22 |
respectively.
The Born rule is sometimes raised to the level of an axiom of the theory, but
it is plausible to expect that, given a full understanding of how measurements
work, it can be derived from the more fundamental axioms of the previous
section. Such an understanding though of how classical behavior emerges in
experiments is a very challenging topic, with the notion of “decoherence” playing
an important role. See the end of this chapter for some references that discuss
these issues in detail.
Note that the statec|ψ〉will have the same eigenvalues and probabilities as
the state|ψ〉, for any complex numberc. It is conventional to work with states