QuantumPhysics.dvi

4.1 Conservation of probability

The combination of Principles 2 and 3 specifies the nature of measurement of a general

observableA. If a quantum system has been prepared in an arbitrary state|φ〉, the mea-

surement of the observableAproceeds as follows.

By Principle 2, only a single eigenvalueaiof Awill be recorded during a single mea-

surement of the observable associated withA. (In the photon polarization experiment, any

single photon has polarization either along thexoryaxis, but not both.) By principle 3,

the probability for eigenvalueaito be recorded is given by

p

(

|φ〉→|φi〉

)

=|〈φi|φ〉|^2 (4.5)

provided the eigenvalueai is non-degenerate, i.e. has only a single eigenvector|φi〉. Both

states|φi〉and|φ〉are assumed to be normalized here||φi||=||φ||= 1. More generally,

if the eigenvalueaiis degenerate, letPibe the projection operator onto the eigenspaceEi

associated with the eigenvalueai. The total probabilityp(ai|φ) to record the eigenvalueai

for the observableAin the state|φ〉is then given by

p(ai|φ) =p

(

|φ〉→Ei

)

=||Piφ||^2 (4.6)

The property ofself-adjointnessof any observable Aguarantees that it admits a decom-

position into mutually orthogonal projection operatorsPi weighted by the corresponding

eigenvaluesaiofA,

A=

∑

i

aiPi

∑

i

Pi=IH (4.7)

and the second identity guarantees that this decomposition is complete. As a result, the

sum of the probabilities for the state|φ〉to be recorded in all possible outcomes is 1, since,

∑

i

p(ai|φ) =

∑

i

p

(

|φ〉→Ei

)

=

∑

i

||Piφ||^2 =

∑

i

〈φ|Pi†Pi|φ〉

=

∑

i

〈φ|Pi|φ〉=〈φ|φ〉= 1 (4.8)

This result is, of course, a very important consistency check on the probabilistic interpreta-

tion of quantum mechanics.

4.2 Compatible versus incompatible observables

Two observablesAandB are said to becompatibleprovided [A,B] = 0; if [A,B] 6 = 0,

the observable areincompatible. For example, the momentum componentspxandpyare