QuantumPhysics.dvi

4.3 Expectation values and quantum fluctuations

If|φ〉is an eigenstate of an observableAwith eigenvaluea, then the probability for the

outcomeain a measurement ofAis unity. If|φ〉is not an eigenstate ofA, then|φ〉cannot

be associated with any one of the eigenvalues ofA, but there is still a probabilistic expectation

value for the observableA. For a general state|φ〉, any of the eigenvaluesaiofAmay be

the outcome of a measurement ofA, but only with a certain probabilityp(ai|φ), computed

in the preceding subsection. The expected outcome in the state|φ〉of a measurement onA

is then the probabilistic expectation value for any of the eigenvaluesai, given by

∑

i

aip(ai|φ) =

∑

i

ai〈φ|Pi|φ〉=〈φ|A|φ〉 (4.15)

The quantity〈φ|A|φ〉, also denoted by〈A〉φ, is referred to asthe expectation value ofAin

the state|φ〉. As before, it is being assumed that||φ||= 1.

In probability theory one is interested in the standard deviation away from an average.

Similarly, in quantum physics one is interested in theaverage quantum fluctuations away

from an expectation value. One defines the strength of these quantum fluctuations as follows.

For any given state|φ〉, subtract the expectation value of the observable to define a new

observable which has vanishing expectation value (still in the state|φ〉),

Aφ≡A−〈A〉φIH 〈φ|Aφ|φ〉= 0 (4.16)

The magnitude of the quantum fluctuations is then defined by

(∆φA)^2 ≡〈φ|(Aφ)^2 |φ〉=

∑

i

(

ai−〈A〉φ

) 2

p(ai|φ) (4.17)

Of course, for|φ〉an eigenstate of the observable A, we have ∆φA= 0, and there are no

quantum fluctuations of the observableAin this state.

4.4 Incompatible observables, Heisenberg uncertainty relations

Compatible observables may be measured simultaneously on all states. In particular, this

means that we can simultaneously have

∆φA= ∆φB= 0 (4.18)

for one and the same state|φ〉, which is then an eigenstate of bothAandB.

How about two incompatible observables,AandB, characterized by [A,B] 6 = 0? Al-

though incompatible observables cannot be measured simultaneously on all states,it may or

may not be possible to measureAandBsimultaneously on some subset of all the states.