Mathematical Methods for Physics and Engineering : A Comprehensive Guide

QUANTUM OPERATORS

IfA|an〉=a|an〉for allN 1 ≤n≤N 2 ,then

|ψ〉=

∑N^2

n=N 1

dn|an〉satisfiesA|ψ〉=a|ψ〉for any set ofdi.

For a general state|ψ〉,

|ψ〉=

∑∞

n=0

cn|an〉,wherecn=〈an|ψ〉. (19.10)

This can also be expressed as the operator identity,

1=

∑∞

n=0

|an〉〈an|, (19.11)

in the sense that

|ψ〉=1|ψ〉=

∑∞

n=0

|an〉〈an|ψ〉=

∑∞

n=0

cn|an〉.

It also follows that

1=〈ψ|ψ〉=

(∞

∑

m=0

c∗m〈am|

)(∞

∑

n=0

cn|an〉

)

=

∑∞

m,n

c∗mcnδmn=

∑∞

n=0

|cn|^2.

(19.12)

Similarly, the expectation value of the physical variable corresponding toAis

〈ψ|A|ψ〉=

∑∞

m,n

c∗m〈am|A|an〉cn=

∑∞

m,n

c∗m〈am|an|an〉cn

=

∑∞

m,n

c∗mcnanδmn=

∑∞

n=0

|cn|^2 an. (19.13)

19.1.1 Commutation and commutators

As has been noted above, the productABof two linear operators may or may

not be equal to the productBA.Thatis

AB|ψ〉is not necessarily equal toBA|ψ〉.

IfAandB are both purely multiplicative operators, multiplication byf(r)

andg(r) say, then clearly the order of the operations is immaterial, the result

|f(r)g(r)ψ〉being obtained in both cases. However, consider a case in whichA

is the differential operator∂/∂xandBis the operator ‘multiply byx’. Then the

wavefunction describingAB|ψ〉is

∂

∂x

(xψ(x))=ψ(x)+x

∂ψ

∂x

,