QuantumPhysics.dvi

3.2 Linear operators on Hilbert space

A linear operator (or simply operator)AonHis a linear map fromHtoH. Its action on

a state|φ〉∈His denoted as follows,

A|φ〉=|Aφ〉 (3.24)

Successive application of operators involves taking products of operators. The product of op-

erators is generally associative,A(BC) = (AB)C=ABC, but is not generally commutative,

allowing forAB 6 =BA.

3.2.1 Operators in finite-dimensional Hilbert spaces

Every linear operatorAon anN-dimensional Hilbert spaceHis equivalent to anN×N

matrix, whose entriesAmnmay be obtained as matrix elements in a given basis, such as the

orthonormal basis{|n〉}n=1,···,Nconstructed earlier,

Amn=〈m|A|n〉 〈m|n〉=δmn (3.25)

The product of operators maps to matrix multiplication. IfAmnandBmnare the matrix

elements of the operatorsAandBin a certain orthonormal basis{|n〉}ofH, then the matrix

elements (AB)mnof the productABare analogously defined by (AB)mn=〈m|AB|n〉. Upon

inserting the completeness relation betweenAandB, we find,

(AB)mn=

∑

p

〈m|A|p〉〈p|B|n〉=

∑N

p=1

AmpBpn (3.26)

which is nothing but the rule of matrix multiplication. The product is associative and

generally non-commutative. There is an identity operatorIH which is represented by the

unit matrix. A Hermitian operator is represented by a Hermitian matrixA†=A, whose

matrix elements satisfy

Amn=A∗nm (3.27)

for allm,n= 1,···,N, while a unitary operatorUis defined to satisfyU†U=I.

3.2.2 Operators in infinite-dimensional Hilbert spaces

When the dimension ofHis infinite, a linear operator may not be well-defined on all elements

ofH, but only on adense subset, which is referred to as thedomainD(A) of the operator

A. Take for example the Hilbert spaceL^2 , and consider the linear operatorDwhich acts as

follows on the states|n〉,

D|n〉=n|n〉 n∈N (3.28)