Physical Chemistry Third Edition

686 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

Theproduct of two operatorsis defined as successive application of the operators, and

is denoted by writing the two operator symbols adjacent to each other. If we write the

operator equation

̂CÂB̂ (16.2-7)

then this is equivalent to

̂Cf(q)̂ABf̂ (q)Â(Bf̂ (q))Aĝ (q) (16.2-8)

whereg(q) is the function produced when̂Boperates onf(q). The operator written

next to the symbol for the function (the operator on the right) always operates first, so

operators operate from right to left. The square of an operator means operating twice

with that operator.

Operator multiplication isassociative, which means that

̂A(B̂Ĉ)(̂AB̂)Ĉ (16.2-9)

This means that if the operator productB̂Ĉequals the operatorF̂and the operator

product (Â̂B) equals the operatorĜ, thenÂF̂ĜĈ.

Operator multiplication and addition aredistributive, which means that

Â(B̂+Ĉ)Â̂B+ÂĈ (16.2-10)

Operator multiplication is not necessarilycommutative. It can happen that

̂AB̂̂BÂ (in some cases) (16.2-11)

IfÂ̂B̂BÂ, the operatorsÂandB̂are said tocommute.

Thecommutatorof two operatorsÂandB̂is denoted by [̂A,B̂] and is defined by

[Â,B̂]ÂB̂−̂BÂ (definition of the commutator) (16.2-12)

If two operators commute, their commutator vanishes.

EXAMPLE16.1

Find the commutator

[

x,

d

dx

]

.

Solution

We let the commutator act on an arbitrary differentiable function,f(x):

[

x,

d

dx

]

f(x)x

df

dx

−

d(xf)

dx

x

df

dx

−x

df

dx

−f(x)−f(x)

As an operator equation, [

x,

d

dx

]

−Ê− 1

Exercise 16.1

Find the commutator

[

x^2 ,

d

dx

]

.