Physical Chemistry Third Edition

16.3 The Operator Corresponding to a Given Variable 689

The operator forp^2 xmust equal the square of the operator forpx. Since the operator for

p^2 xis negative, the operator forpxmust be purely imaginary.

px
̂px±ih ̄

d

dx

(16.3-8)

whereistands for the imaginary unit,i

√

−1. Since any quantity has two square

roots, we have a choice of signs. We take the negative sign and the operator forpxis

̂px−ih ̄

d

dx



h ̄

i

d

dx

(16.3-9)

The sign that we choose in Eq. (16.3-9) gives the momentum the correct sign when a

particle is moving in a known direction (see Problem 16.20).

We complete the third postulate by the additional assertion that the pattern of

Eq. (16.3-9) holds for all Cartesian momentum components and make the following

assertion:The quantum mechanical operator for any mechanical variable is obtained

by (1) expressing the quantity classically in terms of Cartesian coordinates and Carte-

sian momentum components and (2) replacing the momentum components byh/i ̄ times

the derivative with respect to the corresponding Cartesian coordinate.The validity of

the operator must be verified by comparison of the consequences of its action with

experimental fact, and this recipe has passed this test in every case.

For a particle moving in three dimensions

K 

1

2 m

(

p^2 x+p^2 y+p^2 z

)

(16.3-10)

so that

K̂−h ̄

2

2 m

(

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

−

h ̄^2

2 m

∇^2 (16.3-11)

where we write partial derivatives since we have more than one coordinate and where

∇^2 is called theLaplacian operator.

EXAMPLE16.3

Construct the operator for thezcomponent of the angular momentum of one particle.

Solution

The angular momentum is defined in Appendix E as thevector product(cross product)

Lr×p (16.3-12)

From Eq. (B-42) in Appendix B, thezcomponent is

Lzxpy−ypx (16.3-13)

The operator for this component is

̂Lzh ̄

i

[

x

∂

∂y

−y

∂

∂x

]

(16.3-14)

The expressions for̂LxandL̂ycan be obtained similarly.