Mathematical Methods for Physics and Engineering : A Comprehensive Guide

19.2 PHYSICAL EXAMPLES OF OPERATORS

quantum-mechanical operators are those corresponding to positionrand mo-

mentump. One prescription for making the transition from classical to quantum

mechanics is to express classical quantities in terms of these two variables in

Cartesian coordinates and then make the component by component substitutions

r→multiplicative operatorr and p→differential operator−i
∇.

(19.22)

This generates the quantum operators corresponding to the classical quantities.

For the sake of completeness, we should add that if the classical quantity contains

a product of factors whose corresponding operatorsAandBdo not commute,

then the operator^12 (AB+BA) is to be substituted for the product.

The substitutions (19.22) invoke operators that are closely connected with the

two that we considered at the start of the previous subsection, namelyxand

∂/∂x.One,x, corresponds exactly to thex-component of the prescribed quantum

position operator; the other, however, has been multiplied by the imaginary

constant−i
,where
is the Planck constant divided by 2π. This has the (subtle)

effect of converting the differential operator into thex-component of anHermitian

operator; this is easily verified using integration by parts to show that it satisfies

equation (17.16). Without the extra imaginary factor (which changes sign under

complex conjugation) the two sides of the equation differ by a minus sign.

Making the differential operator Hermitian does not change in any essential

way its commutation properties, and the commutation relation of the two basic

quantum operators reads

[px,x]=

[

−i

∂

∂x

,x

]

=−i
. (19.23)

Corresponding results hold whenxis replaced, in both operators, byyorz.

However, it should be noted that if different Cartesian coordinates appear in the

two operators then the operators commute, i.e.

[px,y]=[px,z]=

[

py,x

]

=

[

py,z

]

=[pz,x]=[pz,y]=0.

(19.24)

As an illustration of the substitution rules, we now construct the Hamiltonian

(the quantum-mechanical energy operator)Hfor a particle of massmmoving

in a potentialV(x, y, z) when it has one of its allowed energy values, i.e its

energy isEn,whereH|ψn〉=En|ψn〉. This latter equation when expressed in a

particular coordinate system is the Schr ̈odinger equation for the particle. In terms

of position and momentum, the total classical energy of the particle is given by

E=

p^2

2 m

+V(x, y, z)=

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

2 m

+V(x, y, z).

Substituting−i
∂/∂xforpx(and similarly forpyandpz) in the first term on the