Physical Chemistry Third Edition

692 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

Linear Operators

Quantum mechanical operators arelinear operators. A linear operator̂Aobeys the two

properties

Â[f(q)+g(q)]̂Af(q)+Aĝ(q)

(

property 1 of

linear operators

)

(16.3-26)

and

Â[cf(q)]cAf̂ (q)

(

property 2 of

linear operators

)

(16.3-27)

wherecis an arbitrary constant and wherefandgare well-behaved functions (for

example, ifÂis a derivative operator they must be differentiable).

Hermitian Operators

Quantum mechanical operators arehermitian operators. A hermitian operator̂Aobeys

the relation

∫

f∗Agdq̂ 

∫

(Af̂ )∗gdq

∫

(Â

∗

f∗)gdq

(

property of

hermitian operator

)

(16.3-28)

whereqis an abbreviation for all of the independent variables on which the func-

tionsfandgdepend and where the integrations are over all values of the independent

variables. For example, ifqrepresents the Cartesian coordinates of a particle that can

move in three dimensions, the integral is a three-fold integral anddqstands for (dxdydz).

The symbolf* denotes thecomplex conjugateof the functionf, andÂ* denotes the

complex conjugate of the operatorÂ.

Complex quantities are surveyed briefly in Appendix B. Ifzis a complex quantity

it can be written

zx+iy (16.3-29)

where the real quantityxis called thereal partofzand the real quantityyis called

theimaginary partofz. The complex conjugate of any complex number, function, or

operator is obtained by changing the sign of its imaginary part:

z∗x−iy (16.3-30)

A real quantity or a real operator is equal to its complex conjugate, and an imaginary

quantity or an imaginary operator is equal to the negative of its complex conjugate.

If the particle can move in all of space, the integration limits in Eq. (16.3-28) are

−∞to∞for each Cartesian coordinate. The functionsfandgmust obey boundary

conditions such that the integral converges. The functionsfandgmust approach zero

for large magnitudes of the Cartesian coordinates.

Hermitian operators have several important properties:

1. Hermitian operators are linear.


2. Two hermitian operators are not required to commute with each other.