Physical Chemistry Third Edition

F. Some Mathematics Used in Quantum Mechanics

F. 1 The Classical Wave Equations for

Electromagnetic Radiation

In 1865, Maxwell developed a mathematical theory of electromagnetism. In this theory,

there are four important vector quantities, theelectric fieldE, theelectric displacement

D, themagnetic field strengthH, and themagnetic inductionB. The magnetic induction

Bis generally called themagnetic fieldby chemists. The dependence of these quantities

on time and position is described byMaxwell’sequations, which Maxwell deduced from

empirical laws. He found that the electric and magnetic fields can oscillate like waves,

constituting electromagnetic radiation. All other waves known at the time of Maxwell

were oscillations of some physical object, so it was thought that light consisted of

oscillations in a medium called the “luminiferous ether.” The assumption that such

a medium exists was abandoned after Michelson and Morley demonstrated that the

speed of light has the same value for observers moving with different velocities. We

now think of electromagnetic waves as oscillations that do not require any supporting

medium.

Aplane polarizedelectromagnetic wave traveling in theydirection can have an

electric field that oscillates in theyzplane and a magnetic field that oscillates in the

xyplane. In a medium with zero electrical conductivity (a perfect insulator or a vacuum),

the following equations for such a wave follow from Maxwell’s equations.^8

∂^2 Ez

∂y^2

+

1

c^2

∂^2 Ez

∂t^2

 0 (F-1)

∂^2 Hx

∂y^2

+

1

c^2

∂^2 Hx

∂t^2

 0 (F-2)

where

c

1

√

εμ

(F-3)

Thepermittivityof the medium is denoted byεand thepermeabilityof the medium is

denoted byμ. The values of these quantities for a vacuum are denoted byε 0 andμ 0.

In SI units

ε 0  8. 8542 × 10 −^12 C^2 N−^1 m−^2 (F-4)

(^8) See J. C. Slater and N. H. Frank,Electromagnetism, McGraw-Hill, New York, 1947, p. 90ff, or any
other textbook on electricity and magnetism.
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