Physical Chemistry Third Edition

1276 F Some Mathematics Used in Quantum Mechanics

and

μ 0  4 π× 10 −^7 newton ampere−^2  4 π× 10 −^7 NA−^2 (exact value by definition)

 12. 566370614 × 10 −^7 NA−^2 (F-5)

The quantitycis the speed of propagation of the wave. The theory of Maxwell correctly

predicts the value of the speed of light in a vacuum:

c

1

√

(8. 8542 × 10 −^12 C^2 N−^1 m−^2 )(12. 56637 ...× 10 −^7 NA−^2 )

 2. 9979 × 108 ms−^1

There is an additional condition from Maxwell’s equations that makes Eqs. (F-1)

and (F-2) interdependent:

Ez

Hx

±

√

μ

ε

(F-6)

The electric field cannot oscillate without oscillation of the magnetic field, and vice

versa.

A traveling-wave solution to Eqs. (F-1) and (F-2) is

Ez(y,t)E 0 sin[2π(y−ct)/λ] (F-7)

Hx(y,t)H 0 sin[2π(y−ct)/λ] (F-8)

whereE 0 andH 0 are constants that obey Eq. (F-6). The wavelengthλcan take on any

real value.

An electromagnetic wave cannot penetrate a perfect conductor (one with zero resis-

tance) since a finite electric field would produce an infinite current. Electromagnetic

waves must have nodes at perfectly conducting walls, and will approximately vanish

at a real conducting wall. Reflection between walls in a conducting cavity can produce

standing electromagnetic waves.

F. 2 The Particle in a Three-Dimensional Box

In Section 15.3 we discussed a model system consisting of a single point-mass particle

absolutely confined in a three-dimensional rectangular box. We now go through the

mathematics of the Schrödinger equation for this system. We choose the potential

energy function:

V 

{

0if0<x<aand 0<y<band 0<z<c

V 0 otherwise (outside the box)

(F-9)

and then take the limit thatV 0 approaches+∞.

We divide our space into two regions: region I, inside the box, and region II, outside

the box. For reasons exactly the same as in the one-dimensional case, the coordinate

wave function vanishes in region II. The time-independent Schrödinger equation for

the interior of the box is

(

∂^2 ψ

∂x^2

+

∂^2 ψ

∂y^2

+

∂^2 ψ

∂z^2

)

−

2 mE

h ̄

2 ψ (F-10)