Physical Chemistry Third Edition

632 14 Classical Mechanics and the Old Quantum Theory

These derivatives are ordinary derivatives sinceηandφare functions of one variable.

We divide Eq. (14.3-6) byφηto complete the separation of variables:

1

φ(x)

d^2 φ

dx^2



1

c^2 η(t)

d^2 η

dt^2

(14.3-7)

Each term now depends on only one independent variable.

Sincexandtare independent variables, it is possible to keeptfixed while we allowx

to range. The right-hand side of Eq. (14.3-7) is then equal to a constant (the separation

constant), so the term containingxmust be a constant function ofx. Similarly, the

term containingtmust be a constant function oftand must be equal to the separation

constant:

1

φ(x)

d^2 φ

dx^2

constant−κ^2 (14.3-8)

1

c^2 η(t)

d^2 η

dt^2

−κ^2 (14.3-9)

The separation constant must be negative to give an oscillatory solution. We denote

this constant by−κ^2 so thatκwill be a real quantity.

Exercise 14.4

If the separation constant is assumed to be positive, we have the equation

1

φ(x)

d^2 φ

dx^2

constanta^2

whereais a real constant.

a.Show that the following function is a solution to this equation:

φLeax+Me−ax

whereLandMare constants. This is not a periodic function.

b.Show that

ηPeact+Qe−act

which is not an oscillating function.

Multiplying Eq. (14.3-8) byφand Eq. (14.3-9) byc^2 ηgives

d^2 φ

dx^2

−κ^2 φ(x) (14.3-10)

d^2 η

dt^2

−κ^2 c^2 η(t) (14.3-11)

These equations have the same form as Eq. (14.2-14) with different symbols. We

transcribe the solution to that equation with appropriate replacements of symbols:

φ(x)Bcos(κx)+Dsin(κx) (14.3-12)

η(t)Fcos(κct)+Gsin(κct) (14.3-13)