Physical Chemistry Third Edition

672 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

and a frequency given by

ν

2 πE

h ̄



E

h

 ̄

hκ^2

4 π^2 m



hκ^2

2 πm

(15.3-34)

A nonzero value of the constantFcorresponds to a traveling wave moving to the left.

IfDandFare equal, the two traveling waves interfere to produce a standing wave:

ψ(x)D(eiκx+e−iκx) 2 Dcos(κx) (15.3-35)

which follows from the identity

cos(α)

1

2

(

eiα+e−iα

)

(15.3-36)

The time-dependent wave function corresponding to Eq. (15.3-35) is

Ψ(x,t) 2 Dcos(κx)e−iEt/h ̄ (15.3-37)

IfD−F, a different standing wave results.

If the constantsDandFare not equal to each other the complete wave function is

Ψ(x,t)Dei(κx−Et/h ̄)+Fe−i(κx+Et/h ̄) (15.3-38)

which represents a combination of traveling waves, one moving to the right and one

moving to the left. A single wave function corresponds to motion in two different direc-

tions. This behavior is rather different from that found in classical mechanics, in which

one state always corresponds to only one kind of behavior. The idea that a single particle

can have a single state corresponding to motion in two different directions at the same

time seems unreasonable, but it is permitted in quantum mechanics. A possible inter-

pretation is that since some predictions of quantum mechanics are statistical in nature,

a wave function should be thought of as representing the behavior of a large collection

(an ensemble) of objects, all in the same state but capable of different outcomes of a

particular measurement. We will return to this question in the next chapter.

The Free Particle in Three Dimensions

The time-independent Schrödinger equation for a free particle moving in three dimen-

sions is

(

∂^2 ψ

∂x^2

+

∂^2 ψ

∂y^2

+

∂^2 ψ

∂z^2

)

−

2 mE

h ̄^2

ψ (15.3-39)

This is the same as for a particle inside a three-dimensional box, and it can be solved

in the same way by separation of variables. Each factor in the solution can be written

either in the form of Eq. (15.3-6) or in the form of Eq. (15.3-26), withκreplaced by

κx,κy,orκz. For the special case of a traveling wave with definite values ofκx,κy, and

κz, we can write a one-term energy eigenfunction as

ψ(x,y,z)Deiκxxeiκyyeiκzz (15.3-40)

The vectorκwith componentsκx,κy, andκzpoints in the direction in which the

traveling wave moves and is called thewave vector.

The energy eigenvalue is given by the sum of three terms:

EEx+Ey+Ez

h ̄^2

2 m

(κ^2 x+κ^2 y+κ^2 z)

h ̄^2 κ^2

2 m

(15.3-41)