Physical Chemistry Third Edition

668 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

The Time-Dependent Wave Function for a Particle in a Box

Equation (15.2) can be used to obtain the time-dependent wave function for a particle

in a one-dimensional box:

Ψn(x,t)Csin

(nπx

a

)

e−iEnt/ ̄h

Csin

(nπx

a

)(

cos(Ent/h ̄)−isin(Ent/h ̄)

)

(15.3-17)

where we use the fact that the cosine is an even function (cos(−α)cos(α)) and that

the sine is an odd function (sin(−α)−sin(α)).

The periodτof an oscillation is the time required for the argument of the sine or

cosine function in the time factor to change by 2π.

Ent/h ̄ 2 π

τ 2 πh/E ̄ nh/En 8 ma^2 /h n^2 (15.3-18)

The frequency is

v 1 /τEn/hhn^2 / 8 ma^2 (15.3-19)

Although the coordinate wave functions for the classical vibrating string and the particle

in a box are identical, the time-dependent wave functions are not. The frequency of a

vibrating string was proportional ton. The frequency of the de Broglie wave given in

Eq. (15.3-19) is proportional ton^2. However, the frequency of a de Broglie wave is

not physically meaningful. It can have a constant added to it by adding a constant to

the potential energy, as seen in the following exercise.

Exercise 15.7

a.Calculate the frequency of the de Broglie wave for then2 andn3 states of an electron

in a box of length 1.000 nm. What happens to these frequencies if a constantV 0 is added to

the potential energy in the box?

b.Calculate the difference between these frequencies. What happens to the difference between

these frequencies if a constantV 0 is added to the potential energy in the box?

c.Compare these frequencies and their difference with the photon frequency in Exercise 15.6a.

Do you think there is any simple relationship between these frequencies?

Specification of the State of a Particle in a Box

Instead of specifying the position and velocity of the particle, the state of the quantum-

mechanical particle is specified by saying what the wave function of the particle is. We

frequently say that a wave function is occupied by a system. There is a wave function

for each state and a state for each wave function. We say that there is aone-to-one

correspondencebetween a state and a wave function, and we will sometimes use the

terms “state,” “state function,” and “wave function” interchangeably.

There are two general classes of wave functions:

1. The wave function of the system is known to be an energy eigenfunction times the
   appropriate time-dependent factor as in Eq. (15.3-17). The wave function in this
   case corresponds to a standing wave. Chemists are usually interested in this case.