BioPHYSICAL chemistry

state is also symmetrical but with the wave-

function having a change in sign, correspond-

ing to negative parity. The change in sign for

the wavefunction as it passes through x=L/2

means that the wavefunction must beexactly

zero at x=L/2. The higher-order wavefunc-

tions all have parity, alternating between

positive and negative.

Wavelength

Each solution to Schrödinger’s equation pos-

sesses a wavelength given by:

(10.17)

According to the de Broglie relation, the wave-

length is related to the momentum:

(10.18)

Since the momentum is also related to kinetic energy, the de Broglie rela-

tion predicts the following values of energy:

(10.19)

As expected, these energies agree exactly with those derived using

Schrödinger’s equation.

Probability

The probability of finding a particle at any given position in the box

varies depending upon the position and the quantum number of the

wavefunction. For the ground-state wavefunction, the probability is zero

at x=0 and increases until it reaches a maximum at x=L/2. Due to the

symmetry, the probability is equal for finding a particle at equal distances

from the center. The total probability of finding the particle is set to one

so the probability of finding the particle in a smaller region must be less

than one. We can calculate the probability for any region: for example,

the probability between x=0 and x=l is given by:

ψψ (10.20)

π

*dxxx d()() sin

L

nx

L

x

ll

0

2

0

2

∫∫=

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

E

p

mm

nh

L

nh

mL

==

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

2 2 22

2 2

1

28

p

hnh

L

==

λ 2

Ln

L

n

==

λ

λ

2

2

or

202 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

ψ 5 (x)  ^2 sin^5 πx  parity

n  5

aa

ψ 2 (x)  ^2 sin^2 πx  parity

n  2

aa

ψ 1 (x)  ^2 sin πx  parity

n  1

0 a

aa

x

Figure 10.3
Symmetry of the
wavefunctions for
the particle in a box
for n=1, 2, and 5.