Physical Chemistry Third Edition

698 16 The Principles of Quantum Mechanics. II. The Postulates of Quantum Mechanics

If we choose a value of a constant multiplying a wave function such that

∫

Ψ∗Ψdq1 (definition of normalization) (16.4-5)

the wave functionΨis then said to benormalized. If a normalized wave function is

used in Eq. (16.4-1), the denominator in the equation equals unity and

〈A〉

∫

Ψ∗ÂΨdq (ifΨis normalized) (16.4-6)

Some of the coordinate wave functions that we have written are multiplied bynormal-

ization constantssuch that the wave functions are normalized.

EXAMPLE16.10

Show that if

Ψ(q,t)ψ(q)e−iEt/h ̄

and ifΨis normalized thenψis also normalized.

Solution

∫

Ψ∗Ψdq 1 

∫

ψ∗(q)eiEt/h ̄ψ(q)e−iEt/ ̄hdq

e−iEt/h ̄eiEt/h ̄

∫

ψ∗(q)ψ(q)dq

∫

ψ∗(q)ψ(q)dq

EXAMPLE16.11

Show that the particle-in-a-box energy eigenfunction given in Eq. (16.4-10) is normalized.

Solution

Outside of the range 0<x<a, the wave function vanishes. Therefore, we integrate only

over the range 0<x<a:

∫∞

−∞

ψn∗ψndx

∫a

0

ψn∗ψndx

(

2

a

)∫a

0

sin^2 (nπx/a)dx

(

2

a

)(

a

nπ

)∫nπ

0

sin^2 (y)dy



(

2

nπ

)∫nπ

0

sin^2 (y)dy

2

nπ

[

y

2

−

sin(2y)

4

]nπ

0



2

nπ

[

nπ

2

−

sin(2nπ)

4

− 0 + 0

]



2

nπ

nπ

2

 1

where we have looked up the integral in Appendix C.

The Predictable Case and the Statistical Case

Part a of the fourth postulate states that if a mechanical variable is measured without

experimental error, the outcome must be an eigenvalue of the operator corresponding

to that variable. It does not tell which eigenvalue will occur in a single measurement