Physical Chemistry Third Edition

17.5 Expectation Values in the Hydrogen Atom 751

Solution

Since the wave function is normalized we can omit the denominator in the formula for the

expectation value shown in Eq. (16.4-1). We can factor the integral:

〈 1

r

〉



∫∞

0

R∗ 10

1

r

R 10 r^2 dr

∫π

0

Θ∗ 00 Θ 00 sin(θ)dθ

∫ 2 π

0

Φ∗ 0 Φ 0 dφ (17.5-7)

By our separate normalizations, the second and third integrals both equal unity, so that

〈 1

r

〉



∫∞

0

R∗ 10

1

r

R 10 r^2 dr 4

(

Z

a

) 3 ∫∞

0

e−^2 Zr/ardr 4

(

Z

a

) 3 (

a

2 Z

) 2 ∫∞

0

e−uudu

 4

(

Z

a

) 3 (

a

2 Z

) 2

(1)

Z

a

(17.5-8)

where we looked up the integral in Appendix C. Since theθand theφfunctions are both

separately normalized, we could have omitted them from the outset and used only the

radial factorRin calculating〈 1 /r〉. The same thing is true for the expectation value of any

function ofr.

We can use the result for〈 1 /r〉to calculate〈V〉:

〈V〉−

Ze^2

4 πε 0

〈

1

r

〉

−

Z^2 e^2

4 πε 0 a

 2 E 1 s (17.5-9)

Note that〈 1 /r〉is proportional toZand〈V〉is proportional toZ^2.

Exercise 17.13

Obtain numerical values for〈 1 /r〉and〈V〉for a hydrogen atom (Z1) in the 1sstate.

The expectation value of the potential energy of a hydrogen-like atom equals twice

the total energy. Therefore, the kinetic energy is half as large as the magnitude of the

potential energy, and is equal in magnitude to the total energy (the kinetic energy must

be positive while the total energy and the potential energy are negative). This behavior

occurs in all systems of particles interacting only with the Coulomb potential energy,

and is a consequence of thevirial theoremof mechanics.^4

The Radial Distribution Function

The radial distribution function,fr(r), is defined as the probability per unit value ofr

for finding the electron at a given distance from the nucleus. Ifr′is a particular value

ofr, then

⎛

⎝

probability that the distance

from the nucleus to the electron

lies betweenr′andr′+dr

⎞

⎠fr(r′)dr (17.5-10)

(^4) I. N. Levine,op. cit., p. 459ff (note 2).