Physical Chemistry Third Edition

15.4 The Quantum Harmonic Oscillator 679

EXAMPLE15.7

Find the classical amplitude of vibration of a hydrogen molecule with an energy equal to that

of thev0 quantum state. Express it as a percentage of the bond length, 0. 74 × 10 −^10 m.

The force constantkis equal to 576 N m−^1 and the reduced mass is equal to 8. 363 × 10 −^28 kg

(half the mass of a hydrogen nucleus).

Solution

From Table A.22 of Appendix A, the frequency of vibration is

ν(4401.2cm−^1 )(2. 9979 × 1010 cm s−^1 ) 1. 319 × 1014 s−^1

For thev0 state,

E 1 

1

2

hv

1

2

(6. 6261 × 10 −^34 J s)(1. 319 × 1014 s−^1 ) 4. 371 × 10 −^20 J

The amplitude of oscillation corresponds to

Vmax

1

2

kx^2 tE

wherextis the value ofxat the turning point.

xt

√

2 E

k



(

2(4. 371 × 10 −^20 J)

576 N m−^1

) 1 / 2

 1. 23 × 10 −^11 m

This value is 16.6% of the bond length.

PROBLEMS

Section 15.4: The Quantum Harmonic Oscillator

15.19a.Construct an accurate graph of thev2 wave
function of a harmonic oscillator as a function of

√

ax.

b.Express the value of the wave function at the turning

point in terms of the parametera.

c.Ifmis equal tomp/2, the reduced mass of the H 2

molecule, and ifk 574 .75 N m−^1 , the value for the

H 2 molecule, find the value ofψ 22 at the turning point.

Express your value forψ^22 in m−^1 and in

Å

− 1

(1 Å 10 −^10 m).

15.20a.Construct an accurate graph of thev3 wave
function of a harmonic oscillator as a function of
√
ax.
b.Find the value of the wave function at the turning
point.

15.21Using the recursion relation, Eq. (15.4-6), obtain the
unnormalized energy eigenfunctionsψ 4 andψ 5 for the
harmonic oscillator.

15.22 a.Sketch a graph of the product ofψ 0 andψ 1 , the first

two energy eigenfunctions of a harmonic oscillator

and argue from the graph that the two functions are

orthogonal, which means for these real functions

that

∫∞

−∞

ψ 1 (x)ψ 2 (x)dx 0

b.Work out the integral and show thatψ 0 andψ 1 are

orthogonal.

15.23 a.Construct an accurate graph ofψ^22 , the square of the

third energy eigenfunction of the harmonic

oscillator.

b.Find the locations of the relative maxima and minima

inψ^22 in terms of the parametera.

c.Express the locations of the relative maxima as

fractions of the classical turning point for the energy

corresponding to this energy eigenfunction.