15.4 The Quantum Harmonic Oscillator
679
E X A M P L E 15.7
Find the classical amplitude of vibration of a hydrogen molecule with an energy equal to that
of the v 0 quantum state. Express it as a percentage of the bond length, 0.74 × 10 −10 m.
The force constant k is 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 (4401.2 cm −1 )(2.9979 × 10 10 cm s −1 ) 1.319 × 10 14 s −1
For the v 0 state,
E 1
1
2
hv
1
2
(6.6261 × 10 −34 J s)(1.319 × 10 14 s −1 ) 4.371 × 10 −20 J
The amplitude of oscillation corresponds to
V max
1
2
kx 2
t E
where x t is the value of x at the turning point.
x t
√
2E
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.
P R O B L E M S
Section 15.4: The Quantum Harmonic Oscillator
15.19 a. Construct an accurate graph of the v 2 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 parameter a.
c. If m is equal to m p /2, the reduced mass of the H 2
molecule, and if k 574.75 N m −1 , the value for the
H 2 molecule, find the value of ψ 2
2 at the turning point.
Express your value for ψ 2
2 in m −1 and in
Å
−1 (1 Å 10 −10 m).
15.20 a. Construct an accurate graph of the v 3 wave
function of a harmonic oscillator as a function of
√
ax.
b. Find the value of the wave function at the turning
point.
15.21 Using the recursion relation, Eq. (15.4-6), obtain the
unnormalized energy eigenfunctions ψ 4 and ψ 5 for the
harmonic oscillator.
15.22 a. Sketch a graph of the product of ψ 0 and ψ 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 that ψ 0 and ψ 1 are
orthogonal.
15.23 a. Construct an accurate graph of ψ 2
2 , the square of the
third energy eigenfunction of the harmonic
oscillator.
b. Find the locations of the relative maxima and minima
in ψ 2
2 in terms of the parameter a.
c. Express the locations of the relative maxima as
fractions of the classical turning point for the energy
corresponding to this energy eigenfunction.
679
E X A M P L E 15.7
Find the classical amplitude of vibration of a hydrogen molecule with an energy equal to that
of the v 0 quantum state. Express it as a percentage of the bond length, 0.74 × 10 −10 m.
The force constant k is 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 (4401.2 cm −1 )(2.9979 × 10 10 cm s −1 ) 1.319 × 10 14 s −1
For the v 0 state,
E 1
1
2
hv
1
2
(6.6261 × 10 −34 J s)(1.319 × 10 14 s −1 ) 4.371 × 10 −20 J
The amplitude of oscillation corresponds to
V max
1
2
kx 2
t E
where x t is the value of x at the turning point.
x t
√
2E
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.
P R O B L E M S
Section 15.4: The Quantum Harmonic Oscillator
15.19 a. Construct an accurate graph of the v 2 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 parameter a.
c. If m is equal to m p /2, the reduced mass of the H 2
molecule, and if k 574.75 N m −1 , the value for the
H 2 molecule, find the value of ψ 2
2 at the turning point.
Express your value for ψ 2
2 in m −1 and in
Å
−1 (1 Å 10 −10 m).
15.20 a. Construct an accurate graph of the v 3 wave
function of a harmonic oscillator as a function of
√
ax.
b. Find the value of the wave function at the turning
point.
15.21 Using the recursion relation, Eq. (15.4-6), obtain the
unnormalized energy eigenfunctions ψ 4 and ψ 5 for the
harmonic oscillator.
15.22 a. Sketch a graph of the product of ψ 0 and ψ 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 that ψ 0 and ψ 1 are
orthogonal.
15.23 a. Construct an accurate graph of ψ 2
2 , the square of the
third energy eigenfunction of the harmonic
oscillator.
b. Find the locations of the relative maxima and minima
in ψ 2
2 in terms of the parameter a.
c. Express the locations of the relative maxima as
fractions of the classical turning point for the energy
corresponding to this energy eigenfunction.