146
3 Quantum Mechanics – II
3.2.4 Simple Harmonic Oscillator
3.51 Show that the wavefunction ψ 0 (x) = A exp(−x
2
/2a
2 ) is a solution to the
time- independent Schrodinger equation for a simple harmonic oscillator
(SHO) potential.
−
2
2m
d
2
ψ/dx
2
+
1
2
mω 0 x
2
ψ = Eψ
with energy E 0 =
1
2
ω 0 , and determine a in terms of m and ω 0 .
The corresponding dimensionless form of this equation is
−d
2
ψ/dR
2
+ R
2
ψ = εψ
where R = x/a and ε = E/E 0 .
Show that putting ψ(R) = AH (R) exp(−R
2
/2) into this equation leads to
Hermite’s equation
d
2 H
dR 2 − 2R
dH
dR
+ (ε − 1) H = 0
H (R) is a polynomial of order n of the form a n R
n
+a n−2 R
n−2
+a n−4 R
n−4
+. . .
Deduce that ε is a simple function of n and that the energy levels are equally
spaced.
[Adapted from the University of London, Royal
Holloway and Bedford New College 2005]
3.52 Show that for a simple harmonic oscillator in the ground state the probability
for finding the particle in the classical forbidden region is approximately 16%
3.53 Determine the energy of a three dimensional harmonic oscillator.
3.54 Show that the zero point energy of a simple harmonic oscillator could not be
lower than ω/2 without violating the uncertainty principle.
3.55 Show that when n → ∞ the quantum mechanical simple harmonic oscillator
gives the same probability distribution as the classical one.
3.56 Derive the probability distribution for a classical simple harmonic oscillator
3.57 The wave function (unnormalized) for a particle moving in a one dimensional
potential well V (x) is given by ψ(x) = exp(−ax
2
/2). If the potential is to
have minimum value at x = 0, determine (a) the eigen value (b) the potential V (x).
3.58 Show that for simple harmonic oscillator Δx.Δ p x = (n + 1/2), and that this
is in agreement with the uncertainty principle.
3.59 In HCl gas, a number of absorption lines have been observed with the following wave numbers (in cm
−1 ): 83.03, 103.73, 124.30, 145.03, 165.51, and
185.86.
Are these vibrational or rotational transitions? (You may assume that transitions involve quantum numbers that change by only one unit). Explain your
3 Quantum Mechanics – II
3.2.4 Simple Harmonic Oscillator
3.51 Show that the wavefunction ψ 0 (x) = A exp(−x
2
/2a
2 ) is a solution to the
time- independent Schrodinger equation for a simple harmonic oscillator
(SHO) potential.
−
2
2m
d
2
ψ/dx
2
+
1
2
mω 0 x
2
ψ = Eψ
with energy E 0 =
1
2
ω 0 , and determine a in terms of m and ω 0 .
The corresponding dimensionless form of this equation is
−d
2
ψ/dR
2
+ R
2
ψ = εψ
where R = x/a and ε = E/E 0 .
Show that putting ψ(R) = AH (R) exp(−R
2
/2) into this equation leads to
Hermite’s equation
d
2 H
dR 2 − 2R
dH
dR
+ (ε − 1) H = 0
H (R) is a polynomial of order n of the form a n R
n
+a n−2 R
n−2
+a n−4 R
n−4
+. . .
Deduce that ε is a simple function of n and that the energy levels are equally
spaced.
[Adapted from the University of London, Royal
Holloway and Bedford New College 2005]
3.52 Show that for a simple harmonic oscillator in the ground state the probability
for finding the particle in the classical forbidden region is approximately 16%
3.53 Determine the energy of a three dimensional harmonic oscillator.
3.54 Show that the zero point energy of a simple harmonic oscillator could not be
lower than ω/2 without violating the uncertainty principle.
3.55 Show that when n → ∞ the quantum mechanical simple harmonic oscillator
gives the same probability distribution as the classical one.
3.56 Derive the probability distribution for a classical simple harmonic oscillator
3.57 The wave function (unnormalized) for a particle moving in a one dimensional
potential well V (x) is given by ψ(x) = exp(−ax
2
/2). If the potential is to
have minimum value at x = 0, determine (a) the eigen value (b) the potential V (x).
3.58 Show that for simple harmonic oscillator Δx.Δ p x = (n + 1/2), and that this
is in agreement with the uncertainty principle.
3.59 In HCl gas, a number of absorption lines have been observed with the following wave numbers (in cm
−1 ): 83.03, 103.73, 124.30, 145.03, 165.51, and
185.86.
Are these vibrational or rotational transitions? (You may assume that transitions involve quantum numbers that change by only one unit). Explain your