1000 Solved Problems in Modern Physics

146 3 Quantum Mechanics – II

3.2.4 Simple Harmonic Oscillator .........................

3.51 Show that the wavefunctionψ 0 (x) = Aexp(−x^2 / 2 a^2 ) is a solution to the

time- independent Schrodinger equation for a simple harmonic oscillator

(SHO) potential.

(

−

^2

2 m

)

d^2 ψ/dx^2 +

(

1

2

)

mω 0 x^2 ψ=Eψ

with energyE 0 =

( 1

2

)

ω 0 , and determineain terms ofmandω 0.

The corresponding dimensionless form of this equation is

−d^2 ψ/dR^2 +R^2 ψ=εψ

whereR=x/aandε=E/E 0.

Show that puttingψ(R)=AH(R)exp(−R^2 /2) into this equation leads to

Hermite’s equation

d^2 H

dR^2

− 2 R

(

dH

dR

)

+(ε− 1 )H= 0

H(R) is a polynomial of ordernof the formanRn+an− 2 Rn−^2 +an− 4 Rn−^4 +...

Deduce thatεis a simple function ofnand 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 thanω/2 without violating the uncertainty principle.

3.55 Show that whenn→∞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 wellV(x) is given byψ(x)=exp(−ax^2 /2). If the potential is to

have minimum value atx =0, determine (a) the eigen value (b) the poten-

tialV(x).

3.58 Show that for simple harmonic oscillatorΔx.Δpx=(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 fol-
lowing 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 tran-
sitions involve quantum numbers that change by only one unit). Explain your