1000 Solved Problems in Modern Physics

(Tina Meador) #1
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

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