3.2 Problems 147
reasoning briefly. (a) If the transitions are vibrational, estimate the spring con-
stant (in dyne/cm) (b) If the transitions are rotational, estimate the separation
betweenHand Cl nuclei. WhatJvalues do they correspond to, and what is
the moment of inertia of HCl (in g-cm^2 )?
[Arizona State University 1996]3.60 Determine the degeneracy of the energy levels of an isotropic harmonic oscil-
lator.
3.61 At timet=0, particle in a harmonic oscillator potentialV(x)=mω
(^2) x 2
2 has a
wavefunction
ψ(x,0)=
(
1
√
2
)
[ψ 0 (x)+ψ 1 (x)]whereψ 0 (x) andψ 1 (x) are real ortho-normal eigen functions for the ground
and first-excited states of the oscillator. Show that the probability density
|ψ(x,t)|^2 oscillates with angular frequencyω.3.62 The quantum state of a harmonic oscillator has the eigen-functionψ(x,t)=
(
1
√
2
)
ψ 0 (x)exp(
−
iE 0 t
)
+
(
1
√
3
)
ψ 1 (x)exp(
−
iE 1 t
)
+
(
1
√
6
)
ψ 2 (x)exp(
−
iE 2 t
)
whereψ 0 (x),ψ 1 (x) andψ 2 (x) are real normalized eigen functions of the har-
monic oscillator with energyE 0 ,E 1 andE 2 respectively. Find the expectation
value of the energy.
3.63 (a) Show that the wave-functionψ 0 (x)=Aexp(−x^2 / 2 a^2 ) with energyE=
ω/2 (whereAandaare constants) is a solution for all values ofxto the
one-dimensional time-independent Schrodinger equation (TISE) for the
simple harmonic oscillator (SHO) potentialV(x)=mω^2 x^2 / 2
(b) Sketch the functionψ 1 (x)=Bxexp(−x^2 / 2 a^2 )
(whereB=constant), and show that it too is a solution of the TISE for
all values ofx.
(c) Show that the corresponding energyE=(3/2)ω
(d) Determine the expectation value<px>of the momentum in stateψ 1
(e) Briefly discuss the relevance of the SHO in describing the behavior of
diatomic molecules.3.2.5 Hydrogen Atom...................................
3.64 Find the expectation value of kinetic energy, potential energy, and total energy
of hydrogen atom in the ground state. Takeψ 0 = e
−r/a 0
(πa^30 )^1 /^2
, wherea 0 =Bohr’s
radius