3.2 Problems
147
reasoning briefly. (a) If the transitions are vibrational, estimate the spring constant (in dyne/cm) (b) If the transitions are rotational, estimate the separation
between H and Cl nuclei. What J values 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 oscillator.
3.61 At time t = 0, particle in a harmonic oscillator potential V (x) =
mω
2 x
2
2
has a
wavefunction
ψ(x, 0) =
1
√
2
[ψ 0 (x) + ψ 1 (x)]
where ψ 0 (x) and ψ 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 frequency ω.
3.62 The quantum state of a harmonic oscillator has the eigen-function ψ(x, t) =
1
√
2
ψ 0 (x) exp
−
i E 0 t
+
1
√
3
ψ 1 (x) exp
−
i E 1 t
+
1
√
6
ψ 2 (x) exp
−
i E 2 t
where ψ 0 (x), ψ 1 (x) and ψ 2 (x) are real normalized eigen functions of the harmonic oscillator with energy E 0 , E 1 and E 2 respectively. Find the expectation
value of the energy.
3.63 (a) Show that the wave-function ψ 0 (x) = A exp(−x
2
/2a
2 ) with energy E =
ω/2 (where A and a are constants) is a solution for all values of x to the
one-dimensional time-independent Schrodinger equation (TISE) for the
simple harmonic oscillator (SHO) potential V (x) = mω
2 x
2
/2
(b) Sketch the function ψ 1 (x) = Bx exp(−x
2
/2a
2 )
(where B = constant), and show that it too is a solution of the TISE for
all values of x.
(c) Show that the corresponding energy E = (3/2)ω
(d) Determine the expectation value < p x > of the momentum in state ψ 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. Take ψ 0 =
e
−r/a 0
(π a
3
0 )
1/2 , where a 0 = Bohr’s
radius
147
reasoning briefly. (a) If the transitions are vibrational, estimate the spring constant (in dyne/cm) (b) If the transitions are rotational, estimate the separation
between H and Cl nuclei. What J values 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 oscillator.
3.61 At time t = 0, particle in a harmonic oscillator potential V (x) =
mω
2 x
2
2
has a
wavefunction
ψ(x, 0) =
1
√
2
[ψ 0 (x) + ψ 1 (x)]
where ψ 0 (x) and ψ 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 frequency ω.
3.62 The quantum state of a harmonic oscillator has the eigen-function ψ(x, t) =
1
√
2
ψ 0 (x) exp
−
i E 0 t
+
1
√
3
ψ 1 (x) exp
−
i E 1 t
+
1
√
6
ψ 2 (x) exp
−
i E 2 t
where ψ 0 (x), ψ 1 (x) and ψ 2 (x) are real normalized eigen functions of the harmonic oscillator with energy E 0 , E 1 and E 2 respectively. Find the expectation
value of the energy.
3.63 (a) Show that the wave-function ψ 0 (x) = A exp(−x
2
/2a
2 ) with energy E =
ω/2 (where A and a are constants) is a solution for all values of x to the
one-dimensional time-independent Schrodinger equation (TISE) for the
simple harmonic oscillator (SHO) potential V (x) = mω
2 x
2
/2
(b) Sketch the function ψ 1 (x) = Bx exp(−x
2
/2a
2 )
(where B = constant), and show that it too is a solution of the TISE for
all values of x.
(c) Show that the corresponding energy E = (3/2)ω
(d) Determine the expectation value < p x > of the momentum in state ψ 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. Take ψ 0 =
e
−r/a 0
(π a
3
0 )
1/2 , where a 0 = Bohr’s
radius