Elements of Quantum Theory
99
3. Show that the frequency of radiation emitted when the particle inside a
one-dimensional box undergoes a transition from (n + 1) state to n state,
tends to the classical frequency of motion inside the box, for n → ∞. This
is another illustration of Bohr’s correspondence principle.
4. Consider a wave function Ae
–r/a
for the ground state of the hydrogen
atom, r being the separation between the electron and the proton. Determine
A, a, and the ground state energy. What are the classical and quantum
mechanical probabilities of finding the electron at a separation greater
than 2a?
5. If a three-dimensional harmonic oscillator has a solution of the form AY 1
0
(θ, φ) re
–ar2
, determine a, A, and the energy in terms of mass and forceconstant of the oscillator.
6. For a particle inside a one-dimensional square well defined by V (x) = 0
for |x| ≥ a and V(x) = –V 0 for |x| < a, obtain a relationship between
the binding energy, a, and V 0 . Show that for V 0 → 0, there is a bound state
with energy E → –2ma
2
V 0
2
/
2
(this is a shallow bound state in the sense
that E/V 0 → 0 as V 0 → 0).
7. For a particle in a one-dimensional box, obtain the standard deviations
σ(x) and σ(p) for position and momentum respectively. Show that σ(x)
σ(p) =
1/ 2
2 2
1
12
2
n
π
−
, and that it is greater than / 2
.
8. For a particle in a one-dimensional potential well defined by V (x) = ∞ for
x < 0, V (x) = –V 0 for 0 ≤ x ≤ a and V (x) = 0 for x > a, obtain a relation
between the binding energy, a, and V 0 . Show that these states are the
same as the odd states in Problem 6.
9. Show that p z and L z are hermitian operators. Show that the commutator
[p z , L z ] = 0 and [p y , L z ] =
x
i p
. Also show that the wave function in
Problem 5 is an eigenstate of L z but not of p z or p x . Calculate the expectation
values of L z , p z , p x and p
2
for this wave function.
10. Obtain the first order perturbation to the energy of the ground state for a
one-dimensional harmonic oscillator in the presence of a perturbing potential
λx
4
.
11. Obtain the first order perturbations to the energy of the ground state and
the first excited states for a particle in a cubic box with the centre at the
origin and the edges parallel to the coordinate axes, due to a perturbing
potential λ xy .
99
3. Show that the frequency of radiation emitted when the particle inside a
one-dimensional box undergoes a transition from (n + 1) state to n state,
tends to the classical frequency of motion inside the box, for n → ∞. This
is another illustration of Bohr’s correspondence principle.
4. Consider a wave function Ae
–r/a
for the ground state of the hydrogen
atom, r being the separation between the electron and the proton. Determine
A, a, and the ground state energy. What are the classical and quantum
mechanical probabilities of finding the electron at a separation greater
than 2a?
5. If a three-dimensional harmonic oscillator has a solution of the form AY 1
0
(θ, φ) re
–ar2
, determine a, A, and the energy in terms of mass and forceconstant of the oscillator.
6. For a particle inside a one-dimensional square well defined by V (x) = 0
for |x| ≥ a and V(x) = –V 0 for |x| < a, obtain a relationship between
the binding energy, a, and V 0 . Show that for V 0 → 0, there is a bound state
with energy E → –2ma
2
V 0
2
/
2
(this is a shallow bound state in the sense
that E/V 0 → 0 as V 0 → 0).
7. For a particle in a one-dimensional box, obtain the standard deviations
σ(x) and σ(p) for position and momentum respectively. Show that σ(x)
σ(p) =
1/ 2
2 2
1
12
2
n
π
−
, and that it is greater than / 2
.
8. For a particle in a one-dimensional potential well defined by V (x) = ∞ for
x < 0, V (x) = –V 0 for 0 ≤ x ≤ a and V (x) = 0 for x > a, obtain a relation
between the binding energy, a, and V 0 . Show that these states are the
same as the odd states in Problem 6.
9. Show that p z and L z are hermitian operators. Show that the commutator
[p z , L z ] = 0 and [p y , L z ] =
x
i p
. Also show that the wave function in
Problem 5 is an eigenstate of L z but not of p z or p x . Calculate the expectation
values of L z , p z , p x and p
2
for this wave function.
10. Obtain the first order perturbation to the energy of the ground state for a
one-dimensional harmonic oscillator in the presence of a perturbing potential
λx
4
.
11. Obtain the first order perturbations to the energy of the ground state and
the first excited states for a particle in a cubic box with the centre at the
origin and the edges parallel to the coordinate axes, due to a perturbing
potential λ xy .