3.2 Problems
143
Fig. 3.2 Bound states in a
square well potential
3.36 In Problem 3.25 express the normalization constant A in terms of α, β and a.
3.37 A particle of mass m is trapped in a square well of width L and infinitely deep.
Its normalized wave function within the well for the nth state is
ψ n =
2
L
1/2
sin
nπ x
L
(a) Show that its mean position is L/2 and the variance is
L
2
12
1 −
6
n 2 π 2
(b) Show that these expectations are in agreement with the classical values
when n → ∞.
3.38 The quantum mechanical Hamiltonian of a system has the form
H = (−
2
/2m)∇
2
+ ar
2
1 −
5
6
sin
2
θ cos
2
ϕ
Find the energy eigen value of the two lowest lying stationary states.
3.39 (a) Write down the three-dimensional time-independent Scrodinger equation
in Cartesian co-ordinates. By separating the variables, ψ (x, y, z) =
X (x)Y (y)Z (z), solve this equation for a particle of mass m confined to
a rectangular box of sides a, b and c, with zero potential inside.
(b) Show that the particle has energy given by
E =
2
8m
n
2
x
a 2 +
n
2
y
b 2 +
n
2
z
c 2
3.40 In Problem 3.39, consider the special case of a cube a = b = c. Draw up a
table listing the first six energy levels, stating the degeneracy for each level.
3.41 A particle of mass m is moving in a region where there is a potential step at
x = 0 : V (x) = 0 for x < 0 and V (x) = U 0 (a positive constant) for x ≥ 0
(a) Determine ψ(x) separately for the regions x 0 and x 0 for the cases:
(i) U 0 < E
(ii) U 0 > E.
(b) Write down and justify briefly the boundary conditions that ψ(x) must satisfy at the boundary between the two adjacent regions. Use these
143
Fig. 3.2 Bound states in a
square well potential
3.36 In Problem 3.25 express the normalization constant A in terms of α, β and a.
3.37 A particle of mass m is trapped in a square well of width L and infinitely deep.
Its normalized wave function within the well for the nth state is
ψ n =
2
L
1/2
sin
nπ x
L
(a) Show that its mean position is L/2 and the variance is
L
2
12
1 −
6
n 2 π 2
(b) Show that these expectations are in agreement with the classical values
when n → ∞.
3.38 The quantum mechanical Hamiltonian of a system has the form
H = (−
2
/2m)∇
2
+ ar
2
1 −
5
6
sin
2
θ cos
2
ϕ
Find the energy eigen value of the two lowest lying stationary states.
3.39 (a) Write down the three-dimensional time-independent Scrodinger equation
in Cartesian co-ordinates. By separating the variables, ψ (x, y, z) =
X (x)Y (y)Z (z), solve this equation for a particle of mass m confined to
a rectangular box of sides a, b and c, with zero potential inside.
(b) Show that the particle has energy given by
E =
2
8m
n
2
x
a 2 +
n
2
y
b 2 +
n
2
z
c 2
3.40 In Problem 3.39, consider the special case of a cube a = b = c. Draw up a
table listing the first six energy levels, stating the degeneracy for each level.
3.41 A particle of mass m is moving in a region where there is a potential step at
x = 0 : V (x) = 0 for x < 0 and V (x) = U 0 (a positive constant) for x ≥ 0
(a) Determine ψ(x) separately for the regions x 0 and x 0 for the cases:
(i) U 0 < E
(ii) U 0 > E.
(b) Write down and justify briefly the boundary conditions that ψ(x) must satisfy at the boundary between the two adjacent regions. Use these