100
2 Quantum Mechanics – I
2.65 The J = 0 → J = 1 rotational absorption line occurs at wavelength 0.0026
in C
12 O
16 and at 0.00272 m in C
x O
16 . Find the mass number of the unknown
Carbon isotope.
2.66 Assuming that the H
2 molecule behaves like a harmonic oscillator with force
constant of 573 N/m. Calculate the vibrational quantum number for which the
molecule would dissociate at 4.5 eV.
2.2.7 Commutators
2.67 (a) Show that e
i pα/ x e
−i pα/
= x + α
(b) If A and B are Hermitian, find the condition that the product AB will be
Hermitian
2.68 (a) If A is Hermitian, show that e
i A is unitary
(b) What operator may be used to distinguish between
(a) e
ikx and e
−ikx (b) sin ax and cos ax?
2.69 (a) Show that exp (iσ xθ) = cos θ + iσ x sin θ
(b) Show that
d
dx
† = −
d
dx
2.70 Show that
(a) [x, p x ] = [y, p y ] = [z, p z ] = i
(b) [x
2
, p x ] = 2ix
2.71 Show that a hermitian operator is always linear.
2.72 Show that the momentum operator is hermitian
2.73 The operators P and Q commute and they are represented by the matrices
1 2
2 1
and
3 2
2 3
. Find the eigen vectors of P and Q. What do you notice
about these eigen vectors, which verify a necessary condition for commuting
operators?
2.74 An operator ˆ
A is defined as ˆ
A = α ˆ
x + iβ ˆ
p, where α , β are real numbers
(a) Find the Hermitian adjoint operator ˆ
A
†
(b) Calculate the commutators [ ˆ
A, ˆ
x], [ ˆ
A, ˆ
A] and [ ˆ
A, ˆ
P]
2.75 A real operator A satisfies the lowest order equation.
A
2
− 4A + 3 = 0
(a) Find the eigen values of A (b) Find the eigen states of A (c) Show that A
is an observable.
2.76 Show that (a) [x, H ] =
i p
μ
(b) [[x, H ], x] =
2
μ
where H is the Hamiltonian.
2.77 Show that for any two operators A and B,
[A
2
, B] = A[ A, B] + [A, B]A
2 Quantum Mechanics – I
2.65 The J = 0 → J = 1 rotational absorption line occurs at wavelength 0.0026
in C
12 O
16 and at 0.00272 m in C
x O
16 . Find the mass number of the unknown
Carbon isotope.
2.66 Assuming that the H
2 molecule behaves like a harmonic oscillator with force
constant of 573 N/m. Calculate the vibrational quantum number for which the
molecule would dissociate at 4.5 eV.
2.2.7 Commutators
2.67 (a) Show that e
i pα/ x e
−i pα/
= x + α
(b) If A and B are Hermitian, find the condition that the product AB will be
Hermitian
2.68 (a) If A is Hermitian, show that e
i A is unitary
(b) What operator may be used to distinguish between
(a) e
ikx and e
−ikx (b) sin ax and cos ax?
2.69 (a) Show that exp (iσ xθ) = cos θ + iσ x sin θ
(b) Show that
d
dx
† = −
d
dx
2.70 Show that
(a) [x, p x ] = [y, p y ] = [z, p z ] = i
(b) [x
2
, p x ] = 2ix
2.71 Show that a hermitian operator is always linear.
2.72 Show that the momentum operator is hermitian
2.73 The operators P and Q commute and they are represented by the matrices
1 2
2 1
and
3 2
2 3
. Find the eigen vectors of P and Q. What do you notice
about these eigen vectors, which verify a necessary condition for commuting
operators?
2.74 An operator ˆ
A is defined as ˆ
A = α ˆ
x + iβ ˆ
p, where α , β are real numbers
(a) Find the Hermitian adjoint operator ˆ
A
†
(b) Calculate the commutators [ ˆ
A, ˆ
x], [ ˆ
A, ˆ
A] and [ ˆ
A, ˆ
P]
2.75 A real operator A satisfies the lowest order equation.
A
2
− 4A + 3 = 0
(a) Find the eigen values of A (b) Find the eigen states of A (c) Show that A
is an observable.
2.76 Show that (a) [x, H ] =
i p
μ
(b) [[x, H ], x] =
2
μ
where H is the Hamiltonian.
2.77 Show that for any two operators A and B,
[A
2
, B] = A[ A, B] + [A, B]A