Indian Institute of Technology Bombay

PHY-403 Quantum Mechanics-I 13.10.2023

Assignment 7

Notes:

1. Some problems will be discussed on 19.10.2023 (Thursday).

2. Please make sure that you do the assignment by yourself. You can con-
sult your classmates and seniors and ensure you understand the concept.
However, do not copy assignments from others.

3. During the tutorial session, TA will randomly ask a student to come to the
board to solve tutorial problems.

1. The expectation value of x̂, p̂ in x−representation are:

Z
hx̂i = dx ψ ∗ (x, t)xψ(x, t) (1)
Z  
∗ ∂
hp̂i = dx ψ (x, t) −ih̄ ψ(x, t) . (2)
∂x

Using the time-dependent Schrodinger equation, show that the expectation

value of x̂ and p̂ are related by:

dhx̂i
m = hp̂i (3)
dt

2. Consider the wave packet represented by

Z k+∆k
Ψ(x, t) = A cos (k 0 x − k 0 t/c) dk 0 (4)
k−∆k

where A, c are constants. Integrate over k 0 and show that

Ψ(x, t) = S(x − ct) cos k(x − ct) (5)

where
sin [∆k(x − ct)]
S(x − ct) = 2A∆k (6)
∆k(x − ct)
Describe the propagation properties of this wave-packet.

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3. The wavefunction of a particle of mass m is given by:

x2
 
Ψ(x) = N exp − 2 (7)
2L

where N, L are constants and L has units of Length.

(a) Show that the expectation values of the position and the square of the
position are
L2
hx̂i = 0 hx̂2 i = (8)
2
(b) Show that the expectation values of the momentum and the square of
the momentum are
h̄2
hp̂i = 0 hp̂2 i = (9)
2 L2

(c) Obtain the uncertainty in position and momentum and show that

h̄
∆x ∆p = (10)
2
4. A particle confined to a region 0 ≤ x ≤ L and has a wave function of the
form:
Ψ(x) = N x (L − x), (11)
where N is a constant.

(a) Normalize the wave function and find the mean position of the particle.
(b) Find the uncertainties in the position and momentum of the particle.
(c) What is the value of ∆x∆p?

5. Consider a particle with normalized wave function

(
N x e−αx/2 if x ≥ 0,
Ψ(x) = (12)
0 if x < 0
p
where α is a positive real constant and N = α3 /2

(a) Write down an expression for the probability of finding the particle
between x and x + dx. Illustrate how this probability depends on x
and find the most probable value of x.
(b) Find the expectation values for the position and the square of the
position, hxi and hx2 i.

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 (c) Find the expectation values for the position and the square of the
momentum, hpi and hp2 i.
(d) Show that these expectation values yield uncertainties for position
and momentumwhich are consistent with the Heisenberg uncertainty
relation.
6. Stationary and non-stationary states
A particle with the wave function
ψn (x, t) = ψn (x) e−i En t (13)
which represents a quantum state with a well-defined energy En .
(a) Show that the probability of these states do not change with time.
[Hence, they are referred to as stationary states.]
(b) Show that the probability of superposition of two energy eigen states
change with time. [Hence, they are referred to as non-stationary
states.]
(c) Calculate the energy expectation value and uncertainty in energy for
non-stationary states.
(d) Define periodicity of the oscillation (δt) between these two states from
the uncertainty in energy.
Note: δt∆E ∼ h̄ is referred to Energy-time uncertainty relation.
7. Two states of a particle with definite energy E1 and E2 are represented by
the following normalized, orthogonal solutions of the Schroedinger equation:
Ψ1 (x, t) = ψ1 (x) exp (−iE1 t) Ψ2 (x, t) = ψ2 (x) exp (−iE2 t) (14)
(a) Write down a linear superposition of Ψ1 and Ψ2 which represents the
state for which the expectation value of the energy is 41 E1 + 43 E2
(b) Find the uncertainty in energy for the state written down.
(c) Show, for the state written down, that the probability density oscillates
with time. Find the relation between the period of these oscillations
and the uncertainty in the energy.
8. The general wave function of a particle of mass m in a one-dimensional
infinite square well with width L at time t is
∞
X
Ψ(x, t) = cn ψn (x) e−i En t (15)
n=1

where ψn (x) is an eigenfunction with energy En = n2 π 2 h̄2 /(2mL2 ). After

how much time the wave function returns to its original form?

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 9. Free particle in Momentum representation

(a) Write down the Hamiltonian of a free particle.

(b) Write down the eigen-value equation for the Hamiltonian operator
interms of Momentum states. Is this equation valid for all times?
(c) Obtain hx, t|p, 0i.
(d) Obtain hx, t|ψi.
(e) Show that
i. hP (t)i = hP (0)i
t
ii. hX(t)i = hX(0)i + M
hP (0)i
(f) Calculate hP 2 (t)i, hX 2 (t)i.
(g) Calculate h∆P 2 (t)i, h∆X 2 (t)i.
R∞
(h) Calculate −∞ dxψ ∗ (x, 0)ψ(x, t)

10. A particle of mass m is subjected to the one dimensional potential


−gδ(x) x ≤ a
V (x) =
∞ x>a
(a) Find the implicit equation to find a bound state with E < 0 and the
conditions on parameters (m, g) for the existence of such a state.
(b) Discuss the limit a → ∞
11. Check that the ground-state wave function of the harmonic oscillator ψ0 (x)
satisfies the time-independent Schrödinger equation for the harmonic oscil-
lator, i.e.
h̄2 d2 ψ(x) mω 2 2
− + x ψ(x) = Eψ(x) (16)
2m dx2 2
12. Show that any initial quantum state evolving under the harmonic oscillator
Hamiltonian with frequency ω acquires, after a period τ = 2π/ω, a phase
equal to −π.
13. Time-dependence of Harmonic Oscillator
An Harmonic Oscillator in an energy state is in a stationary state. It will
not exhibit the oscillatory behavior of a classical oscillator. Let us consider
the initial state of the Harmonic oscillator to be a superposition of two
neighbouring energy states, i. e.,
|ψi = cn |ni + cn+1 |n + 1i (17)
Show that the average position and momentum of this state is given by
hx̂i = A cos(ωt + δ) hp̂x i = −mω A sin(ωt + δ) (18)