Assignment 1
Quantum Mechanics (PHS 2203)
Course Instructors: Dr. Mintu Mondal
Teaching assistant: Mr. Deep Singha Roy, Mr. Soham Das, Mr. Tuhin
Debnath
1. A particle of mass m is in the ground state of one dimensional oscillator
with frequency ω. If the spring constant suddenly quadruples, without
initially changing the wave function, then find out the probability that
a measurement of energy would return the value ℏω.
2. A particle of mass m is in the ground state of a one dimensional symmetric infinte potential box spanning from −a to a. What is the probability of finding the particle between − na to na , where n ≥1.
3. A particle of mass is trapped in a potential of the form,
V (x) = V0 Lδ(x3 − 7x + 6), [− inf ≤ x ≥]
where V0 and L are constants. Check the continuity of
(0.1)
dψ
dx .
4. The wavefunction of a particle is real and is given by the following figure
4.Calculate the expectation value of position operator and momentum
operator.
1
5. If a lowering operator is defined as
r
â =
mω
ip̂
x̂ + √
2ℏ
2ℏmω
(0.2)
If f (x) is any function of x, then calculate [â, f (x)].
6. Calculate the ground state energy for a particle of mass m sliding frictionlessly around a circular ring of circumference L.
7. Consider a particle bound in a potential V (x) = αx4 . Calculate the
rate of expectation value of an operator i.e. defined as Ô = αtx̂
8.
9. Consider a particle of mass m that is bouncing vertically and elastically
on a perfectly reflecting hard floor under the following potential
(
V (x) =
if 0 ≤ x,y,z ≤ 2L
otherwise
mgz
∞,
where α is a parameter Calculate the ground state energy.
10. The eigenstates of a system are described by
r
ψn (x) =
1
nπx
cos
; |x| ≤ a[n = 1, 3, 5.....]0; otherwise
a
2a
(0.3)
Calculate the probability of finding the particle within a distance
1) from the centre.
a
m (m >
11. A particle is in an infinite potential well
V (x) = 0[0 < x < a]
(0.4)
and ∞ otherwise The particle starts out in a state ψ(x) = A[ √1 ψ1 (x)−
2
√
3
2 ψ3 (x) + ψ5 (x)],
where ψn denote its eigenstates. Calculate the ex2
pectation value of energy of the particle at t = 2ma
ℏπ 2 .
2
12. Consider 11 non-interacting spin − 23 particles placed in an infinite potential well of width L. How much energy is required to take the system
from its ground state to first excited state ?
13. Consider a particle of mass m in a 1D infinte potential well spanning
from x = 0 to x = 4L. If the wavefunction of the particle is
(
Ψ (x) =
A sin3 πx
L
0,
in between 0 and 4L
otherwise
Then calculate the difference between the maximum and minimum
possible values of energy for the particle.
14. When 900 particles of mass m of energy E(≥ V0 ) are incident on a
π 2 ℏ2
1D barrier of width a and height V0 = 8ma
2 , a hundred of them are
reflected back. What is the energy of the incident beam.
3