1. Suppose your friend drops a rock off the “Chittipara” cliff of height h. As it falls, you snap
a million photographs from “Tholicode Juntion”, at random intervals. On each picture you
measure the distance the rock has fallen. What is the average of all these distances? That is
to say, what is the time average of the distance traveled?
2. (a) Find the Fourier transform of φ(k) = Ae−a|k|−ibk, where a and b are real numbers.
(b) Find the normalization constant A.
(c) Calculate ∆x∆p.
3. Find ψ(x) for a Gaussian wave packet φ(k) = Ae−a
2
(k−k0)
2/4
, where A is the normalization
factor to be found. Calculate the probability of finding the particle in the region −a/2 < x <
a/2.

4. In Problem 3, it was shown that the Fourier transform of a Gaussian function is also a Gaus-
sian function. Using the wavefunctions ψ(x) and φ(k) shows that the Gaussian wave packet

achieves the minimum uncertainty product according to Heisenberg’s uncertainty principle.
5. Consider a one-dimensional particle which is confined within the region 0 < x < a and whose
wave function is ψ(x, t) = sin πx
a

e
−iωt
.

(a) Find the potential V (x).
(b) Calculate the probability of finding the particle in the interval a/4 < x < 3a/4.
6. Consider a physical system whose Hamiltonian H and initial state |Ψ0⟩ are given by

H = ε


0 i 0
−i 0 0
0 0 −1

 , |Ψ0⟩ =
1
√
5


1 − i
1 − i
1



where ε has the dimensions of energy.
(a) What values will we obtain when measuring the energy, and with what probabilities?
(b) Calculate ⟨Hˆ ⟩, the expectation value of the Hamiltonian.
7. The wavefunction of a particle in a state is ψ(x) = N sin (πx/a) for 0 < x < a. Evaluate
∆x∆p.
8. Consider the wave function ψ(x) = A exp(−
x
2
a
2 ) exp(ikx) where A is a real constant.

(a) Find the value of A.
(b) Calculate ⟨p⟩ for this wave function.
9. For the one-dimensional motion of a particle of mass m in a potential V (x), prove the following
relation:

d⟨x⟩
dt =
⟨px⟩
m

(1)

Explain the physical significance of these results also.

1

PHY/MSP-314

10. A particle has the wave function ψ(r) = Ne−αr, where N is a normalization factor and α is
a known real parameter.
(a) Calculate the factor N.
(b) Calculate the expectation values ⟨x⟩, ⟨r⟩.
11. Prove the famous “PHY-314 uncertainty principle”, relating the uncertainty in position to
the uncertainty in energy :

σxσH ≥ (ħ/2m)|⟨p⟩| (2)
12. The angular frequency for a wave propagating inside a waveguide is given by the following
relation;

ω = kc 
1 −
π
2
b
2k
2
−1/2

where b is the width of the waveguide. Find the phase and the group velocities of the wave.
13. Consider the following operators:

Lx =
1
√
2


0 1 0
1 0 1
0 1 0

 , Ly =
1
√
2


0 −i 0
i 0 −i
0 i 0

 , Lz =


1 0 0
0 0 0
0 0 −1



(a) What possible values can one obtain if one measures Lz?
(b) Take the state in which Lz = +1, then find ⟨Lx⟩,⟨L
2
x
⟩ and ∆Lx.

(c) Find the eigenvalues and eigenstates of Lx.
(d) Check that these operators follow the commutation relation [Li

, Lj ] = iεijkLk.

14. Compute ⟨x⟩,⟨x
2
⟩,⟨p⟩,⟨p
2
⟩,⟨T⟩,⟨V ⟩ for the state ψ0(x) =
mω
πħ
1/4
e
− mωx2
2ħ

15. Consider a quantum system described by the Hamiltonian H and initially in the state |ψ0⟩,
where;

H =
1
√
2


0 −i 0
i 3 3
0 3 0

 , |Ψ0⟩ =
1
√
5


4 − i
−2 + 5i
3 + 2i



If a measurement of the Hamiltonian H is performed, what are the possible values that can
be obtained, and what are the probabilities associated with each value?

solve it