DEPARTMENT OF PHYSICSINDIAN INSTITUTE OF TECHNOLOGY, MADRASPH5100 Quantum Mechanics  1
Assignment 1(30.7.2014) To be discussed on: 7.8.2014
One Dimensional Problems in Quantum Mechanics
1. At time t = 0 the state of a free particle is speciﬁed by a wavefunctionΨ(
x,
0) =
Ae
−
x
2
a
2
+
ik
0
x
. Find the factor A and the region where the particle is localized. Determine the probabilty current density J. Find thefourier transform of the wavefunction and the width of the wavepacketin kspace.2. Normalize the wavefunction Ψ =
e
−
x

sinαx
.3. Find the eigenfunctions and eigenvalues for sin
ddφ
4. Find the wavefunction and the allowed energy levels for a particle in apotential ﬁeld V(x) of the form
V
(
x
) = 0 for 0
≤
x
≤
a,
0
≤
y
≤
b,
0
≤
z
≤
c,
=
∞
for
x <
0
,x > a,y <
0
,y > b,z <
0
,z > c
5. Consider the onedimensional timeindependent Schrodinger equationfor some arbitrary potential V(x). Prove that if a solution Ψ(
x
) hasthe property that Ψ(
x
)
→
0 as
x
→ ±∞
, then the solution must benondegenrate and therefore real, apart from a possible overall phasefactor.6. Consider the onedimensional problem of a particle of mass m in a potential
V
=
∞
,x <
0;V = 0, 0
≤
x
≤
a
;V = V
0
,x > a.
(a) Show that the bound state energies (
E < V
0
) are given by the equation tan
√
2
mEa
=
−
E V
0
−
E
. (b) Without solving any further, sketch theground state wavefunction.1
7. A particle of mass m moves nonrelativistically in one dimension in apotential given by
V
(
x
) =
−
aδ
(
x
). The particle is bound. Find thevalue of
x
0
such that the probability of ﬁnding the particle

x

< x
0
isexactly equal to 1/2.8. A particle of mass m moving in one dimension is conﬁned to the region0
< x < L
by an inﬁnite square well potential. In addition, the particleexperiences a delta function potential
V
=
λδ
(
x
−
L/
2). Find thetranscendental equation for the energy eigenvalues E in terms of themass m, the potential strength
λ
, and the size L of the system.9. A particle, moving in one dimension, has a ground state function (notnormalized and do not normalize) given by Ψ
0
(
x
) =
e
−
α
4
x
44
(Where
α
is a real constant) belonging to the energy eigenvalue
E
0
=
2
α
2
m
.Determine the potential in which the particle moves.10. A particle moves in a potential, in one dimension, of the formV(x) =
∞
if
x
2
> a
2
else V(x) =
γδ
(
x
), where
γ >
0. For suﬃcientlylarge
γ
, calculate the time required for the particle to tunnel from beingin the ground state of the well extending from x = a to x = 0 to theground state of the well extending from x=0 to x=a.2