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Physics 580
Handout 12
19 October 2010
Quantum Mechanics I
webusers.physics.illinois.edu/
∼
goldbart/
Homework 8
Prof. P. M. Goldbart
3135 (& 2115) ESB
University of Illinois
1) Free particles – optional
:
a) Write down, in the
x
basis, the timeindependent Schr¨odinger equation for a particle
of mass
m
moving in one dimension in the presence of a potential
V
(
x
)?
b) Show that, when
V
(
x
) = 0, the wave function
ψ
E,α
(
x
) =
N
exp(
iα
√
2
mEx/
¯
h
)
describes a particle with energy eigenvalue
E
and momentum eigenvalue
α
√
2
mE
,
where
E
≥
0 and
α
=
±
1.
c) The wave function in part (a) is
⟨
x

E,α
⟩
(
i.e.
, the projection on to the
x
basis of the
energy and momentum eigenket

⟩
). Derive the normalisation
N
such that the
eigenkets
{
⟩}
are orthonormal (
i.e.
,
⟨

E
′
,α
′
⟩
=
δ
(
E
−
E
′
)
δ
α,α
′
).
d) A particle is in the state
ψ
(
x
) =
N
+
e
ipx/
¯
h
+
N
−
e
−
ipx/
¯
h
.
Give the possible outcome of the observation of the energy? What possible results fol
low from observation of the momentum? What is the probability of ﬁnding momentum
p
? What is the expectation value of the current operator at the position
x
(
i.e.
, the
current density at
x
) in this state? Brieﬂy discuss the major diﬀerence between this
state and a classical state of the same energy.
2) Expansion of a box (after Shankar, 5.2.1)
: A particle is in the ground state of a
onedimensional box. Suddenly the box (symmetrically) doubles its length, leaving the state
undisturbed. Determine the probability of ﬁnding the particle in the new ground state.
3) Expansion of a box (after Shankar, 5.2.4)
: A particle is in the
n
th
excited state

n
⟩
of a onedimensional box of length
L
. Determine the force encountered as the walls are
slowly pushed in. Assume
adiabaticity
,
i.e.
, that the quantum number(s) of the state do not
vary as the properties (here, the length) of the system are adjusted. Compare your result
for the force with that obtained for a classical particle of the same energy. [Hint: You may
compute the latter via the frequency of collisions and the momentum transfer per collision.]
4) Delta function potential (after Shankar, 5.2.3)
: Consider the attractive potential
V
(
x
) =
−
aδ
(
x
). Show that it admits a bound state, and determine the the corresponding
energy eigenvalue. Sketch the wave function, indicating its essential features.
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