PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 6
1. [10 points] The quantum state of a freeparticle of mass,
M
, at time
t
is a wavepacket of the form
ψ
(
x,t
) =
1
radicalbig
Γ(5
/
4)
σ
0
e
−
(
x

x
0
)
4
2
σ
4
0
+
ip
0
x/
planckover2pi1
,
We can safely predict that the width of the wave packet will grow in time. Clearly, the spreading
velocity
v
s
must be determined by the initial conditions
x
0
,
p
0
, and
σ
0
, as well as the mass,
M
, and
Planck’s constant,
planckover2pi1
, as there are no other parameters around to use.
(a) [2 pts] If there are no external forces acting on the particle, which parameters can we rule out
based on symmetry arguments ?
The parameters
x
0
and
p
0
are depend on choice of coordinate system and inertial frame. The
spread velocity of the wavefunction is not frame dependent.
The basic equations for a free
particle are invariant under boost and translation, therefore the spreading dynamics should not
depend on framedependant parameters. Thus we conclude
v
s
does not depend on
x
0
and
p
0
.
(b) [2 pts] Of the remaining parameters, how many unique ways are there to combine them to make
an object with units of a velocity?
The remaining parameters are
planckover2pi1
,
M
, and
σ
0
. Thus units of
planckover2pi1
are kg m
2
s
−
1
, thus any combination
giving a velocity must depend only on
planckover2pi1
/M
, so that kg is cancelled.
planckover2pi1
/M
has units m
2
s
−
1
, and
the only parameter left is
σ
0
, which has units of length. Since we need a s
−
1
for velocity, the
only possibility is
v
=
planckover2pi1
Mσ
0
.
(1)
(c) [2 pts] Based on this result alone, give a unitsbased estimate for the velocity at which the
wavepacket should spread.
The spread velocity must be
v
s
∼
planckover2pi1
Mσ
0
,
(2)
as there are no other possibilities.
(d) [2 pts] Again, by considering units alone, what energy scale,
E
s
, would you associate with a
wavepacket of width
σ
0
?
From the width
σ
0
, and the constants
planckover2pi1
and
M
, the only energy we can form is
E
s
=
planckover2pi1
2
Mσ
2
0
.
(3)
(e) [2 pts] We can assign a temperature to the wavepacket by setting
E
s
=
k
B
T
. Solve this equation
for
σ
0
as a function of temperature,
T
. This is known as the thermal de Broglie wavelength,
or the thermal coherence length, usually denoted as
λ
coh
. It gives the lengthscale on which a
particle at temperature
T
exhibits spatial coherence (quantum superposition).
If we set
k
B
T
=
planckover2pi1
2
/
(
Mσ
2
0
), we find
λ
coh
=
planckover2pi1
√
Mk
B
T
.
(4)
This increases as the temperature decreases, which makes some kind of intuitive sense.
1