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Physics 137B, Fall 2007
Problem set 10
: Fermi gases, molecules, and scattering
Assigned Friday, 23 November. Due
in box
Friday, 30 November.
1. Write out for yourself that the probability current
j
=
¯
h
2
im
(
ψ
*
∇
ψ

(
∇
ψ
*
)
ψ
)
(1)
satisﬁes the “continuity equation”
∂

ψ

2
∂t
+
∇ ·
j
= 0
.
(2)
If you get stuck, you can refer to class notes or the derivation in Bransden 3.2. Use this expression
for
j
to show that any wavefunction that is completely real (has no imaginary part) has vanishing
probability current.
2. In class we said that the total cross section seen by classical small particles scattering oﬀ a large
stationary hard sphere of radius
r
0
is
σ
tot
=
πr
0
2
. Use classical physics to calculate the diﬀerential
cross section
dσ
d
Ω
in this case, as a function of the scattering angle
θ
. Hint: ﬁrst show by explaing
an appropriate diagram that the impact parameter
b
is related to
θ
via
b
r
0
= cos(
θ/
2)
(3)
Now obtain
dσ
d
Ω
by writing
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This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at University of California, Berkeley.
 Fall '07
 MOORE
 mechanics, Current

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