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Physics 498/MMA
Handout 5
Fall 2007
Mathematical Methods in Physics I
http://w3.physics.uiuc.edu/
∼
mstone5
Prof. M. Stone
2117 ESB
University of Illinois
1) Missing State
: In Homework Set 4 you found that the Schr¨
odinger equation
±

d
2
dx
2

2 sech
2
x
²
ψ
=
E ψ
has eigensolutions
ψ
k
(
x
) =
e
ikx
(

ik
+ tanh
x
)
with eigenvalue
E
=
k
2
.
•
For
x
large and positive
ψ
k
(
x
)
≈
A e
ikx
e
iδ
(
k
)
, while for
x
large and negative
ψ
k
(
x
)
≈
A e
ikx
e

iδ
(
k
)
, the (complex) constant
A
being the same in both cases. Express
δ
(
k
) as
the inverse tangent of an algebraic expression in
k
.
•
Impose periodic boundary conditions
ψ
(

L/
2) =
ψ
(+
L/
2) where
L
±
1.
Find the
allowed values of
k
and hence an explicit expression for the
k
space density,
ρ
(
k
) =
dn
dk
,
of the eigenstates.
•
Compare your formula for
ρ
(
k
) with the corresponding expression,
ρ
0
(
k
) =
L/
2
π
, for
the eigenstate density of the zeropotential equation and compute the integral
Δ
N
=
Z
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This homework help was uploaded on 01/29/2008 for the course PHYS 598 taught by Professor Stone during the Fall '07 term at University of Illinois at Urbana–Champaign.
 Fall '07
 Stone
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