PHYS 507
Homework VI (Fall ’05)
Assigned:
November 23, 2005, Monday.
Due:
November 30, 2005, Wednesday, at 5:00 pm.
1.
In the Hydrogen atom problem
H
=
~
p
2
/
2
m

Ze
2
/r
, the particular gauge chosen is
~
A
= 0 and
φ
=
Ze/r
. In another possible gauge (
~
˜
A,
˜
φ
), we might choose
˜
φ
= 0. In
that case the Hamiltonian becomes
˜
H
=
1
2
m
~
p

(

e
)
c
~
˜
A
¶
2
,
i.e., consisting of kinetic energy only. One price paid is
~
˜
A
=
~
˜
A
(
~
r, t
) and hence
˜
H
=
˜
H
(
t
)
is now time dependent. But this Hamiltonian still provides a valid description of the
Hydrogen atom problem.
(a)
What is
~
˜
A
(
~
r, t
)?
(b)
What is the scalar function Λ used in the gauge transformation from (
~
A, φ
) to
(
~
˜
A,
˜
φ
)?
(c)
If
ψ
is a positionspace wavefunction for a particular state of the electron in the
gauge (
~
A, φ
), what is the positionspace wavefunction
˜
ψ
that needs to be used to
describe the same state in the gauge (
~
˜
A,
˜
φ
)?
2.
Consider a particle in 1D having charge
Q
moving under a time and position inde
pendent electric field
E
. There are two frequently used Hamiltonians to describe the
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 Spring '11
 starg
 Magnetic Field, Fundamental physics concepts, Hamiltonians, Hydrogen atom problem, p2 /2m

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