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Transcript of Townsend, Problem 10
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8/17/2019 Townsend, Problem 10
1/22
Physics
412
Homework Set No.
t.
Townsend,
Problem
10.3,
page
303;
Townsend,
Problem
10.6,
page
303.
3
. Evaluate explicitly the energy eigenfunctions for the
|
ls)
,
I
Zsl
,lZ
pl,
|
3s)
bound
states
of
the
hydrogen atom.
Verify
that
these
eigenfunctions are
orthogonal to each other
(it's
enough
to
check
this result for
just
two of
these eigenfunctions). Don't
forget to
normalize
the eigenstates Note: You must
use
the
recursion relation
derived
in
class.
4. Evaluate the mean value of the
(measurements
of
the)
position
of the
electron for
the
ground
state
(n:l)
and first
excited
state(s)
(n:2)
of
the hydrogen
atom.
(We
can
,
interpret
the result
as
providing
a
measure
of
the
"mean distance"
between
the
electron
and the
proton,
for both
states.)
Note the dependence
of
your
result on the
state.
5. a)
Wite
down
the
energy eigenfuncti ons
Qn1.(;)
=
(;
I
nlml for the
bound
states
of
a
spinless
particle
in
the
presence
of
a
three-dimensional
"spherical
square-well
,"
i.e.,
a
central
potential
ofthe
form
,
-Vo,
r
Ea
v
(;):
0, rla,
where Vo > 0
(i.e.,
the
potential
is attractive
-it
tries to bind the
particle).
No
need
to
normalize
the
eigenfunctions. Obtain
the
transcendental equation from which
the
corresponding
energy
eigenvalues
would
be obtained
(numerically).
Note:
Remember
that
the boundary
condition
which
must be satisfied by
the radial wave
function
at
the
origin
is
equivalent
to thinking
that there
is
an
infinite
barrier
for
r
=0.
b)
Obtain
an
equation
relating the
parameters
of
the
well
(its
depth and
its
width)
such that the
ground
state
is barely
bound, i.e., its
binding energy
approaches zero.
(Hint:
Obtain first the
general
transcendental
equation from which
this eigenvalue
is
to
be determined,
and
then
take the appropriate
limit.) Recall
that a similar
problem
was solved in the case
of
a one-dimensional square
well
potential
9'
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