Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #7
Due Wednesday, Oct 15 at 2pm
Problem
7. 1
Qualitative methods.
[20 pts]
A) The potential energy V(x) for a particle is given by:
V
(
x
)
=
V
0
for x < 0
0
for 0 < x < a
V
0
/2
for a < x < 2a
V
0
for 2a < x
Sketch this potential. Assume “V
0
” and “a” have been chosen so that
0 < E
1
<
V
0
/2 <
E
2
< V
0
.
Now draw two separate sketches, one for the ground state (call it u
1
(x), with energy E
1
),
another one for the 1
st
excited state (u
2
(x), with energy E
2
).
Think carefully about, and comment on,
the features of the curves everywhere, including what happens at x=0, a, and 2a.
(As always, your job
is to convince us that you understand. A sloppy sketch without explanation may receive little or no credit.)
B) Let’s make the potential simpler, just a “finite square well”:
V
(
x
) =
+
V
0
for x < 0
0
for 0 < x < a
+V
0
for x > a
Derive an
approximate
simple expression for the number
of bound states in the well, in terms of a
and V
0
.
Your expression does NOT have to be exact, in fact no long or messy calculations are
permitted, just make a quick “intelligent crude guess” (
Hint: Just think about the energies in an
infinite square well!
)
Please also guess/briefly discuss whether your estimate is more likely to
over
estimate, or
under
estimate, the correct answer.
C) Sketch a few energy eigenstates u
n
(x) for the asymmetric potential well shown below.
In particular, sketch n = 1 (ground state), n = 2, and some large n (like
n = 10 or
so).
ALSO
, please invent/ describe a physical example
of
a system that has a potential energy graph like this.
In your sketches, clearly explain in words the important features of
the wavefunctions. (Don’t forget to discuss “wavelength”,
“amplitude”, and “curvature” behaviors)
(Note: You can check yourself
with phet.colorado.edu, go to the “Quantum bound states” sim.
Another more
complicated sim is available at
www.falstad.com/qm1d/
which might be useful for
part a, though it’s considerably trickier to use than Phet )
Important: we are
not
interested in having you copy sim outputs – we want your basic physics explanations. (Just use
sims to
check
you haven’t missed any key features!)
Problem