Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #10
Due Wednesday, Nov 5 at 2pm
Problem
10. 1
Generalized Uncertainty
[15 pts]
A) If operators Â and
B
are Hermitian, what must be true about the constant
α
in order to ensure
that
α
A, B
is
also
a
Hermitian operator?
This result is relevant to the generalized uncertainty principle (GUP, Griffiths Eq 3.62): setting
α
=1/2i, show that the right-hand-side of the GUP will indeed always be real and nonnegative.
(If the right side
weren't
real and nonnegative, would the GUP be of any use at all?)
B) Show that the generalized uncertainty principle for
x
and
H
(position and Hamiltonian) is
σ
x
s
H
h
2m
p
.
(1)
What can you
deduce
about <p> for any stationary state?
Is this uncertainty principle
useful
for stationary states?
Lastly, show that result (1) also follows immediately from Griffiths' form of the "energy-time
uncertainty principle", namely his Eq 3.72 (with an obvious choice for Q in that equation)
Problem
10. 2
Time dependence
[15 pts]
A) Apply Griffiths Eq 3.71 (page 115) to the following special cases:
Q=1, Q=H, and Q=p.
(Note that you already DID this for Q=x on last week's homework.)
In
each
case (including the Q=x case from last week), comment on the result, with particular
reference to Griffiths Equations 1.27, 1.33, 1.38, and conservation of energy (which Griffiths talks
about on the bottom of page 37)
B) Consider a particle which is a 50-50 mixture of ground state and first excited states in the infinite
square well,
Ψ
(
x
,0) =
A
(
u
1
(
x
) +
u
2
(
x
))
.
(Choose A to normalize the state.
What is the time dependence of this state?)
Compute <H> and <H
2
>, and thus the uncertainty in energy,
H
.
Simplify your result as much as possible - in particular, write your final expression so it is easy to
see how big it is compared to (E
2
-E
1
)
Problem
10. 3
Uncertainty principle for small but still macro objects.
[15 pts]
We never notice the Uncertainty Principle (
x
p
∆
∆
h
) for macroscopic objects, because Planck's
constant is so small.
Let's see how big an effect the Uncertainty Principle produces for an object that
is very small, but still large compared to atoms.
Consider a 1-micrometer diameter droplet of oil
suspended in a vacuum chamber. (You can suspend an oil droplet by charging it and applying an
electric field, like Millikan did in his famous experiment)