PHY662 - Quantum Mechanics II
HWK #2, Due Tues., Jan. 27, at the
start
of class
•
Read the section from Baym to be handed out on Thursday.
•
Complete Ch. 14 of Shankar, though you can skip pp. 393-395 (sections
on Paramagnetic Resonance and Negative Absolute Temperature).
1.
Practice on a spinor.
Make up a state for a spin-
1
2
particle. This is your
very own spinor; it has two amplitudes in it. All I ask is that it is not
an eigenstate for
Sx
or
Sy
or
Sz
and that at least one of the amplitudes
is complex. So it has a bit of character. Answer the following questions
about your spinor:
(a) Is your spinor properly normalized? If not, show how to normalize it
and carry out the rest of the exercise with the normalized spinor.
(b) What are
hSxi
,
hSyi
, and
hSzi
?
(c) What are
hS2
xi
,
hS2
yi
, and
hS2
zi
? [Check that the total is
3
4¯h2
.]
(d) What do you get if you apply
S+
to your spinor? What if you apply
S−
?
(e) What direction
ˆn
gives
hˆn·~
Si= ¯h/2
? (That is, what direction is
your spin pointing in?)
(f) What would the elements of your spinor be if you rotated it by an
angle of
π
3
about the
ˆz
-axis?
2.
Using real units.
This problem is an exercise in using real numbers.
(a) Exercise 14.4.4 in Shankar.
(b) Repeat (a) using a neutron, rather than an electron.
(c) Use your spinor from problem #1. Suppose your spinor represents a
proton. If you turn on a magnetic eld of 0.5 Tesla in the
ˆx
-direction
how long do you need to leave it on to rotate the spin direction of
your proton by
π
2
?
(d) Assume silver atoms are entering a Stern-Gerlach apparatus. They
have been heated in an oven to a temperature of 1200 K.
i. What is the average speed of the silver atoms?
ii. If these atoms pass through a S-G apparatus of length 1m, with a
eld gradient of about 100 Tesla/m transverse to the velocity of
the atoms, what would be the approximate separation between
the two beams? How easily noticeable is this separation?
3.
Detecting snoopers.
An exercise in cryptography and measurement theory.
Suppose Alice and Bob try to establish a one-time pad key using the pro-
tocol described in class. Discuss how Alice and Bob can detect snoopers,
as suggested in item #5 in the list in Sec 5.2 of the class meeting outline
for Tuesday, Jan. 20 (#03).
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