Introduction to Quantum Mechanics II
Quiz 1
Name:
August 25, 2012
Consider a two-level system, for example, a spin-1/2 system. Operators
may be represented by 2 2 matrices in this system. Consider the following
matrices:
01
0 i
0 1
,
,
.
10
i0
10
Which tw
Introduction to Quantum Mechanics II
Quiz 13
Name:
November 30, 2012
Use the generating function for the spherical harmonics
1
z
2
2y
2
2x
(+ + ) + (i+ i ) + (2+ )
l l!
2
r
r
r
l
l+m lm
+
4
=
Ylm (, )
,
2l + 1
(l + m)!(l m)!
m=l
l
where + and are two arb
Introduction to Quantum Mechanics II
Quiz 12
Name:
November 16, 2012
Let
h
h
Jy () = eiJx / Jy eiJx / ,
and
h
h
Jz () = eiJx / Jz eiJx / .
By dierentiating with respect to , and solving the resulting dierential
equations, compute Jy () and Jz () in terms
Introduction to Quantum Mechanics II
Quiz 11
Name:
November 9, 2012
The general rotation operator is given in terms of Eulerian angles , ,
by
h
h
h
U (, , ) = eiJz / eiJy / eiJz / .
Compute for j = 1 the 3 3 rotation matrix
j m|U (, , )|jm
for the specia
Introduction to Quantum Mechanics II
Quiz 10
Name:
November 2, 2012
In this problem, consider the hydrogen atom, but include the spin of the
electron, a spin-1/2 particle, but disregard the spin of the nucleus. Also,
ignore the eect of the electron spin o
Introduction to Quantum Mechanics II
Quiz 9
Name:
October 26, 2012
We can construct the spin-1 states from combining two spin-1/2 systems
from the bottom up. Assume that
|j = 1, m = 1 = |m1 = 1/2, m2 = 1/2 .
Then by applying the raising operator,
1
J+ |jm
Introduction to Quantum Mechanics II
Quiz 8
Name:
October 19, 2012
The ground-state wavefunction of the hydrogen atom is characterized by
the properites
L100 (r) = 0, A100 (r) = 0.
Here
L = r p,
A=
r
1
(p L ihp),
r Ze2
where on wavefunctions p is realized
Introduction to Quantum Mechanics II
Quiz 7
Name:
October 5, 2012
Consider r, p variables satisfying
[rk , pl ] = ihkl .
The orbital angular momentum is dened by
L = r p.
Show that
pLLp
is Hermitian, but
pL+Lp
is not. Evaluate the latter in terms of p. [H
Introduction to Quantum Mechanics II
Quiz 6
Name:
September 28, 2012
Given the commutators,
[Rk , Rl ] = 0,
[Pk , Pl ] = 0,
[Rk , Pl ] = ihkl ,
and the constructions for the angular momentum and the boost generator,
J = R P + S,
N = Pt M R,
and the fact t
Introduction to Quantum Mechanics II
Quiz 5
Name:
September 21, 2012
In terms of harmonic oscillator variables satisfying
[q, p] = i,
compute
f (q ) = eiq p qeiq p ,
where q is a number, by dierentiating f (q ) with respect to q . Then,
evaluate
eiq p q n
Introduction to Quantum Mechanics II
Quiz 4
Name:
September 14, 2012
The states |n of the harmonic oscillator are eigenstates of the energy
operator or Hamiltonian,
p2 + q 2
H=
2
What are the eigenvalues of H in these states (dont derive this, just give
t
Introduction to Quantum Mechanics II
Quiz 3
Name:
September 7, 2012
Using the realization of the position and momentum operators on eigenstates of position,
q |q = q q |,
q |p =
1
q |,
i q
compute
q |[q, p].
Is this consistent with the canonical commutato
Introduction to Quantum Mechanics II
Quiz 2
Name:
August 31, 2012
Starting from (J = Jx iJy )
1
J |jm =
h
(j
m)(j m + 1)|jm 1 ,
specialize to the spin-1/2 case, where J = h /2, and where we denote
|1/2, +1/2 = |+ ,
|1/2, 1/2 = |
. Evaluate
1
(x + iy )|+ ,
Introduction to Quantum Mechanics II
Quiz 14
Name:
December 7, 2012
The Zeeman eect is due to the interaction of the magnetic dipole moment
of the atom with an external magnetic eld,
E = B.
For the orbital motion of the electron,
=
e
L,
2mc
where m is the