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
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 (,
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
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
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 th
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
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
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,
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 b
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 dierentiat
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 eigenval
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 |
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/
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 or