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Phys 321
Fall 2006
Quantum Theory I : Final Exam
13 December 2006
Name: Total: /
Instructions
- There are 5 questions on 11 pages.
- Show your reasoning and calculations and always motivate your answers.
Physical constants and useful formulae
Charge of an electron e = − 1. 60 × 10
− 19 C
Planck’s constant h = 6. 63 × 10
− 34
Js ℏ = 1. 05 × 10
− 34
Js
Mass of electron m e
= 9. 11 × 10
− 31
kg = 511 × 10
3
eV/c
2
Mass of proton m p
= 1. 673 × 10
− 27 kg = 938. 3 × 10
6 eV/c
2
Mass of neutron m n
= 1. 675 × 10
− 27 kg = 939. 6 × 10
6 eV/c
2
Spherical coordinates nˆ = sin θ cos φxˆ + sin θ sin φyˆ + cos θ ˆz
Spin 1/2 state |+nˆ〉 = cos (θ/2) |+ˆz〉 + e
iφ sin (θ/2) |−zˆ〉
Spin 1/2 state |−nˆ〉 = sin (θ/2) |+zˆ〉 − e
iφ cos (θ/2) |−zˆ〉
sin (ax) sin (bx) dx =
sin ((a − b)x)
2(a − b)
sin ((a + b)x)
2(a + b)
if a 6 = b
sin (ax) cos (ax) dx =
2 a
sin
2 (ax)
sin
2
(ax) dx =
x
sin (2ax)
4 a
x sin
2
(ax) dx =
x
2
x sin (2ax)
4 a
cos (2ax)
8 a
2
x
2
sin
2
(ax) dx =
x
3
x
2
4 a
sin (2ax) −
x
4 a
2
cos (2ax) +
8 a
3
sin (2ax)
Formulae continued...
Angular Momentum
L
x
= iℏ
sin φ
∂θ
cos θ cos φ
sin θ
∂φ
Ly = iℏ
− cos φ
∂θ
cos θ sin φ
sin θ
∂φ
L
z
= −iℏ
∂φ
L
|l, m〉 = ℏ
l(l + 1) − m(m + 1) |l, m + 1〉
L
−
|l, m〉 = ℏ
l(l + 1) − m(m − 1) |l, m − 1 〉
Spherical Harmonics Y 0 , 0
(θ, φ) =
4 π
Y
1 , 0
(θ, φ) =
4 π
cos θ
Y
1 ,± 1
(θ, φ) = ∓
8 π
e
±iφ sin θ
sin (ax) sin (bx) dx =
sin ((a − b)x)
2(a − b)
sin ((a + b)x)
2(a + b)
if a 6 = b
sin (ax) cos (ax) dx =
2 a
sin
2 (ax)
sin
2 (ax) dx =
x
sin (2ax)
4 a
x sin
2
(ax) dx =
x
2
x sin (2ax)
4 a
cos (2ax)
8 a
2
x
2
sin
2
(ax) dx =
x
3
x
2
4 a
sin (2ax) −
x
4 a
2
cos (2ax) +
8 a
3
sin (2ax)
∞
−∞
e
−αx
2 +βx dx =
π
α
e
β
2 / 4 α
∞
−∞
xe
−αx
2 +βx dx =
β
π
2 α
3 / 2
e
β
2 / 4 α
∞
−∞
x
2 e
−αx
2 +βx dx =
(β
2
π
4 α
5 / 2
e
β
2 / 4 α
∞
−∞
x
3 e
−αx
2 +βx dx =
β(β
2
π
8 α
7 / 2
e
β
2 / 4 α
c) Although |Ψ i
〉 is unknown, the fact that the SG ˆn measurement gave S n
= +ℏ/2 does
restrict the possible initial states. Determine one state in which the particle could not
have been prior to the measurement and express this in terms of the {|+zˆ〉 , |−zˆ〉} basis.
/XX
Question 2
A spin-1/2 particle of charge q and g-factor g is placed in a magnetic field B = B ˆz for time t.
The resulting Hamiltonian is
H =
ℏω
σˆ z
where ω = −gqB/ 2 m and in the basis {|+zˆ〉 , |−zˆ〉} ,
a) Find the matrix representation, in the basis {|+zˆ〉 , |−zˆ〉} , of the evolution operator,
U (t).
Question 2 continued...
Question 3
A particle of mass m is in an infinite square well potential
V (x) =
0 0 6 x 6 L
∞ otherwise,
for which the energy eigenstates are
ψ n
(x) =
L
sin
nπx
L
0 6 x 6 L
0 otherwise.
corresponding to energy eigenvalue E n
ℏ
2 n
2 π
2
2 mL
2
for n = 1, 2 ,.... The particle is in the state
ψ(x) =
L
sin
2 πx
L
0 6 x 6
L
0 otherwise.
Note that the wavefunction is only non-zero in the region 0 6 x 6
L
The energy of the particle is measured. Determine the probability that the outcome of this
measurement is E 2
/XX
Question 4
Consider an ensemble of charged harmonic oscillators, each of mass m and frequency ω.
Suppose that the oscillators are initially each in the state
|Ψ(t = 0)〉 =
where |n〉 is the n
th energy eigenstate.
a) Prove that at a later time t, the state of each oscillator is
|Ψ(t)〉 = e
−iωt/ 2
| 0 〉 + e
−i 2 ωt
Question 4 continued...
Question 5
An ensemble of rigid rotators, such as the CO molecule, are each prepared so that each
ensemble member is in the state
|ψ〉 =
where the notation denotes standard angular momentum states |l, m〉. The ensemble is sub-
jected to measurements of Lx.
a) Determine the expectation value, 〈L x
Question 5 continued...
b) Determine the uncertainty, ∆Lx.
c) Suppose that one more molecule in the state |Ψ〉 is subjected to a measurement of L x
What can you predict regarding the outcome of the measurement?
/XX
