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The final exam for phys 321: quantum theory i, which was held on december 13, 2006. The exam covers various topics related to quantum mechanics, including physical constants, angular momentum, spherical harmonics, and spin. It consists of five questions and includes instructions, physical constants, and useful formulae.
Typology: Exams
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Phys 321
Fall 2006
13 December 2006
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
− 31
kg = 511 × 10
3
eV/c
2
Mass of proton m p
− 27 kg = 938. 3 × 10
6 eV/c
2
Mass of neutron m n
− 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
x
= iℏ
sin φ
∂θ
cos θ cos φ
sin θ
∂φ
Ly = iℏ
− cos φ
∂θ
cos θ sin φ
sin θ
∂φ
z
= −iℏ
∂φ
|l, m〉 = ℏ
l(l + 1) − m(m + 1) |l, m + 1〉
−
|l, m〉 = ℏ
l(l + 1) − m(m − 1) |l, m − 1 〉
Spherical Harmonics Y 0 , 0
(θ, φ) =
4 π
1 , 0
(θ, φ) =
4 π
cos θ
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
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
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...
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) =
sin
nπx
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) =
sin
2 πx
0 6 x 6
0 otherwise.
Note that the wavefunction is only non-zero in the region 0 6 x 6
The energy of the particle is measured. Determine the probability that the outcome of this
measurement is E 2
/XX
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...
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