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Phys 321
Fall 2006
Quantum Theory I : Class Exam 2
30 October 2006
Name: Total: /
Instructions
- There are 3 questions on 7 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
Spherical coordinates ˆn = sin θ cos φxˆ + sin θ sin φyˆ + cos θ zˆ
Spin 1/2 state |+nˆ〉 = cos
θ
2
|+zˆ〉 + e
iφ sin
θ
2
|−zˆ〉
Spin 1/2 state |−nˆ〉 = sin
θ
2
|+zˆ〉 − e
iφ
cos
θ
2
|−zˆ〉
Rotation Generators ˆσn = |+ˆn〉 〈+nˆ| − |−nˆ〉 〈−nˆ|
Rep. in |±zˆ〉 basis
R(ϕn) =
cos
ϕ
2
− i sin
ϕ
2
cos θ −i sin
ϕ
2
e
−iφ sin θ
−i sin
ϕ
2
e
iφ sin θ cos
ϕ
2
ϕ
2
cos 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)
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b) After the particle leaves the magnetic field, the z-component of spin is measured. De-
termine the probability with which the outcome Sz = −ℏ/2 occurs.
/
Question 2
The observables corresponding to components of spin for a spin-1/2 particle are
Sx =
σˆx
Sy =
ˆσy
Sz =
σˆz
where, in terms of representations in the {|+zˆ〉 , |−zˆ〉} basis,
σ x
σ y
0 −i
i 0
σ z
a) Verify that
[
Sy ,
Sx
]
= −iℏ
Sz.
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
ψn(x) =
L
sin
nπx
L
0 6 x 6 L
0 otherwise
corresponding to energy eigenvalues
En =
π
2 ℏ
2 n
2
2 mL
2
for n = 1, 2 ,....
a) Consider an ensemble of particles that are prepared so that, at some instant, they are
each in the state
|ψ〉 =
e
iπ/ 4 |ψ 1
e
−iπ/ 4 |ψ 4
The energy of each particle is measured. List the possible outcomes and determine the
probabilities with which they occur.
/
