Assessment item 1—Assignment 1
Due date:
On-campus/Off-campus students—20 April 2012
Weighting:
30%
ASSESSMENT
1
Objectives
This assessment item relates to the course learning outcomes 1 and 2 as stated in Part A.
Assessment criteria
Each question in this assignment will be assessed separately for the criterion accuracy and
correct results and given a mark as specified in each question.
The marking for this assignment are based on accuracy and correct results, including:
 Correct application of maths and arithmetic
 Answers clearly identified
 Correct results
In addition, the assignment as a whole will be assessed against the following criteria:
Evidence of correct procedures
 All necessary steps in analysis are present on correct order
 Clear presentation of mathematical and arithmetical working linking given details of the
problem to the results obtained.
 Evidence of checking results (mathematical, graphical, logic-common sense)
Evidence of understanding of the topic
 Explanation of choices made in the analysis (why is procedure required, why this particular
procedure)
 Interpretation of results, eg limitations, direction of vectors
Professional presentation
 The work is clearly identified (problem, date, analyst)
 Clear statement of each problem and its details and requirements
 Logical layout of analysis
 Appropriate use of diagrams, clear diagrams
 Correct use of terminology, conventions
 Clear English in the explanation of procedure and interpretation of results.
 Referencing of authoritative sources of equations and data
Question 1 (4%)
J.S. Townsend, Chapter 1: Light, Problem 1.6
An AM radio station broadcasts at 1000 kHz with an output power of 50,000 watts. Assuming the
broadcast antenna is located 100 km away from receiver and, for ease of calculation, the antenna
radiates isotropically, estimate the number of photons per cubic metre at the receiver’s location.
Question 2 (4%)
J.S. Townsend, Chapter 1: Light, Problem 1.26
Assume that the first beam splitter at A in the Mach-Zehnder interferometer is a “third-silvered
mirror”, that is, a mirror that reflects one-third the light and transmits two-thirds. The two mirrors at B
and C reflect 100% of the light, and the second beam splitter at D is a traditional half-silvered mirror
that reflects one-half the light and transmits one-half. The probability of detecting a photon in either
photomultiplier PM1 or PM2 varies with the position of the movable mirror, say mirror B. Determine
that maximum probability and the minimum probability of obtaining a count in, say, PM 1. What is the
visibility
P  Pmin
V  max
Pmax  Pmin
Of the interference fringes, where Pmax and Pmin are the maximum and minimum probabilities,
respectively, that a photon is counted by the detector, as the position of the movable mirror varies?
(Note: In the experiment of Aspect et al. described in Section 1.5 the visibility of the fringes is 0.987
0.005.
Question 3 (4%)
J.S. Townsend, Chapter 2: Wave Mechanics, Problem 2.8
Through what potential difference must an electron be accelerated so that the electron’s wavelength is
1 nm = 10-9 m? Repeat the calculation for  = 1 pm = 10-12 m and  = 1 fm = 10-15 m.
Question 4 (4%)
J.S. Townsend, Chapter 2: Wave Mechanics, Problem 2.15
Verify that 𝜓(𝑥, 𝑡) = 𝐴𝑐𝑜𝑠(𝑘𝑥 − 𝜔𝑡) and 𝜓(𝑥, 𝑡) = 𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡) are not solutions to the
Schrodinger equation for a free particle:
ℏ2 𝜕 2 𝜓(𝑥, 𝑡)
𝜕𝜓(𝑥, 𝑡)
−
= 𝑖ℏ
2
2𝑚 𝜕𝑥
𝜕𝑡
Question 5 (4%)
J.S. Townsend, Chapter 2: Wave Mechanics, Problem 2.19
For the wave function  ( x)  Aeikx  Beikx evaluate the probability current
jx 
 * 
 * 





2mi 
x
x 
Question 6 (8%)
J.S. Townsend, Chapter 2: Wave Mechanics, Problem 2.20
(a) Normalize the wave function
Ae x
x0
 ( x )   x
Ae
x0
Note: This wave function is the ground-state wave function for the Dirac delta function potential
energy well to be discussed in Section 4.4.
(b) What is the probability that the particle will be found within 1/ of the origin if a measurement of
its position is carried out?
Question 7 (14%)
J.S. Townsend, Chapter 2: Wave Mechanics, Problem 2.29
(a) Show that 𝜓(𝑥, 𝑡) = 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡) is a solution to the Klein-Gordon equation
𝜕 2 𝜓(𝑥, 𝑡) 1 𝜕𝜓(𝑥, 𝑡) 𝑚2 𝑐 2
− 2
− 2 𝜓(𝑥, 𝑡) = 0
𝜕𝑥 2
𝑐
𝜕𝑡
ℏ
if
𝑚2 𝑐 4
)
ℏ2
𝜔 = √𝑘 2 𝑐 2 + (
The Klein-Gordon equation is a relativistic quantum mechanical wave equation for a free particle.
(b) Determine the group velocity of a wave packet made of waves satisfying the Klein-Gordon
equation.
(c) The Klein-Gordon equation describes the motion of particles of mass m. Using your result from
(a), show that
𝐸 = √𝑝2 𝑐 2 + 𝑚2 𝑐 4
for these particles.
(d) Show that the speed v of these particles is equal to the group velocity that you determined in (b).
Question 8 (8%)
J.S. Townsend, Chapter 2: Wave Mechanics, Problem 2.30
Suppose the wave function for a particle is given by the symmetric “tent” wave function in Fig. 2.27
12 𝑎
𝜓(𝑥, 𝑡) = √ 3 ( − |𝑥|)
𝑎 2
= 0
|𝑥| ≤
𝑎
2
|𝑥| ≥
𝑎
2
Show that 𝜓(𝑥, 𝑡) is properly normalized. What is ⟨𝑥⟩ for the particle? Calculate the uncertainty x in
the particle’s position. Note: The wave function is an even function.
Question 9 (8%)
J.S. Townsend, Chapter 3: The Time-Independent Schrodinger Equation, Problem 3.4
At time t=0 the wave function for a particle in a box is given by
2
1
𝜓(𝑥) = √ 𝜓1 (𝑥) + √ 𝜓2 (𝑥)
3
3
where 𝜓1 (𝑥) and 𝜓2 (𝑥) are the ground-state and first-excited-state wave functions with corresponding
energies E1 and E2, respectively. What is 𝜓(𝑥, 𝑡)? What is the probability that a measurement of the
energy yields the value E1? What is ⟨𝐸⟩ ? How would you go about testing these predictions?
Question 10 (14%)
J.S. Townsend, Chapter 3: The Time-Independent Schrodinger Equation, Problem 3.6
At time t=0 the normalized wave function for a particle mass m in the one-dimensional infinite well
0
0 x L
V ( x) 

elsewhere
is given by
1 i 2
x 1 2
2 x
 ( x) 
sin

sin
0 xL
2 L
L
L
2 L

elsewhere
(a) What is (x,t)?
(b) What is the probability that a measurement of the energy at time t will yield the result 2 2 / 2mL2
?
(c) What is E for the particle at time t? Suggestion: This result can be obtained by inspection. No
integrals are required.
(d) Is x time dependent? Justify your answer.
Question 11 (4%)
J.S. Townsend, Chapter 4: One-Dimensional Potentials, Problem 4.1
Figure 4.29 shows four wave functions in the region x > 0. Indicate for each wave function whether
the wave function is an acceptable or unacceptable wave function for an actual physical system. If the
wave function is not acceptable explain why.
Question 12 (8%)
J.S. Townsend, Chapter 4: One-Dimensional Potentials, Problem 4.20
At time t = 0 a position measurement locates a particle in the potential energy box
0
0 x L
V ( x) 

elsewhere
to be in the vicinity of the centre of the box, x = L/2.
(a) Assume that we approximate the particle’s (unnormalized) wave function to be (xL/2), a Dirac
delta function. That is, we take (x) = (xL/2). Find the relative probabilities Pn that a
measurement of the particles energy will yield En, for all n.
(b) Determine (x,t), the (unnormalized) wave function of the particle at time t. Do the probabilities
that you determined in (a) vary with time? Explain why or why not.
Question 13 (8%)
J.S. Townsend, Chapter 4: One-Dimensional Potentials, Problem 4.24
Verify that the wave function   Ce xiEt / (see Example 4.3) in the region x > 0 for the step potential
of Section 4.6 leads to zero probability current in this region.
Use the conservation of probability equation
j
 *
 x
t
x
to argue that the probability current vanish in the region x < 0 as well for this energy eigenfunction.
What can you therefore conclude about the magnitude of the reflection coefficient?
Question 14 (8%)
J.S. Townsend, Chapter 4: One-Dimensional Potentials, Problem 4.25
Solve the time-independent Schrodinger equation for a particle of mass m and energy E > V0 incident
from the left on the step potential
𝑉(𝑥) = 𝑉0
𝑥<0
=0
𝑥>0
See Fig. 4.37. Determine the reflection coefficient R and the transmission coefficient T. Verify that
probability is conserved.