Assessment item 2—Assignment 2
Due date:
On-campus/Off-campus students—26 May 2012
Weighting:
30%
ASSESSMENT
2
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 (5%)
J.S. Townsend, Chapter 5: Principles of Quantum Mechanics, Problem 5.4
1
A particle of mass m moves in the potential energy V ( x)  m 2 x2 . The ground-state wave
2
function is
1/ 4
2
a
 0 ( x)    e  ax / 2
 
and the first excited-state wave function is
1/ 4
 4a 3 
 1 ( x)  

  
xe  ax
2
/2
where a  m / . What is the average value of the parity for the state
 ( x) 
3
1 i
 0 ( x) 
 1 ( x)
2
2 2
Question 2 (5%)
J.S. Townsend, Chapter 5: Principles of Quantum Mechanics, Problem 5.8
Let the operator 𝐴𝑜𝑝 correspond to an observable of a particle. It is assumed to have just tow
eigenfunctions 𝜓1 (𝑥) and 𝜓2 (𝑥) with distinct eigenvalues. The function corresponding to an arbitrary
state of the particle can be written as
𝜓(𝑥) = 𝑐1 𝜓1 (𝑥) + 𝑐2 𝜓2 (𝑥)
An operator 𝐵𝑜𝑝 is defined according to
𝐵𝑜𝑝 𝜓(𝑥) = 𝑐2 𝜓1 (𝑥) + 𝑐1 𝜓2 (𝑥)
Prove that 𝐵𝑜𝑝 is Hermitian.
Question 3 (5%)
J.S. Townsend, Chapter 5: Principles of Quantum Mechanics, Problem 5.12
Use definitions (5.90) and (5.91) of the right and left circular polarized states to show that the twophoton state (5.89) becomes (5.92) when expressed in terms of the linearly polarized states. Caution:
Since the photons are travelling back to back and photon 1 is travelling in the positive z direction,
photon 2 is travelling in the negative z direction. Consequently, for photon 2
1
𝜓𝑅 (2) =
[𝜓𝑥 (2) − 𝑖𝜓𝑦 (2)]
√2
and
1
𝜓𝐿 (2) =
[𝜓𝑥 (2) + 𝑖𝜓𝑦 (2)]
√2
Question 4 (5%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 6.2
Solve the three-dimensional harmonic oscillator for which
1
𝑉(𝑟) = 𝑚𝜔2 (𝑥 2 + 𝑦 2 + 𝑧 2 )
2
by separation of variables in Cartesian coordinates.
Assume that the one-dimensional oscillator has eigenfunctions 𝜓𝑛 (𝑥) with corresponding energy
1
eigenvalues 𝐸𝑛 = (𝑛 + 2) ℏ𝜔.
What is the degeneracy of the first excited state of the oscillator?
Question 5 (10%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 6.4
The normalized energy eigenfunctions for the first excited state of the one-dimensional harmonic
oscillator is given by
3/4
4𝑚3 𝜔3
2
𝜓1 (𝑥) = (
) 𝑥𝑒 −𝑚𝜔𝑥 /2ℏ
3
𝜋ℏ
with corresponding energy 𝐸1 = 3ℏ𝜔/2.
What is the energy of the first excited state of the three-dimensional isotropic harmonic oscillator?
What is the degeneracy for this energy eigenvalues?
What is the orbital angular momentum of the particle in this excited state? Explain your reasoning.
Question 6 (10%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 6.6
Suppose that
1
1 i
 ( ) 

e
3
6
Show that () is properly normalized. What are the possible results of a measurement of LZ for a
particle whose wave function is () and what are the probabilities of obtaining those result?
Suggestion: See Example 6.1.
Question 7 (10%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 6.11
The normalized angular wave function for a three-dimensional rigid rotator is given by
3
𝜓(𝜃, 𝜙) = √8𝜋(−𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙 + 𝑖𝑐𝑜𝑠𝜃)
Show that this wave function is an eigenfunctions of
ℏ
𝜕
𝜕
𝐿𝑦𝑜𝑝 = (𝑐𝑜𝑠𝜙
− 𝑐𝑜𝑡𝜃𝑠𝑖𝑛𝜙 )
𝑖
𝜕𝜃
𝜕𝜙
the operator corresponding to the y component of the orbital angular momentum.
What is the corresponding eigenvalues for this wave function?
Question 8 (5%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 6.20
Verify that
1 𝑍 3/2 𝑍𝑟 −𝑍𝑟/2𝑎
0
𝑅2,1 (𝑟) =
(
)
𝑒
𝑎0
√3 2𝑎0
Is an energy eigenfunctions for the hydrogen atom with the appropriate eigenvalues. Suggestion: See
Example 6.3.
Question 9 (5%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 6.26
Show that the matrix operators
ℏ 0 1
ℏ 0 −𝑖
ℏ 1 0
𝑆𝑥 𝑜𝑝 = (
)
𝑆𝑦 𝑜𝑝 = (
)
𝑆𝑦 𝑜𝑝 = (
)
2 1 0
2 𝑖 0
2 0 −1
satisfy the commutation relation
[𝑆𝑦 𝑜𝑝 𝑆𝑦 𝑜𝑝 ] = 𝑖ℏ𝑆𝑥 𝑜𝑝
Notes: The three 22 matrices
0 1
0 −𝑖
1 0
𝜎𝑥 = (
)
𝜎𝑦 = (
)
𝜎𝑧 = (
)
1 0
𝑖 0
0 −1
are often referred to as the Pauli spin matrices.
Question 10 (5%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 6.28
Determine the eigenvalues and eigenvectors of the operator
 0 i 
S yop  

2i 0 
What is the probability of obtaining S z  / 2 if a measurement is carried out on a silver atom that is
known to be in a state with S y   / 2 ?
Question 11 (5%)
J.S. Townsend, Chapter 6: Quantum Mechanics in Three Dimensions, Problem 7.1
Verify that the states
  (1)   (2)
1
  (1)   (2)    (1)   (2)
2
  (1)   (2)
are eigenstates of the z component of total spin S Z  S1Z  S 2 Z with eigenvalue ,0, respectively.
Question 12 (5%)
J.S. Townsend, Chapter 7: Identical Particles, Problem 7.3
The spatial wave functions for two identical particles in the one-dimensional box (see Section 7.1) are
given by
1
[𝜓1 (𝑥1 )𝜓2 (𝑥2 ) + 𝜓2 (𝑥1 )𝜓1 (𝑥2 )]
Ψ𝑆 (𝑥1 , 𝑥2 ) =
√2
and
1
[𝜓1 (𝑥1 )𝜓2 (𝑥2 ) − 𝜓2 (𝑥1 )𝜓1 (𝑥2 )]
Ψ𝐴 (𝑥1 , 𝑥2 ) =
√2
where one of the particles is in the ground state and one is in the first excited state.
Calculate the probability that measurement of the positions of the two particles finds them both in the
left-hand side of the box, that is measurement yield 0 < 𝑥1 < 𝐿/2 and 0 < 𝑥2 < 𝐿/2.
Notice how the probability is significantly larger for the symmetric state Ψ𝑆 than for the antisymmetric
state Ψ𝐴 . Thus the particles behave as if they attract each other in the symmetric state and repel each
other in the antisymmetric state eventhough the Hamiltonian for the two particles does not include any
interaction term. Heisenberg called these fictitious forces of attraction and repulsion exchange forces.
Question 13 (10%)
J.S. Townsend, Chapter 7: Identical Particles, Problem 7.6
(a) Calculate (i) the Fermi energy, (ii) the Fermi velocity, and (iii) the Fermi temperature for gold at
0K. The density of gold is 19.32 g/cm3 and the molar weight is 194 g/mole. Assume each gold
atom contributes one “free” electron to the Fermi gas.
(b) In a cube of gold ` mm on an edge, calculate the approximate number of conduction electrons
whose energies lie in the range from 4.000 to 4.025 eV.
Question 14 (10%)
J.S. Townsend, Chapter 7: Identical Particles, Problem 7.12
Consider the following two microstates for ten identical particles. In one of the microstates there are
ten particles in the ground state and none in the excited state, while in the other microstate there are
five particles in the ground state and five in the excited state. The “statistical weight” of these two
microstates is 1 to 1. If the particles are distinguishable, there is still just one microstate corresponding
to the ten particles being in the ground state. Calculate the number of microstates for the case of five
particles in the ground state and five in the excited state if the particles are distinguishable. What is the
statistical weight of these two configurations for distinguishable particles?
Question 15 (5%)
J.S. Townsend, Chapter 7: Identical Particles, Problem 7.28
Determine the value of T such that 𝑘𝐵 𝑇 is roughly equal to the spacing between the ground state and
the first excited state for a cubic box 10 m on a side. How does this value for T compare with the
critical temperature for Bose-Einstein condensation for a gas of 5105 sodium atoms confined in the
box? These are roughly the conditions that existed in the Ketterle experiment discussed Section 7.7.