THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF PHYSICS
FINAL EXAMINATION
JUNE 2016
PHYSZ 111 Quantum Physics
Time Allowed - 3 hours
Total number of questions - 6
Use separate booklets for Questions 1 & 2, Questions 3 & 4, and Questions 5 & 6
Total marks: 180 - Questions are all of equal value (30 marks)
This paper may be retained by the candidate.
Students must provide their own UNSW approved calculators.
Answers must be written in ink Except where they are expressly required, pencils may only
be used for drawing, sketching or graphical work
The following information may be useful
Planck's constant h = 6.626 x 10-34 Js
Fundamental charge unite= 1.60 x 10-19 C
Speed oflight (vacuum) c = 3.0 x 108 m/s
Electron mass= 9.1 x 10-31 kg= 0.511 MeV/c2
Neutron mass= 1.675 x 10-27 kg= 939.6 MeV /c2
Proton mass= 1.672 x 10-27 kg= 938.3 MeV /c2
Boltzmann's constant k = 1.38 x 10-23 JK-1
Angstrom (A) = 1.0 x 10-10 m
Permittivity constant so= 8.85 x 10-12 Fm-1
Gravitational constant G = 6.6 7 x 10-11 Nm2 /kg2
h/mec = 2.43 x 10-12 m
1 eV = 1.60 x 10-19 J
1 J = 6.24 x 1018 eV
Time-independent Schrodinger Equation: - Ii d
2
1f/~X) + V lf/(x) = E lf/(x)
2m dx
Time-dependent Schrodinger Equation: - ri2 a
2
\jf(~,t) + V(x)\jf(x,t) = iii B\jf(x,t)
2m ax at
Bragg's law: nA = 2dsin 0
Compton Shift: ~,i = ~(1- cos 0)
me
Bohr-Sommerfeld equation: p pdx = nh
f . 2 (b )d _ x sin(2bx) sm x x----~~ 2 4b
f . 2( )d- _ x
2
xsin(2x) cos(2x)
xsm x x--------~~
4 4 8
f 2 . 2(b )d _ x
3 (x2 1 J . (2b ) xcos(2bx) x sm x x--- --- sm x --~~
6 4b 8b3 4b2
f e-bx dx = 1t oo 2 ~
-00 b
ro I
f n -bxdx n. xe =-bn+1
0
f b dzg dglb f b df dg f-dx = f- - --dx a dx2 dx a a dx dx
f (a - bx) 1l 2 dx = _2_ (a - bx) 312 3b
d dg df
dx (f g) = f dx + g dx
sin(20) = 2 sin0 cos0
P 1. . . (0 1) _ (0 -i) (1 0 ) au 1 spm matrices: r5x = 1 0 , r5y = i O , r5z = o -1
Visible light 11, - 400 - 700 nm
Ez =(pcf +(mczf
p=mv/ ~1-(v/cf
¢(p)= ~ J e-ipx/nV/(x)dx
\I 21rri -w
-00
Question 1 (Marks 30)
(a) A photon with energy 662 keV collides with an electron in a metal sample and is
scattered causing the electron to recoil. The electron is assumed to be stationary before
the collision. Write down the relativistic equation of conservation of energy and the
equations for the conservation of the horizontal and vertical components of momentum
for this collision but do not solve these.
(b) A gamma ray photon of wavelength 1.88 pm strikes a metal target and the wavelength of
the scattered radiation is measured at an angle 0 = 90° to the incident beam. Calculate:
(i) the wavelength of the scattered radiation,
(ii) the kinetic energy of the recoiling electron,
(iii) the fraction of the energy of the incident photon lost in the collision with the
electron, and
(iv) the angle made by the trajectory of the recoiling electron with the incident beam.
(c) Write down an expression for the photon's fractional loss in energy K/E in the type of
collision considered in (b) and hence demonstrate with a numerical estimate that the
fractional energy loss for visible light is orders of magnitude less than for gamma-ray
photons. (n.b., Kis kinetic energy, Eis total energy)
Question 2 (Marks 30)
(a) A particle is described by the one-dimensional wavefunction
lfl(x) = Ae _a2x212
(i) Normalise the wavefunction
(ii) Find the momentum space wavefunction ¢(p) corresponding to l,ll(x)
(iii) Evaluate the expectation values (1J1lx2 llfl) and (1J1IP2 i1J1)
(iv) Comment on the value of the product of your answers to (iii) above
(b) The uncertainty principle can be used to estimate time and frequency scales governing
optical transitions. Make estimates for the following:
(i) Calculate the percentage uncertainty in the frequency ~ v/ v of the emitted light
when a hydrogen atom makes a transition from the fourth excited state to the
ground state when the lifetime of the fourth excited state is -c = 2.75 x 10-9 s.
(ii) What is the lifetime of the ground state? Give the reason for your answer.
(iii) A laser pulse has a linewidth defined as the uncertainty in the output (the spectral
line) wavelength ~11,. Find an expression relating the linewidth of the laser's
spectral line to the bandwidth ~v, which is the uncertainty in the frequency of the
spectral line.
(iv) If the duration of the laser pulse in (iii) is 1.5 x 10-9 s, estimate the linewidth of the
11, = 632.8 nm beam from a HeNe laser.
Question 3 (Marks 3 0)
The Schrodinger equation in one dimension is described mathematically by:
. aw ti2 a2w lfi- = ----+ vw
dt 2m ax2 (3.1)
for a specific potential energy function, V(x,t), and for a particle of mass m. W is the wave
function, while x and t represent space and time variables respectively. Other symbols have
their usual meanings.
After some mathematical manipulation, we get the time independent Schrodinger equation
(TISE):
li2 a2'1/J
---+ V'ljJ = E'ljJ
2m ax2 (3.2)
(a) While there are some mathematical manipulations required to get from Equation (3.1)
to Equation (3.2), there are also some assumptions about physical quantities required.
(i) What assumptions do we make about time dependence to get from Equation (3.1)
to Equation (3.2)?
(ii) Considering that Equation (3.2) is derived from Equation (3.1), how would you
best describe the role of the separation constant E in Equation (3.2), taking into
account the assumptions needed to get from Equation (3.1) to (3.2)?
(iii) Write down the Hamiltonian operator contained in Equation (3.2).
(iv) One very important physical implication for the solution to Equation (3.2) is the
existence of stationary states. Define a stationary state in terms of the Hamiltonian,
and explain its importance for the probability density associated with a
wavefunction.
V
-a/2 a/2 X
(b) Consider a particle in square well, as represented in the figure above:
(i) Assume the potential energy barriers (x < -a/2 or x > a/2) are infinite. Give the
value for '4J(X) in Regions I and III, and explain your reasoning, with reference to
the TISE, Equation (3.2).
(ii) Give the value for Vin Region II, and hence write down the TISE for Region II.
(iii) The solution to the equation you wrote down in (ii) can be written as a sum of
sines and cosines:
T(x) = Asin(kx) + Bcos(kx) (3.3)
where A and B are arbitrary constants. Give the values that k will need to take in
this equation.
(iv) Give the boundary conditions that \ll(x) must meet, for the boundaries shown in
the figure. Justify your answer.
(v) Using the boundary conditions from (iv), find the range of values for the constants
A and B for which Equation (3.3) will be true.
(vi) Find the energy eigenvalues for the allowed solutions, and give the first four
energy levels. Sketch the wave functions for these first four energy states, marking
odd and even states on your sketch.
·Question 4 (Marks 30)
(a) A nitrogen molecule Nz vibrates back and forwards along its long axis (x-axis), as shown
in the figure below, resembling two particles on a spring, where the bond takes the place
of the spring. For this part of the question, part (a), we will treat the molecule as a
classical simple harmonic oscillator. Assume that the molecule/bond system is stretched
by 0.1 nm in the x-direction, resulting in an increase in the energy of the system of 1 eV.
Find the spring constant k for this bond.
······~
X
(b) You are given an equation describing a Schrodinger equation with the following
potential:
1 2 2 V(x)=-mco x -bx
2 (4.1)
Rather than solving the full Schrodinger equation, you can take a short cut by rewriting
(4.1) as
1 2 2 V(x)=-mco (x-x0 ) 2 (4.2)
In other words, (4.2) is (4.1) with an offset in the spatial position of the potential, and a
shift in the energy scale. This is useful provided you expand ( 4.2) so that by comparing
terms with (4.1), you can find the new value for the parameter bin (4.1) in terms of the
offset to the equilibrium position xo. Once you expand ( 4.2) you will find an additional
energy term that gives the shift in energy due to the offset in equilibrium position. Give
that shift in energy in terms of the value for b you found in ( 4.2).
( c) For the traditional quantum harmonic oscillator with a potential of
) 1 2 2 V(x =-mco0 x 2 (4.3)
we expect a ground state wave function of the form:
'ljJ0 (x) = Ae (4.4)
Write down the ground state wavefunction expected for the potential in part (b ),
explaining your reasoning.
Question 5 (Marks 30)
(a) Observables are represented by Hermitian operators. Mathematically, this means that:
(flQg) = (Qflg) for allj(x) andg(x)
where Q is the operator corresponding to the observable Q.
(i) In one or two sentences, explain by reference to the required properties of
observables why the corresponding operator must be Hermitian.
(ii) Show that the sum of two Hermitian operators is also Hermitian
(iii) Suppose Q is Hermitian and a is a complex number. Under what condition on a
will the product aQ also be Hermitian?
(iv) Show that the Hamiltonian operator fl= -(h2/2m)d2/dx2 + V(x) is Hermitian (You
should assume f and g obey the normal properties of localized wavefunctions
regarding what happens at x = ±oo; you may also assume V(x) is real).
(b) Quantum mechanical states are defined in an n-dimensional vector space known as a
Hilbert space.
(i) In one or two sentences, explain what a 'basis' is for a vector space.
(ii) What does it mean for a basis to be 'orthonormal'?
(iii) In one or two sentences, explain what the terms eigenfunction and eigenvalue
mean for some arbitrary operator A
(c) A classic 'two level' system in quantum mechanics is the spin-1/2 particle. Here all
possible spin states can be represented in a two-dimensional vector space with basis
vectors lxi) = G) and lx-1,) = (~). (n.b., the relevant Pauli spin matrices are on the
equation sheet).
(i) Suppose a spin-1/2 particle is in the state Ix)=~ c;i). If you make an
observation of the z-component of the spin Sz = - ~ Bz, what are the respective
2
probabilities of measuring Sz = +h/2 and measuring Sz = -h/2? Show these add to
one as expected.
(ii) For the x-component of the spin Sx = - ~ Bx, find the eigenvectors lxJ and lxJ
that correspond to the eigenvalues +h/2 and -h/2 respectively. It may help to
attack this using a generic spin state Ix) = c:). Be sure your final eigenvectors
are properly normalised.
(iii) Calculate the respective probabilities of measuring Sx = +h/2 and measuring
Sx = -h/2 if you make an observation of the x-component of the spin Sx for the spin
state Ix) in (i). Show these also add to one.
(iv) Calculate the expectatation value for Sx, which is obtained as (Sx) = (xlSxlx).
Show you can get the same result using the two probabilities obtained in (iii).
Question 6 (Marks 30)
(a) Consider the quantum mechanical analog to the classical problem of a ball bouncing
elastically on a horizontal plane situated at z = 0. Here we will assume this ball is a
neutron ( as this is an experiment that has actually been done). You may use e V as your
energy units if you find this more convenient than J.
(i) Using your knowledge of gravity from first year physics, come up with a
compelling potential energy function V(z). Be careful to define this function
properly for all z accounting for the plane at z = 0. Sketch a graph of V(z) versus z.
(ii) Use the Bohr-Sommerfeld quantization rule to estimate the energy levels En for
this system. (n.b., do not attempt an exact solution using the Schrodinger equation).
(b) A commutator of two operators A and B is defined as [A, B] = AB - BA
(i) What does it mean mathematically for two operators to 'not commute'? In one or
two sentences, explain what the physical implications are for non-commuting
observables.
(ii) Prove that the following identity is true: [AB, C] = A[B, C] + [A, C]B.
(iii) Show that [t(x), p] = ih df for any function f(x). Hint: you may want to
dx
strategically introduce another functiong(x) to help achieve this.
(c) In the lectures we derived an important relation:
(1)
(i) Using the momentum operator, show that this relation can be used to give you
Ehrenfest's theorem d(p) = -(8
8
v)
dt X
(ii) Briefly explain what Ehrenfest's theorem means physically.
(iii) Think back to the scenario in (a) above. What intuition about the bouncing neutron
can you obtain from a consideration of Ehrenfest's theorem? If this makes sense to
you, explain why? If the result is instead counter-intuitive, explain why?
(iv) Suppose that Q = fl for the Equation 1 above, what principle of physics emerges?
(v) Suppose instead that Q = i, what principle of physics emerges this time? Hint:
think carefully about what an expectation value is.