Download QPexam2012 - QMplus - Queen Mary University of London

BSc and MSci Examination
Thursday 10th May 2012
PHY215
14:30 - 17:00
Quantum Physics
Duration: 2 hours 30 minutes
YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER
UNTIL INSTRUCTED TO DO SO BY AN INVIGILATOR.
Answer ALL questions from Section A and TWO questions from Section B
ONLY NON-PROGRAMMABLE CALCULATORS ARE PERMITTED IN THIS EXAMINATION. PLEASE STATE ON YOUR ANSWER BOOK THE NAME AND TYPE OF
MACHINE USED.
COMPLETE ALL ROUGH WORKINGS IN THE ANSWER BOOK AND CROSS THROUGH
ANY WORK WHICH IS NOT TO BE ASSESSED.
IMPORTANT NOTE:
THE ACADEMIC REGULATIONS STATE THAT POSSESSION OF UNAUTHORISED
MATERIAL AT ANY TIME WHEN A STUDENT IS UNDER EXAMINATION CONDITIONS IS AN ASSESSMENT OFFENCE AND CAN LEAD TO EXPULSION FROM
QMUL.
PLEASE CHECK NOW TO ENSURE YOU DO NOT HAVE ANY NOTES, MOBILE
PHONES OR UNATHORISED ELECTRONIC DEVICES ON YOUR PERSON. IF YOU
HAVE ANY THEN PLEASE RAISE YOUR HAND AND GIVE THEM TO AN INVIGILATOR IMMEDIATELY. PLEASE BE AWARE THAT IF YOU ARE FOUND TO HAVE
HIDDEN UNAUTHORISED MATERIAL ELSEWHERE, INCLUDING TOILETS AND
CLOAKROOMS IT WILL BE TREATED AS BEING FOUND IN YOUR POSSESSION.
UNAUTHORISED MATERIAL FOUND ON YOUR MOBILE PHONE OR OTHER ELECTRONIC DEVICE WILL BE CONSIDERED THE SAME AS BEING IN POSSESSION OF
PAPER NOTES. MOBILE PHONES CAUSING A DISRUPTION IS ALSO AN ASSESSMENT OFFENCE.
EXAM PAPERS CANNOT BE REMOVED FROM THE EXAM ROOM.
Examiners:
Dr G Travaglini and Dr M Bona
c Queen Mary, University of London, 2012
Page 2
SECTION A
PHY215 (2012)
Answer ALL questions in Section A
Question A1
What is the energy, expressed in eV, of a photon of wavelength λ = 0.1 nm?
[5 marks]
Question A2
Write down the Stefan-Boltzmann law for the total emissive power of a blackbody.
[5 marks]
Question A3
A certain star emits radiation mainly in the infrared with wavelength of 900 nm, while a second star
radiates principally at a shorter wavelength of 600 nm. Use Wien’s law to calculate the ratio between the
temperatures of the two stars.
[5 marks]
Question A4
A monochromatic wave of wavelength λ = 700 nm hits a zinc plate. Given that the work function for zinc
is W = 4.31 eV, will electrons be emitted due to the photoelectric effect? Support your conclusion with a
calculation.
[5 marks]
Question A5
Find the de Broglie wavelength of an electron whose kinetic energy is equal to 1 eV. The mass of the
electron is m ≃ 0.5 MeV/c2 . Can you use a non-relativistic approximation to describe the electron?
[5 marks]
Question A6
Consider an idealised single-slit experiment performed with electrons of de Broglie wavelength λ. The
length of the slit is a. What condition should λ obey in order to have diffraction?
[5 marks]
Question A7
For a particle moving in one dimension, state Heisenberg’s uncertainty principle.
[5 marks]
PHY215 (2012)
Page 3
Question A8
Explain why, as a consequence of Heisenberg’s uncertainty principle, the classical concept of trajectory of
a particle no longer makes sense at the quantum level.
[5 marks]
Question A9
Write down the time-independent Schrödinger equation for the wavefunction ψ(x) of a particle of mass m
moving in one dimension x in a potential V (x).
[5 marks]
Question A10
Write down the normalisation condition for the wavefunction ψ(x) of a particle which can move in the
interval −a ≤ x ≤ a.
[5 marks]
Turn over
Page 4
SECTION B
PHY215 (2012)
Answer TWO questions from Section B
Question B1
Planck’s formula for the spectral emittance of a blackbody is
R(λ, T ) =
1
2πhc2
,
5
λ exp [hc/(λkT )] − 1
where T is the temperature of the blackbody and λ is the wavelength of the thermal radiation emitted by
the blackbody.
(i) Explain why the use of Planck’s expression for R(λ, T ) does not lead to the “ultraviolet catastrophe”.
[5 marks]
(ii) Derive an approximated formula, valid at large λ, for the spectral emittance by performing an appropriate Taylor expansion of Planck’s formula for R(λ, T ).
[6 marks]
(iii) Using Planck’s formula for R(λ, T ) given above, prove the Stefan-Boltzmann law for the total emittance
of a blackbody. You may find the following integral useful:
Z
∞
dx
0
π4
x3
=
.
ex − 1
15
Show explicitly that the approximated formula valid at large λ obtained in part (ii) is incompatible with
Stefan-Boltzmann’s law.
[7 marks]
(iv) Assume, as Planck did, that the energies of the oscillators are quantised and distributed with Boltzmann’s probability distribution factor e−E/(kT ) , where E is the energy of the oscillator, T the temperature
and k is Boltzmann’s constant. Find the average energy of the oscillators. You may find the following
formula useful:
∞
X
1
xn =
1
−
x
n=0
(valid for |x| < 1). For which temperatures does your result differ significantly compared to that obtained
according to classical physics?
[7 marks]
PHY215 (2012)
Page 5
Question B2
Consider the Compton scattering of a beam of X-ray photons of wavelength λ = 0.1 nm on a free electron
which is initially at rest.
(i) Write down the momentum and energy conservation equations for the scattering.
[4 marks]
(ii) Determine the shift in the wavelength of the observed radiation when the photons are scattered through
an angle θ = 30◦ . Find also the total energy of the recoiling electron. You may find the following formula
useful:
h
(1 − cos θ) .
λ′ − λ =
mc
[6 marks]
(iii) Using momentum and energy conservation, prove the relation given in part (ii) above, which expresses
the shift λ′ − λ in the wavelength of the outgoing radiation compared to the incoming radiation.
[7 marks]
(iv) Determine the expression of the momentum component p′x of the electron after the scattering as a
function of the energy E of the incident photon and the angle θ. Here x̂ is the direction of the incident
photon.
[8 marks]
Turn over
Page 6
PHY215 (2012)
Question B3
Consider a particle of mass m which is confined to move freely in the one-dimensional interval 0 ≤ x ≤ L
(in other words the potential V is given by V = 0 for 0 ≤ x ≤ L and V = ∞ for x > L and x < 0).
(i) Write down the (time-independent) Schrödinger equation and the boundary conditions on the wavefunction.
[6 marks]
(ii) Find the normalised solutions to the Schrödinger equation in part (i), in particular showing that the
energy levels are discrete. Prove that the energy levels are given by
En =
h̄2 π 2 2
n ,
2mL2
n = 1, 2, . . .
You may find the following integral useful:
Z
dx sin2 (ax) =
1
x
−
sin(2ax) .
2
4a
[7 marks]
(iii) Consider now the normalised wavefunction
2πx i
πx 1 h
ψ(x) = √
+ 3 sin
.
sin
L
L
5L
What is the average value of the energy in the state described by this wavefunction?
[6 marks]
(iv) At the time t = 0 the energy is measured and is found to be equal to 2π 2 h̄2 /(L2 m). Write down the
normalised wavefunction after the measurement has taken place.
[6 marks]
PHY215 (2012)
Page 7
Question B4
Let ψ(x) = a (x − 1)(x + b) be the wavefunction of a particle which is confined to move freely on a line
parameterised by x in the interval −1 ≤ x ≤ 1 (in other words the potential V is given by V = 0 for
−1 ≤ x ≤ 1, and V = ∞ for x > 1 and x < −1). Here a and b are constants.
(i) Determine b such that ψ(x) obeys the correct boundary conditions for a particle constrained to move
for −1 ≤ x ≤ 1.
[5 marks]
(ii) Normalise the wavefunction.
[6 marks]
(iii) Calculate the expectation value of x and the most probable value of x.
[6 marks]
(iv) Verify by an explicit calculation of ∆x ∆p that Heisenberg’s uncertainty principle is satisfied.
[8 marks]
End of Paper - An Appendix of 1 page follows
Turn over
Page 8
PHY215 (2012)
Appendix
Useful data
Speed of light in free space c
Planck’s constant
Boltzmann’s constant
Mass of the electron
h
k
m
h̄c
1 eV
1 MeV
1 fermi
1 nm
= 3.00 × 108
=
=
=
=
=
=
=
=
m·s−1
6.63 × 10−34 J·s
1.38 × 10−23 J·K−1 = 8.63 × 10−5
0.5 MeV/ c2
197 MeV · fermi
1.6 10−19 J
106 eV
10−15 m
10−9 m.
eV·K−1