联系我们： 手动添加方式: 微信>添加朋友>企业微信联系人>13262280223 或者 QQ: 1483266981

Modern Physics 2023/24 Problem Sheet 1
Introducing Workshops and Continuously-assessed hand ins
Welcome to the Modern Physics Workshops. These workshops are designed to help you
in several ways, so please make the most of them! The timetable for the Modern Physics
Problems Sheets and their hand ins is:
Problem sheet 1: handed out 21st September. This problem sheet includes material
presented in Lectures 1-5. It consists of two clearly identified hand in questions,
plus a number of other questions that will be the focus for workshops. The
deadline for submitting the hand in questions is 9am on 9th October
using Gradescope. The hand in will contribute 5% to your overall course mark.
Problem sheet 2: handed out 5th October. This problem sheet includes material
presented in lecture 6-10 and will consists of a number of questions that will be
the focus for workshops.
Problem sheet 3: handed out 26th October. This problem sheet includes material
presented in Lectures 11-15. It consists of two clearly identified hand in questions,
plus a number of other questions that will be the focus for workshops. The
deadline for submitting the hand in questions is 9am on 13th November
using Gradescope. The hand in will contribute 5% to your overall course mark.
Problem sheet 4: handed out 9th November. This problem sheet includes material
presented in Lectures 16-20. It consists of two clearly identified hand in questions,
plus a number of other questions that will be the focus for workshops. The
deadline for submitting the hand in questions is 9am on 27th November
using Gradescope. The hand in will contribute 5% to your overall course mark.
These problem sheets contain a mixture of questions designed to help you review and
consolidate the course material, exam-style questions (sometimes taken from past exam
papers, as shown) and one or two more challenging questions that go beyond the basics
of the course. These ‘challenge’ questions are marked with a *.
In addition, there will be a Class Test in week 6 (during the workshops on the 23rd,
24th and 25th October). This will consist of two unseen Quantum Mechanics exam-type
questions to be attempted in exam conditions in the workshops. Di erent questions
will be used in each of the workshops. The class test should take less than one hour
to complete, with the remainder of the workshop focussing on workshop problems as
normal. The class test will contribute 5% to your overall course mark.
Hand ins need to be entirely your own work. We encourage group learning with your
friends, for example in the workshops, but when it comes to assessment, it should all be
your own work. We keep a close eye out for cases of plagiarism, and treat it very seriously.
If in any doubt, see the note on plagiarism on the University website: https://www.ed.
ac.uk/academic-services/students/conduct/academic-misconduct/plagiarism.
Your hand ins will be marked and returned as soon as possible. If you have any queries,
please ask us. We really do want to help you as much as we can. Of course, the presence
of sta and demonstrators, as well as your fellow students, make the workshops ideal
places to raise other issues too, for example, any problems you might be having with
understanding the lecture material. Good luck with the course, and have fun.
Franz Muheim, Course Organiser.
1
Modern Physics 2023/24 Problem Sheet 1
Hand in Question
Hand in deadline 09:00 Monday 9th October 2023
This week’s handin has two questions, both taken from previous exam papers.
1.1 Electron scattering
A variant of Question B.2. of the Modern Physics Dec 2012 exam paper
A beam of electrons with kinetic energy 50 eV is fired at a crystalline nickel foil.
(a) Calculate the speed of the electrons in the beam.
(b) Based on your answer to part (a), state, with justification, whether classical (New tonian) mechanics correctly describes this system.
(c) Determine the de Broglie wavelength of the electrons.
(d) Electrons incident on the nickel are scattered. The scattered electrons show strong
peaks in intensity at particular orientations of the incident beam to the nickel
target. Explain how this can be accounted for.
[5 marks]
1.2 Uncertainty examples
(a) Write down the momentum/position form of the Heisenberg Uncertainty Principle.
Comment on the role played by Planck’s constant.
(b) A squash ball of mass 24 g is confined to a (1D) court of length 9.75 m. Estimate
the fundamental uncertainty in the speed of the ball. Comment briefly on your
result.
(c) Consider an electron confined within typical atomic dimensions. Use the Heisen berg uncertainty principle to estimate the minimum uncertainty in its momentum
(you can treat this as a 1D problem). What does your result tell you about the
minimum kinetic energy of the electron
[5 marks]
2
Modern Physics 2023/24 Problem Sheet 1
Workshop Questions
1.3 Blackbody radiation: Correspondence Principle
The correspondence principle states that the laws of quantum mechanics must reduce
to the classical result under the conditions where the classical description agrees with
experiment. As an example consider the Planck (quantum) radiation formula:
uq(λ) = 8 hc
λ5
1
ehc/λkT 1
where uq(λ) is the (quantum) energy density, h is Planck’s constant and k is Boltzmann’s
constant.
(a) Sketch this function.
(b) Also sketch the Rayleigh-Jeans (classical) radiation formula:
uc(λ) = 8 kT
λ4
where uc(λ) is the (classical) energy density.
(c) By comparing your sketches determine where the classical and quantum descriptions agree and show formally that uq(λ) approaches uc(λ) in the classical limit.
(d) For a black body the peak intensity occurs at a wavelength λmax, which is inversely
proportional to the temperature T. This is known as Wien’s displacement law and
is given by
λmaxT = 2.898 10 3 m K
Calculate λmax for:
(i) Radiation from the Sun’s surface
(ii) An incandescent lightbulb
(ii) The human body
In each case, comment briefly on your result.
(e) Explain how you could obtain Wien’s displacement law from the Planck Radiation
formula. You only need to show what steps are needed; you do not need to derive
the formula.
1.4 The Davisson Germer experiment
Experimental confirmation of particles exhibiting wavelike behaviour was made by Davisson and Germer (1925, USA). Their experimental setup is shown schematically below.
Electrons are generated in a heated filament and accelerated toward a crystalline Ni target. A detector is set at various angles φ relative to the incident beam and the intensity
of the scattered electrons is measured at di erent accelerating potentials.
There is a strong peak in intensity at a detector angle φ = 50 for electrons accelerated
through a potential of 54 V.
3
(a) The spacing between the di racting planes of Nickel is 0.091 nm. Calculate the
wavelength of the scattered electrons using
(i) Bragg’s law.
(ii) the de Broglie relation.
(b) Explain how this result provided experimental confirmation of the de Broglie hypothesis.
1.5 The Bohr Atom
(a) Consider an atom made up of a nucleus of charge +Ze and mass M. Assuming an
electron executes a circular orbit around the nucleus and that the mass m of the
electron is negligible compared to that of the nucleus, write down the condition
for mechanical stability of the electron orbit.
(b) Show that a consequence of Bohr’s postulates is that the allowed electron orbits
are restricted to those having radii
r = 4 0~2
mZe2 n2 where n = 1, 2, 3,…
(c) Obtain a numerical value for the Bohr radius (i.e. the ground state radius) for
hydrogen.
(d) Show that the orbital speed is given by
v = Ze2
4 0~n
where n = 1, 2, 3,…
(e) Calculate the value of the orbital speed of an electron in the ground state of a
Hydrogen atom. Comment on whether this justifies the use of classical rather
than relativistic mechanics in the Bohr model.
4
1.6 Size of the Hydrogen atom
(a) Write down the classical expression for the total energy of a Hydrogen atom with
an electron of momentum p moving in a circular orbit of radius r. Keep the
potential and kinetic energies separate.
(b) Using the condition that in the lowest energy state the orbit circumference is one
de Broglie wavelength, express p in terms of r and hence obtain an expression for
the total energy as a function of r.
(c) Explore this expression; which term (kinetic energy or potential energy) dominates
at large values of r Which dominates at small r
(d) Find the value of r for which the total energy is a minimum. Compare your result
to the expression for the Bohr radius given in the lecture notes.
1.7 Uncertainty: natural linewidth
The time/energy Heisenberg Uncertainty Principle is the source of a natural linewidth
λ in photons emitted from atoms when electrons change orbitals.
(a) Using the Rydberg formula, calculate the frequency of light emitted in the n2 ! n1
transition for hydrogen.
(b) If the average lifetime of the excited state = 1.6 nsec, estimate the uncertainty
in the energy E of the emitted photon.
(c) Using your result from (b), obtain an estimate of the natural linewidth λ of the
spectral line associated with the n2 ! n1 transition.
1.8 Probability density for classical objects*
A rock of mass m, initially at rest, is dropped o a cli of height H. As the rock falls, a
large number of photographs are taken of it at random times. Each photograph is used
to obtain a measurement of the displacement of the rock with respect to its starting
point.
(a) What is the equation relating the displacement of the rock x(t) to time t You
can assume negligible air resistance.
(b) Qualitatively, would you expect the average of the displacements (as measured on
the photographs) to be less than, the same as, or more than half the height of the
cli
(c) The probability of the measured displacement being in the region x to x + dx can
be expressed as P(x)dx, where P(x) is the probability density for displacement.
Convince yourself that P(x)dx is given by
P(x)dx = dt
T
where dt is the time that the rock spends in the region x to x + dx and T is the
total time taken for the rock to reach the ground.
5
(d) Show that the probability density P(x) is
P(x) = 1
2
p
Hx
where 0