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    Problem Set 1

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    darklordsauron
    This document contains instructions for a quantum physics problem set due on March 2nd. It includes 5 problems related to topics in quantum physics like the stability of the classical hydrogen atom model, Fourier transforms, quantum mechanical treatment of particles at different temperatures, photon absorption, and Gaussian probability distributions. Students are instructed to show their work, include explanations, and submit their solutions as a PDF by the due date. Late assignments will be penalized.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Problem Set 1

    Uploaded by

    darklordsauron
    46 views3 pages
    This document contains instructions for a quantum physics problem set due on March 2nd. It includes 5 problems related to topics in quantum physics like the stability of the classical hydrogen atom model, Fourier transforms, quantum mechanical treatment of particles at different temperatures, photon absorption, and Gaussian probability distributions. Students are instructed to show their work, include explanations, and submit their solutions as a PDF by the due date. Late assignments will be penalized.

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    This document contains instructions for a quantum physics problem set due on March 2nd. It includes 5 problems related to topics in quantum physics like the stability of the classical hydrogen atom model, Fourier transforms, quantum mechanical treatment of particles at different temperatures, photon absorption, and Gaussian probability distributions. Students are instructed to show their work, include explanations, and submit their solutions as a PDF by the due date. Late assignments will be penalized.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    46 views3 pages

    Problem Set 1

    Uploaded by

    darklordsauron
    This document contains instructions for a quantum physics problem set due on March 2nd. It includes 5 problems related to topics in quantum physics like the stability of the classical hydrogen atom model, Fourier transforms, quantum mechanical treatment of particles at different temperatures, photon absorption, and Gaussian probability distributions. Students are instructed to show their work, include explanations, and submit their solutions as a PDF by the due date. Late assignments will be penalized.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    PHYS2013 Quantum Physics Problem Set 1 (2015)

    Due Monday 2nd March, 5pm, please submit your solutions in a PDF document via
    the Wattle site.
    Please attach a completed cover sheet to your work.

    All late submissions of homework will be penalised as explained on the Wattle


    site.
    For an item to be marked correct you must supply working and an explanation as well
    as the correct answer. These must be complete and legible, otherwise zero marks may
    be given.
    1. Stability of a classical atom.
    This problem requires recalling your understanding of classical physics.
    8 marks. 1 mark each for (a) through (h). 1 mark if completely correct. Otherwise 0 or
    at the markers discretion.
    By the end of the 19th century it was known that as a consequence of Maxwells
    equations an accelerating charge q loses energy E at the rate given by the Larmor
    formula:
    dE
    2 q2 a2
    =
    ,
    dt
    3 4 0 c 3
    where a is the magnitude of the acceleration. Consider a classical electron in a
    circular orbit of radius r about a proton, i.e. a hydrogen atom model. Assume nonrelativistic physics.
    (a) Find the kinetic energy of the electron as a function of radius.
    (b) Find the magnitude of the acceleration of the electron as a function of radius.
    (c) Find the total energy of the system as a function of radius.
    (d) Find the power radiated as a function of radius.
    (e) Find the rate of decrease of the radius, as a function of radius. This is a differential
    equation for r. Observe that the power radiated increases as the electron spirals into
    the proton.
    (f) For r = 5x10-10 m calculate values for all these quantities.
    (g) Solve the differential equation you found in (e) for r as a function of time t. Note
    that:
    dr 1 dr 3
    r2
    =3
    .
    dt
    dt
    (h) Hence estimate the time it takes the electron to spiral into the proton from an
    initial orbit of radius r = 5x10-10 m. How does this compare to a typical
    experimentally measured excited 2p state lifetime of about 1.6 ns?

    2. Fourier transforms.
    3 marks. 1 mark each for part completely correct.
    (a) Calculate the Fourier transform ( p ) =

    of the square wave function ( x ) =

    1
    d

    1
    2

    ( x ) eipx/ dx

    for d / 2 x d / 2
    0 elsewhere

    (b) & (c) Plot or sketch each function.

    3. (a) In thermal equilibrium the average kinetic energy of a particle is


    p2
    3
    = kB T
    2m
    2
    where T is in Kelvin and kB is Boltzmanns constant. The lattice spacing in a
    typical solid is 0.3 nm. For what temperatures do we have to treat (a) electrons
    and (b) nuclei in solids quantum mechanically?
    (b) Using the general gas law
    P V = NkB T
    to deduce the interatomic spacing, find out at what temperatures we need to
    treat atoms in an ideal gas at pressure P as quantum mechanical. (Answer:
    T < (1/kB )(h2 /3m)3/5 P 2/5 ). Is the helium inside a childs balloon quantum
    mechanical?
    (c) At Lawrence Livermore National Laboratory in the USA, there is a facility
    for accelerating electrons to energies of about 100 MeV. Ignoring relativistic
    effects, calculate the de Broglie wavelength of these electrons.
    (d) The Large Electron-Positron collider (LEP) at CERN in Europe accelerated
    electrons to energies of around 100 GeV. Again ignoring relativistic effects,
    calculate the de Broglie wavelength of these electrons.
    (e) Do you think the non-relativistic approximation is justified in either case?
    (5 marks)
    4. Show that a stationary electron cannot absorb all the energy of a photon (hint:
    consider conservation of energy and momentum). Is it sensible to worry about
    stationary electrons? (3 marks)

    5. Often we want to calculate the average of a quantity which doesnt take on discrete
    values, such as the position of a particle along the x-axis, or the maximum frequency
    audible to lecturers in the faculty of science. Then the probability of measuring a
    value of x in the interval between a and b is
    P (a < x < b) =

    (x)dx

    (5)

    Here (x) is the probability density. This has to be normalized so that the integral of
    (x) from to + is equal to 1 this is equivalent to requiring that the particle
    has to be somewhere. The equations for the mean etc then become
    hxi =
    hf (x)i =

    Z +

    x(x)dx

    (6)

    f (x)(x)dx

    (7)

    2 = hx i hxi2

    (8)

    Consider a Gaussian distribution,


    2)

    (x) = Ae(c(xa)

    where a, A and c are constants. Such distributions are extremely common in all
    areas of science (and social science), and also play an important role in quantum
    mechanics, so it is good to familiarise yourself with them.
    Looking up any integrals you need (i.e. from sos maths, mathematica, or other similiar
    sources),
    (a) Sketch a graph of the probability density. (1 marks)
    (b) Evaluate A in terms of the other constants. (2 marks)
    (c) Find hxi, hx2i and . (3 marks)

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