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Quantum Mechanics Problems

  1. Apr 6, 2009 #1
    1. The problem statement, all variables and given/known data

    I've actually got a few questions here, some I think I've been reasonably succesful in - others I don't know where to begin. I really am confused by some of this stuff, so a nudge in the right direction would be great. I know my lecturer will probably find and read this - (It's not your fault! I've been preoccupied with other subjects :( Probably too much so.)
    On the other hand - this means that I will certainly be re-writing/calculating these solutions to I remain anonymous. In other words - If you take the time to help me with this I'll certainly be learning from you :)

    Question 1 [20 marks] - Attempted solution below.
    In one of the lectures, a recent experiment showing the interference of buckminsterfullerene (C60)
    molecules using a double slit apparatus was discussed.
    (a) If the average velocity of the C60 molecules is v = 200 m/s, calculate the de Broglie
    wavelength for a C60 molecule.
    (b) If the C60 particles are incident on two 50 nm slits with a center-to-center separation of 100
    nm, what is the distance between the first two maxima deposited on a screen located 2 m
    behind the slits. Given that the resolution limit of optical microscopy is comparable to the
    wavelength, would you be able to resolve the maxima optically using this set-up?
    (c) Suppose one wanted to attempt the Davisson-Germer experiment with C60, and planned to ionise the C60 to C60 2− and then use a potential difference to accelerate the C60 molecule to a velocity sufficient to achieve a wavelength λ = 1.65 Å. How large a potential difference V would be needed to do this?

    Question 2 [10 marks]
    You wish to study a crystal’s structure by diffracting thermal, non-relativistic neutrons from the
    crystal, so you go to a nuclear reactor and set up your diffraction experiment on one of the neutron
    beam-lines. The neutrons emerge from the reactor with a range of wavelengths (i.e., a range of
    energies/momenta), so you use a ‘chopper’ to select neutrons of a certain wavelength.
    The simplest chopper consists of two disks made from a highly neutron absorbing material,
    mounted on an axle. The disks are 2 m apart. In each disk is a small slot (to let the neutrons
    through). There is an offset of 10° between the two slots and the chopper rotates rapidly.
    The distance between the planes of atoms in the diffracting crystal is 1.73Å and you observe strong
    diffraction when the angle between the incident and diffracted neutron beams is 163°. Calculate the
    minimum rotational speed of the chopper in revolutions per minute.
    Question 3 [15 marks]
    (a) Rayleigh’s criterion is used to determine when two objects are resolvable by a lens of
    diameter d. The angular separation of the objects must be at least θ R where:
    θ(subscript r) = 1.22*(λ/d)
    In order to resolve two objects 4000 nm apart at a distance of 20 cm with a lens of diameter 5
    cm, what energy (i) photons and (ii) electrons should be used?
    Is this consistent with the uncertainty principle? Explain briefly and support with numerical
    estimates if required.
    (b) Assume that the uncertainty in the position of a particle is equal to its de Broglie wavelength.
    Show that the uncertainty in its velocity is equal to or greater than 1/(4π) times its velocity.

    2. Relevant equations
    (included in the questions above)

    3. The attempt at a solution
    Question 1:

    A) λ=h/p=h/mv=(6.626 068 96 *(10)^(-34))/(200*(720.66*1.660538782*(10)^(-27) ) )=2.768512669*(10)^(-12) Meters
    B) x=(n*λ*L)/d=(n*2.768512669*(10)^(-12)*2)/(50*(10)^(-9) )
    Sub in n=1 and n=2, and find the difference:
    (2*2.768512669*(10)^(-12)*2)/(100*(10)^(-9) )-(1*2.768512669*(10)^(-12)*2)/(100*(10)^(-9) )=5.537025342*(10)^(-5) Meters

    Given the limit of optical microscopy is comparable to the wavelength of this particular particle then it is reasonable to assume that you would be able to resolve the maxima optically using this setup as the distance is several orders of magnitude larger.

    C) 1/λ=√((2*m*e*V) )/h
    ∴ V = (1/2)*h^2/(λ^2*m*e)=0.00004217267763volts

    As for 2 and 3, I am genuinely stumped for the moment. I'll continue doing some more work on them as I wait for a reply, But I've been stuck on these for quite some time now. Please find it in your soul to help out an undergrad who screwed up his time management :)
    These are due on thursday, so I've got a little time yet.
    Last edited: Apr 6, 2009
  2. jcsd
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