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Transmission coefficient and the reflection coefficient

  1. May 27, 2003 #1

    jby

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    I tried to work out the transmission coefficient and
    the reflection coefficient for a case similar to the
    one refered by this website:
    http://www.chembio.uoguelph.ca/educmat/chm386/rudiment/models/barrier/barsola.htm

    but instead this time, I reverse the situation and
    now, that particle I is heading from the right, ie
    from a higher potential and a possibility that it will
    be transmitted to the left, ie to a lower potential,
    still with the same E > V.

    I've drawn a diagram of the situation which I am
    considering in my question using my own notations.
    (As this is a bmp file, it may take some time to
    load.)

    http://www.geocities.com/ace_on_mark9909/reflection.htm


    My confusion is regarding to the reflection and its
    coefficient, which I've worked in steps here:

    I state here the situation I am refering to: Supposing
    the particle initially is at the potential V = 0,
    heading to the left. At x = 0, there is the sudden
    change in the potential to V = -V'.

    Using p as the wave number, ie (2pi/lambda) for the
    particle when at V = 0, and q as wave number for
    particle at V = -V', I obtained the transmission
    coefficient, T as 4pq/(p+q)^2.

    By the condition of the potential 0 > -V', thus, p <
    q, ie the wavelength at V = 0 > wavelength at V = -V'.


    Let, q = ap, ie a = ratio of final wavenumber to
    initial wave number: q/p. Since, q > p => a > 1.

    We simplify the transmission coefficient to from
    T = 4pq/(p+q)^2
    to
    T = 4a/(1+a)^2 ... (1)

    From equation 1, it states that T is only dependent on
    the ratio of the two wave number and hence dependent
    only on the ratio of both wavelengths, and not on any
    of the wavelength alone

    => the coefficient T does not discriminate on the size
    on any of the wavelength alone but the ratio of the
    magnitude of its wavelengths.

    => T does not distinguish between a particle or a
    macroscopic object, eg, a ball.

    From T = 4a/(1+a)^2, I've drawn a graph of it for a in
    the range 0 <= a <= +infinity. I've uploaded to this
    website:

    http://www.geocities.com/ace_on_mark9909/transmission.htm


    From the graph, it looks like there is a turning point
    at a = 1, corresponding to T = 1, and slowly goes to
    zero, as a -> infinity

    By a -> infinity, we can say that the potential height
    -V' approaches -infinity.

    But, if the potential at x = 0, changes so sharply as
    in approaching infinity, the graph shows T = 0, then,
    it means that if I were to replace a particle with a
    ball/human and is to approach this potential it is
    almost likely to be reflected back...

    Is there anything wrong with my maths? If not, how do
    you interpret this result?
     
    Last edited by a moderator: Feb 5, 2013
  2. jcsd
  3. Jun 11, 2003 #2

    Tom Mattson

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    Staff Emeritus
    Science Advisor
    Gold Member

    This thread got no love in the Homework forum, so I'm kicking it here to Physics.
     
  4. Jun 11, 2003 #3
    The energy of a macroscopic object is going to be so great that the potential isn't going to matter and your going to get complete transmission.

    JMD

    You should rework the solution, beacuse I doubt you get any reflection, but I haven't worked this out for a few years and can't remember right of the top of my head.
     
    Last edited: Jun 11, 2003
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