1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Confused on a Certain Approximation/Expansion

  1. Aug 25, 2008 #1
    Hi, I need a quick bit of help for an approximation/expansion used in a paper by Rolf Landauer (1950). The paper deals with the WKB method (Quantum Mechanics) but that is irrelevant to my question. There is an approximation presented as follows:

    A = 1 - (K/2a)

    K = a - b

    If a>>K, then A will be given by:

    A = (b/a)^(1/2), to the first order in K

    How is this approximation made? This is the entirety of the information provided: it is at the very start of the paper.

    I've change a couple variable names (a and b are infact k1 and k2, the propogation constants in a potential well, if anyone is interested). A is the transmission coefficent of a wavefunction. I appreciate any help!


  2. jcsd
  3. Aug 26, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hmm smells like one of those fishy physicists' "if f(x) = g(x) are equal to first order, they're equal" tricks :)

    Where I got, is:
    if a >> K then [itex]\epsilon = K/a[/itex] is very small.
    And then indeed, in first order,
    [tex]\sqrt{1 - \epsilon} = 1 - \frac12 \epsilon + \mathcal O(\epsilon^2) = A + \mathcal O(\epsilon^2) [/tex].
    Now if there were a plus instead of a minus,
    [tex]1 + \frac{K}{a} = \frac{a + b - a}{a} = \frac{b}{a}[/itex]
    and it would have been explained (sort of).
    Unfortunately, I think it should be a minus... but maybe this puts you on a track.
  4. Aug 26, 2008 #3
    Hmm that could be the right direction.

    Another bit I neglected to mention (but may help with the puzzle!), deals with another value B.

    B = K/2a

    And following the approximation that yields A = (a/b)^(1/2), it implies that this means that B>>A.

    This obviously suggests that K/2a is very small, thus A is approximately 1 and B is approximately 0 (which fits with the following discussion). However, I'm still not sure how the approximation of A is done!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook