1. Not finding help here? Sign up for a free 30min 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!

Making the math easier

  1. Dec 8, 2007 #1
    [SOLVED] Making the math easier

    1. The problem statement, all variables and given/known data

    I am doing a free particle scattering/tunneling by a well and a barrier combination. There are four regions of interest where the wave numbers are different. I have seen the solution inside the well be expressed as a linear combination of exponents or sine and cosine functions.

    Now I have only seen free particles in these problems as a linear combination of exponents... can I rewrite them as a linear combination of sine and cosine. I am finding these exponent forms very very cumbersome b/c I get two terms when I match the wave functions @ x=0, if they were in a sine and cosine combination, then it would sine would be zero @ x=0




    2. Relevant equations
    3. The attempt at a solution

    This is more of a discussion and there is too much work to present here.
     
  2. jcsd
  3. Dec 9, 2007 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    You can use the identity
    [tex]e^{i\theta} = \cos\theta + i \sin\theta[/itex].
    So, for example
    [tex]\sum_k a_k e^{ikx} = \sum_k a_k (\cos(kx) + i \sin(kx))[/tex]
    and then you can try to rewrite this (for example, if the sum runs from -infinity to +infinity, you can use that sin(-kx) = -sin(kx) and cos(-kx) = cos(kx) to simplify).
    If the exponent is real, you can use
    [tex]e^{\theta} = e^{i(-i\theta)} = \cos(-i\theta) + i sin(-i\theta)[/tex]
    and subsequently
    [tex]\sinh x = -i \sin(i\theta), \cosh x = \cos(i\theta)[/tex]
    to write them in hyperbolic sines and cosines.

    Conversely, if you have a combination of sines and cosines, you can always write them in exponentials. All of this is the reason we usually write a plane wave as something like exp(i(kx - wt)) with k the wavevector and w the frequency: you can write it out in sines and cosines to say that it really "waves" :smile:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Making the math easier
  1. Commutator maths (Replies: 8)

  2. Tensor math (Replies: 1)

  3. Math. physics (Replies: 10)

Loading...