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!

Homework Help: Potential barrier problem

  1. Sep 10, 2010 #1
    1. The problem statement, all variables and given/known data
    a particle of kinetic energy E is incident from left on a potential barrier,height U, situated at the origin.the barrier is infinitely wide and E>U

    obtain an expression for the reflection coefficient R of the particle as a fuction ratio e=E/U


    2. Relevant equations



    3. The attempt at a solution

    to left of barrier wavefunctions are free particle waves
    barrier at x=0
    psi(x,t)=Aexp(ikx-wt)+ Bexp(-ikx-wt) x<0

    within barrier wavefunction also is oscillatory

    E=h(cross)*w

    considering case for E<U and using TISE on psi(x,t) within barrier we get

    a=([2m(U-E)]^0.5)/h(cross)

    but now E>U and as a result a becomes imaginary. introducing new wavenumber L and barrier wavefunction becomes

    psi(x,t) =Cexp(-iLx-wt)+Dexp(iLx-wt) x>0 (note this is equation is only for a barrier of finite width)

    but now everywhere to right from origin x=0 is the barrier wavefunction given above . to keep psi(x,t) from diverging for large x we must take D=0 leaving only decaying wave and this is where i am up to i was just wanting to know if i am on the right track
     
  2. jcsd
  3. Sep 11, 2010 #2

    kuruman

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You need two sets of wavefunctions, one for x<0 and another one for x>0. The first set for x<0 has two pieces, one representing the incident wave and one representing the reflected wave. For x>0 you have only one wave, the transmitted wave traveling to the right. Since E > U, all waves are represented by complex exponentials. These are basically sinusoidals and do not decay with x.
     
  4. Sep 11, 2010 #3
    ok i see thanks so now my wavefunctions are

    psi(x,t)=Aexp(ikx-wt)+Bexp(-ikx-wt) x<0
    psi(x,t)=Dexp(iLx-wt) x>0

    so now wavefunctions must be joined smoothly following the conditions
    A+B=D cont of psi
    ikA-ikB=iLD cont of d(psi)/dx

    solving for D i get

    A(1-k/L)=B(-1-k/L)

    B/A= -(1-k/L)/(1+k/L)

    reflection coefficient is given by R=|B^2|/|A^2| but i need to obtain an expression in terms of ratio E/U

    so i tried to substitute k=(2mE/h(cross))^0.5 and L=i(2m(U-E)/h(cross))^0.5 into equation but was unsuccessful to get the ratio out am i on the right track?
     
  5. Sep 11, 2010 #4

    kuruman

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Aren't k and L both real and isn't the ratio L/k related to U/E somehow?
     
  6. Sep 11, 2010 #5
    yes sorry both L and k are real and i get E/U =-1/2 when i put them equal to each other but now im kind of lost
     
  7. Sep 12, 2010 #6

    kuruman

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I don't see how you get E/U = -1/2. What are your (correct) expressions for k and L?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook