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Free particle (spatial part)

  1. Oct 16, 2009 #1
    1. The problem statement, all variables and given/known data

    For a free particle, i have two expressions.

    [tex]\varphi[/tex](x) = [tex]\alpha[/tex]eikx + [tex]\beta[/tex]e-ikx
    and


    [tex]\varphi[/tex](x) = [tex]\gamma[/tex]sin(kx) + [tex]\theta[/tex]cos(kx)

    I have to determine expressions for [tex]\gamma[/tex] and [tex]\theta[/tex] in terms of [tex]\alpha[/tex] and [tex]\beta[/tex].


    2. Relevant equations

    sin(kx) = (eikx - e-ikx)/2i

    cos(kx) = (eix + e-ix)/2


    3. The attempt at a solution

    I replaced sin and cosine in the second equation.
     
  2. jcsd
  3. Oct 16, 2009 #2

    gabbagabbahey

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    That's a good start.....what did you end up with?
     
  4. Oct 16, 2009 #3
    replacing
    alpha with A
    beta with B
    gamma with M
    theta with N

    I have,

    2(Aeikx + Be-ikx) = -Meikx + Me-ikx + Neix + Ne-ix
     
  5. Oct 16, 2009 #4

    gabbagabbahey

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    Good, now just group like terms together:

    [itex]2(\alpha e^{ikx}+\beta e^{-ikx})=(\theta-\gamma)e^{ikx}+(\gamma+\theta)e^{-ikx}[/itex]

    Surely you can see where to go from here?
     
  6. Oct 16, 2009 #5
    actually that's where i'm stuck at. I know its should be something simpler but my system has lasted on me...
     
  7. Oct 16, 2009 #6

    gabbagabbahey

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    Surely you can see that [itex]2\alpha=\theta-\gamma[/itex] and [itex]2\beta=\theta+\gamma[/itex]....can't you?
     
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