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How to solve for D and sketch the wavefunctions

  1. Feb 9, 2017 #1
    1. The problem statement, all variables and given/known data
    See attached image. The potential in question is, ##-V_0## for ##0<r<a,## and ##0## for ##r\geq a.##

    2. Relevant equations
    $$\sinh(x)=\frac{e^x+e^{-x}}{2}$$
    $$\cosh(x)=\frac{e^x-e^{-x}}{2}$$

    3. The attempt at a solution
    I know that the wavefunction for ##r<a## is given by ##Asin(k_1r),## where ##k_1## is $$\frac{\sqrt{(2m(E+V_0)}}{\hbar},$$ and that the wavefunction for ##r\geq a## is $$De^{-k_2r},$$ where $$k_2=\frac{\sqrt{-2mE}}{\hbar}.$$ I can write ##De^{-k_2r}## as ##D(\sinh(k_2r)-\cosh(k_2r)).## Then imposing continuity, I have the system, $$A\sin(k_1a)=D(\sinh(k_2a)-\cosh(k_2a))$$ $$k_1A\cos(k_1a)=k_2D(\cosh(k_2a)-\sinh(k_2a)),$$ such that ##k_1 \cot(k_1a)=-k_2.## Then imposing normalization, on inner wavefunction, as per the text's suggestion, I get ##A=\frac{1}{k_1},## so that $$D=\frac{\frac{1}{k_1}\sin(k_1a)}{\sinh(k_2a)-\cosh(k_2a)}.$$ Naturally, we can rewrite ##k_2## in terms of ##k_1## from the continuity relation, from which we can sketch the wavefunctions. I'm not entirely sure if what I've done so far is correct, and if so how I would even go about sketching these wavefunctions, and doing parts b), c), and d).

    For part b), I'm guessing that we can just equate, $$-\frac{-\sqrt{-2mE}}{\hbar}=k_2$$ and rearrange for ##E.## Though I still have no idea how to semi-quantitatively sketch such a function. As well, by the logic that the probability of the particle to be somewhere in all space, the wavefunction outside the well must decay. (That should describe the behavior outside the well.)

    <Moderator's note: formatting fixed. Please use ## ## for inlined LaTeX.>
     

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    Last edited by a moderator: Feb 10, 2017
  2. jcsd
  3. Feb 17, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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