# How to solve for D and sketch the wavefunctions

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1. Feb 9, 2017

### vbrasic

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.)

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Last edited by a moderator: Feb 10, 2017
2. Feb 17, 2017