- #1

vbrasic

- 73

- 3

## Homework Statement

See attached image. The potential in question is, ##-V_0## for ##0<r<a,## and ##0## for ##r\geq a.##

## Homework Equations

$$\sinh(x)=\frac{e^x+e^{-x}}{2}$$

$$\cosh(x)=\frac{e^x-e^{-x}}{2}$$

## 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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