How to solve for D and sketch the wavefunctions

[vbrasic]

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.)

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[mark726]

Your approach to solving this problem is correct. You have correctly identified the wavefunctions inside and outside the potential well, and have used the continuity and normalization conditions to determine the coefficients A and D.

To sketch the wavefunctions, you can plot the functions $Asin(k_1r)$ and $De^{-k_2r}$ separately, and then combine them to get the overall wavefunction. The wavefunction inside the potential well will have a sinusoidal shape, while the wavefunction outside the potential well will decay exponentially. You can use the continuity condition to determine the values of k1 and k2, and then plot the wavefunction for different values of V0.

For part b), you are correct in assuming that the energy can be determined by equating $-\frac{-\sqrt{-2mE}}{\hbar}=k_2$ and solving for E. To sketch the energy levels, you can plot the energy values for different values of V0 and observe how they change with varying potential depth.

For part c), you can use the wavefunction to calculate the probability density of finding the particle at a particular point. This can be done by taking the square of the absolute value of the wavefunction at that point. You can then plot the probability density for different values of V0 to see how it varies.

For part d), you can use the wavefunction to calculate the expectation value of the position of the particle. This can be done by multiplying the wavefunction by the position operator and integrating over all space. You can then plot the expectation value for different values of V0 to see how it changes.

I hope this helps. Good luck with your calculations.