Two step (spherical) square well potential

[IHateMayonnaise]

Homework Statement

Just need some rough guidance on this one, nothing specific is really needed. The problem:

Given a spherically symmetric potential (V(r))
$$V\left( r \right) = \left\{\begin{gathered} V_0 \hfill \hspace{2}r<a \\ 0 \qquad a<r<b \\ \infty \hfill r>b \\ \end{gathered} \right$$

Find the energies for the ground state and the first excited state. Also find an (approximate) expression for the energy splitting of the levels if $V_0$ is very large compared to these energy levels.

Homework Equations

Must use spherical Bessel and Neumann functions.

The Attempt at a Solution

The wavefunctions:

$$\mathcal{U}_I(r)=A_rJ_{\ell}(k_1r)$$
$$\mathcal{U}_{II}(r)=C_rJ_{\ell}(k_2r)+D_r n_{\ell}(k_2r)$$
$$\mathcal{U}_{III}(r)=0$$

$$k_I=\frac{\sqrt{2m(V_0-E)}}{\hbar}$$

$$k_{II}=\frac{\sqrt{2mE}}{\hbar}$$

where the spherical Neumann function ($$n_{\ell}$$) goes away in the first function since it is singular at the origin.

At this point work with the following boundary conditions:

$$\mathcal{U}_I(a)=\mathcal{U}_{II}(a)$$
$$\mathcal{U}_I^{'}(a)=\mathcal{U}_{II}^{'}(a)$$
$$\mathcal{U}_II(b)=0$$

From here you get a three equations, where the derivatives are taken with respect to K1 and K2 (respectfully). First question: How do I take a derivative of the spherical Bessel and Neumann functions? Should I use the recursion formula and solve for the derivative?

Actually I am just not quite sure where to go from here in terms of finding the ground state energy. If this were a regular potential (two step) barrier, all I would do is eliminate the three constants using the three equations, arriving at a transcendental equation which I would then graph graph each side of the equation separately and find the intersections which represent the energy eigenvalues. Same thing here? Any thoughts?

Thanks yall

IHateMayonnaise

 

[Jhero]

Dear IHateMayonnaise,

Thank you for your post. It seems like you have made a good start on solving this problem. To take derivatives of the spherical Bessel and Neumann functions, you can use the recursion formula as you suggested. Another option would be to use the identities for the derivatives of these functions, which can be found in most mathematical physics textbooks or online resources.

As for finding the ground state energy, you are correct in saying that you can eliminate the three constants using the three boundary conditions and arrive at a transcendental equation. This is the standard approach for solving problems with a spherically symmetric potential. You can then graph both sides of the equation and find the intersections to determine the energy eigenvalues.

In terms of the energy splitting for large V_0, you can use the approximation that the spherical Bessel function is approximately equal to the spherical Neumann function when the argument is large. This can help simplify the transcendental equation and make it easier to solve for the energy splitting.

I hope this helps guide you in the right direction. If you have any further questions or need clarification, please don't hesitate to ask.

Best of luck with your calculations!