Yukawa Potential Homework: Calculating Meson Energy Levels

[raul_l]

Homework Statement

I have been given the following assignment:
"The meson moves within the nucleus under the influence of a screened Coulomb potential known as the Yukawa potential. Given that the mass of a meson is about 270 times the mass ot the electron and that the effective extent of the potential is of the order of the size of a nucleus, calculate the energy levels of mesons within a nucleus consisting of 10 protons and 10 neutrons."

Homework Equations

$$V(r)=- \frac{Ze^2}{4\pi \epsilon_{0} r} e ^{-\frac{r}{a}}$$

The Attempt at a Solution

I have a very simple question. What exactly is r? The distance from the center of the nucleus? What happens if we get closer and closer to the center of the nucleus? Does the potential reach infinity or does it drop to zero (kinda like gravity which usually increases with 1/r but drops to zero a the center of Earth because of the Gauss law)?

I need to understand these basics things before I can move on.

I've tried searching the web but couldn't find anything useful.
I'd really appreciate it if anyone could shed some light on this.

 

[Confundo]

r is just the distance from the centre as in the coulomb potential.

 

[malawi_glenn]

This is not a real situation, so don't bother so much about what "really" happens to the meson.

In reallity you have a hard core and so on..

Yes, I think you should think r as radius. The problem is related to the hydrogenic atom I think..

 

[raul_l]

Still, how does the meson behave inside the nucleus? Will the potential reach infinity? Or should I just consider the nucleus as a hard core and stay out of it?

In my assignment it says "extent of the potential is of the order of the size of a nucleus". Does this mean that it is practically zero outside the nucleus?

I'm sorry if these questions seem silly but I've never studied particle physics before.

 

[malawi_glenn]

yes outside the potential is zero (strong force is short ranged)

Inside, you just treat the system quantum-mechanically.

Do you know how to find the energy levels of the hydrogen atom?

How does the electron behave in the coloumb potential in the hydrogen atom? Recall that.

 

[raul_l]

This is a project for the course "computational physics". I have to create a computer program to solve the problem. I have done this before for the hydrogen atom (I'm using the Shooting method).
Basically, all I need is a Schrödinger equation to describe the system. For hydrogen it was in the form $$\frac{\hbar^2}{2m}\frac{d^2 \psi}{dr^2}+(V-E)\psi=0$$.
Now I would need something similar for the "meson in a nucleus" system.

I guess I could substitute m (electron mass) with the mass of a meson, but I'm not sure I could just substitute V (hydrogen's potential energy) with $$V(r)=- \frac{Ze^2}{4\pi \epsilon_{0} r} e ^{-\frac{r}{a}}$$

 

[malawi_glenn]

Why not? I mean, from a physics point of view you do it. But for computational, no idea.