I Mass for solving the nuclear energy level

[just_mb]

Hi all,
I'm trying to solve a problem of finite square well for the $s$ states graphically. The task is to find energy levels and wavefunctions of proton in a spherically symmetric potential, first for deuteron then $^{48}Ca$. What makes me confused is the mass. For deuteron, the mass used is the reduced mass because it's a two-masses system. Using that, the energy from my calculation is similar to that from literature. But what about the $^{48}Ca$, what mass should I use? When I use reduced mass of $m_{proton}(19 \times m_{proton} + 28 \times m_{neutron})/(20 \times m_{proton} + 28 \times m_{neutron})$ which I found from a literature, the energies don't match with realistic Skyrme Hartree Fock calculations (figure attached). When I use the total mass $(20 \times m_{proton} + 28 \times m_{neutron})$, I found 7 $s$ energy levels as oppose to 2, as shown in the figure. Please help me, I have been searching through tons of books and journals with no luck.

 

[mfb]

Use the reduced mass, or the proton mass - they are nearly the same for larger nuclei.
It is hard to tell what went wrong if you don't show your results.

 

[just_mb]

Use the reduced mass, or the proton mass - they are nearly the same for larger nuclei.
It is hard to tell what went wrong if you don't show your results.

Thanks. That's what I used and this is the graphical solutions that I got.

where $\xi$ is given by $$\xi = \sqrt{\frac{2m(V_0 - |E|)}{\hbar^2}}a$$ where $V_0 = 45 \, MeV$, and $a = ½A^⅓ = 4.36\, fm$. As shown in the graph, the solutions are $\xi = 2.702$ and $5.298$ which according to the equation above correspond to $E = 1.04\, MeV$ and $13.73\, MeV$. The coulomb energy for $^{48}Ca$ is $75\, MeV$. If I substract those energies by Coulomb energy, the results are nowhere near those in Skyrme Hartree Fock calculations. What's wrong with my calculation?

 

[mfb]

It is still not very clear what you did, and it is surprising that you only found two solutions.

Is this part of some textbook problem (so you know this approach should give a useful answer)?

 

[just_mb]

It is still not very clear what you did, and it is surprising that you only found two solutions.

Only the $s$ states need to be solved so I think there are only two solutions, as suggested by Skyrme Hartree Fock calculations. I am also confused with that solution. In the picture, for example, the $0s1$ state of protons only differ by about 1 MeV from that of neutrons. The difference is caused by Coulomb energy. But from my calculation, the Coulomb energy is 75 MeV. Am I understanding it correctly? Can you please explain this to me?

Is this part of some textbook problem (so you know this approach should give a useful answer)?

No, it's not. But we follow the guides from Quantum Mechanics by Schiff.

 

[mfb]

I don't know.
@vanhees71?