What is energy quantization in a finite well? [closed]

The Context to my Question: I have a hydrogen nucleus, a deuteron, a proton and neutron, and I am trying to show that a binding energy of $\beta=2.2 MeV$ would be valid. This is a finite potential well problem. \begin{align} U_0=36MeV\qquad &a=2.6fm\\ \psi(0)=0;\qquad&\psi(x)=-\psi(-x)\\ \end{align} Where $U_0$ is the potential wall of the well, $a$ is the width of the well, $\psi$ is the anti-symmetric wave equation. I think the relationship of binding energy to potential and total is $$E_0=U_0 -\beta$$ Given that $\beta=2.2MeV$ and $U_0=36MeV$, then $E_0 = 33.8MeV$. Although I assume that this deuteron would have quantized energy because it is in a finite well problem, and because the nucleus's of atoms are described with the Schrodinger equation. This energy quantization should be defined by $$2cot(ka)=\frac{k}{\alpha}-\frac{\alpha}{k}$$ Where $k^2=\frac{2m}{\hbar}E$ and $\alpha^2=\frac{2m}{\hbar}(U_0-E)$. This is where I become confused.

Priming: A particle in a a finite well is said have quantized energy. Although, it seems that any value for $U_0$ where $0 < U_0$ and any value for $\beta$ ( the binding energy ) where $0 < \beta < U_0$ will result in the stable bound state. Quantum means a discrete value, although I do not see any condition that makes $E$ discrete.

Question: What condition is making the energy discrete? How do I verify that $E_0 = 33.8 MeV$ is one of these quantized energy values?

closed as off-topic by Gert, user36790, Jon Custer, heather, Bill NOct 17 '16 at 17:28

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Community, Jon Custer, Bill N
If this question can be reworded to fit the rules in the help center, please edit the question.

• hyperphysics.phy-astr.gsu.edu/hbase/quantum/pfbox.html, for a single particle in a box. Your problem appears to be a multiple particle problem though. Please clarify what's in the box and what is being bound there, it's far from clear (at least to me) :-) – Gert Oct 15 '16 at 23:55
• Yes there are two particles, but they can be treated as one. The force bining them is the strong-nuclear force. Because the proton and neutron are of similar mass, I can use relative mass, $m=(m_p*m_n)/(m_p +m_n)$ to treat this problem as a single particle in a finite well. The mass is approximately $m\approx\frac{m_p}{2}\approx\frac{m_n}{2}$. – Tsangares Oct 16 '16 at 3:47
• I meant reduced mass. en.wikipedia.org/wiki/Reduced_mass – Tsangares Oct 16 '16 at 3:57
• I believe if you plot the left side and the right side of your equation simultaneously on the ordinate and $k$ (or $E/U_0$) on the abscissa, you will find there are discrete values of $k$ (or ...) which will satisfy the equation. – Bill N Oct 17 '16 at 17:27