(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The deuterium nucleus (a bound state of a proton and a neutron) has one bound state. The force acting between a proton and a neutron has a strong repulsive component of range 0.4 fm and an attractive component of range ~2.4 fm. The energy needed to seperate the neutron from the proton in a deuterium nucleus is 2.2 MeV. Treat the neutron in deuterium as a particle of mass 1.67*10^-27 kg in a potential well f width 2 fm. Estimate V_0 for this potential.

2. Relevant equations

This potential well starts at Psi(x) = V_0 for x< -a/2, then Psi(x) = 0 for x between -a/2 and a/2, then back up to V_0.

m = 1.67*10^-27 kg

E = energy

hbar = reduced planck's constant

k_2 = sqrt(2*m*E/hbar^2)

alpha = k_2*a/2

V_0 is potential

P = sqrt(m*V_0*a^2/(2*hbar^2))

alpha*tan(alpha) = sqrt(P^2-alpha^2)

-alpha*cot(alpha) = sqrt(P^2-alpha^2)

3. The attempt at a solution

There's one bound state, so it'll be in the alpha*tan(alpha) = sqrt(P^2-alpha^2) equation. I got it down to tan(548000sqrt(E)) = sqrt(V/E-1) (in joules). V should come out to be 66 MeV according to his answers (he gives answers to some questions, just cares about work.) It's possible this was a typo too as it's happened before, but I'll assume it's right.

I have no idea how to find E though. I assume it has to do with that 2.2 MeV used to remove the neutron but I'm not sure how. If my equation is wrong, that's not that big of a deal, I can fix it, I'm mostly worried about how to find E. My only idea is to relate E and V by conservation of energy but I'm not sure how.

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# Finding the potential of a 1d finite square potential well

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