"Use the semi - infinite well potential to model a deuteron, a nucleus consisting of a neutron and a proton. Let the well width L be 3.5 x 10^-15 meters and V - E = 2.2 MeV. Determine the energy E, and determine how many excited states there are."
Since V >> E, I figured the transmission probability will be 0. I used the following equation for tunneling probability T:
T = [1 + (V^2 sinh^2(kL) / ( 4E(V-E) ))]^-1
I substituted V = 2.2MeV + E in the above equation, and used T = 0.
k was found from the following equation:
k = sqrt(2m*(V-E)) / hbar
Plugging everything yielded E = 4.96 MeV, which is wrong. The correct answer is known to be E = 7.28 MeV, with zero excited energy states.
My assumption that T = 0 must be incorrect, so I approached the problem another way. In the attached PDF, I derived an expression for Psi. Looking at this derived value for psi,
I think I need to scrap the term with the positive exponent, as then psi would approach infinity at x> L. From here do I need to normalise psi, then use the energy operator? I am having trouble doing this, as typically our HW problems only require a "plug and chug" approach. The function I found that satisfies the Schrodinger wave equation only depends on x, so using the energy operator ehat = i*hbar d/dt would yield zero, correct?
Any guidance would be greatly appreciated.
The Attempt at a Solution