Challenging Semi-Infinite Potential Well Problem

[mancini0]

Homework Statement

"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."

Homework Equations

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

Done above

 

[vela]

You need to find solutions for inside the well and outside the well and then match them at the boundary. From that, you should come up with a condition that allows you to solve for the energy of the bound states.

 

[mancini0]

Thank you. When x <0,
ψ₁ = Ae^ik₁x+Be^-ik₁x

When 0<x<L, ψ₂ = Ce^kx+ De^-kx

When x> L ,
ψ₃ = Fe^ik₁x+Ge^-ik₁x

Now I can see how that the A term represents the incident wave, the B term represents the reflected wave, The F term represents the transmitted wave, and the G term must be zero because there is no wave moving in the -x direction in region 3. I am having trouble coming up with conditions that describe the C and D coefficents.

The k in region 2 is = sqrt(2m(V-E) / hbar
and k₁ in regions 1, 3 = sqrt(2mE) / hbar.

Now attempting to apply boundary conditions...
The ψ₁ wave must be 0 as x --> infinity, but if I take the limit of ₁ as x --> infinity, the A term goes to infinity, so I assume A must equal 0. But the A term is the incident wave, so I can't see how I can just "delete" it like that. I know I'm thinking of it wrong, because taking the limit as x --> negative infinity would force me to delete the B term as well.

The book explicitly skipped over the application of the boundary conditions, saying the algebra is "too long". Very frustrating, as this is my first encounter with applying boundary
conditions. This stuff is fascinating. Could you give me another nudge in the right direction?

 

[vela]

It's a bound state. There is no incident wave.

 

[Demon117]

Thank you. When x <0,
ψ₁ = Ae^ik₁x+Be^-ik₁x

When 0<x<L, ψ₂ = Ce^kx+ De^-kx

When x> L ,
ψ₃ = Fe^ik₁x+Ge^-ik₁x

So according to the definition of the potential this means that the function ψ₁ does not exist?

In the end I believe you'll end up with two equations and three unknowns. Can someone confirm my thoughts?

 

[Steely Dan]

In the end I believe you'll end up with two equations and three unknowns. Can someone confirm my thoughts?

In these types of problems, you always have one independent coefficient after matching boundary conditions; that coefficient is then determined by normalizing the wavefunction over all space.