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lunamoon_girl
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Homework Statement
Hey, the question is about delta function potentials V(x) = -g[... del(x + 3b/2) + del(x + b/2) + del(x - b/2) ...] going on out to a large x in either direction.
a) sketch the ground-state wave fn, write the form of psi(x) for -b/2 to + b/2
b) show that e^z = (z+z0)/(z-z0) z = qb and z0 = mgb/hbar^2
c) "snip off a string of N of these sites, join the cut ends to make a molecule of length Nb, what is the lowest energy?
Homework Equations
schrodinger's eqn psi'' = 2m/hbar^2[-g*del(x-b/2) - E]psi
The Attempt at a Solution
Dividing it into regions - all of these delta functions will be even fns in the ground state - and they will take the form Ae^qx + Be^-qx
Actually -- in this case A = B
the delta functions will sort of look like telephone poles with wires hanging off of them:
^^^^ where each peak is b/2, 3b/2, etc. Make the bottom parts curved, not pointed.
So I have the following work:
integ[psi''] from just before b/2 to just after b/2 =
2m/hbar^2 (-g)psi. This was from schr. eqn where the energy was infinitesimally small because of the small integration, and the delta function killed all of the potential except at b/2. so = (-2mg/hbar^2)(B(e^q(b/2) + e^-qb/2))
You also know that psi to the right of b/2 = psi to the left of b/2 -- thus:
B(e^qx + e^-qx)(e^-qb) = C(e^qx + e^-qx)
I think... that B = C... ... er. Yeah?
The other side of the eqn is from taking the derivative of psi at b/2 = qB(e^qb/2 - e^-qb/2)(e^-qb) - qB(e^qb/2 - e^-qb/2)
So i set this = to (-2mg/hbar^2)(B(e^q(b/2) + e^-qb/2))
I am now stuck - I cannot find algebra that pops out e^z = z+z0/z-z0
It's a long question. I'm so sorry. If you can offer any guidance/corrections I would really appreciate it.
^_^