Quantum - 1D crystal model using evenly spaced delta functions

[lunamoon_girl]

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.

^_^

 

[Gokul43201]

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

(b) makes no sense to me at all. Maybe someone else will do better.

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

Will it really? First of all, the delta functions do not take any form, it is the wave function whose form you want. What are solutions to the SE for regions where V=0?

 

[lunamoon_girl]

lol. The answer was stupid/trivial for part c. It is apparently as if the crystal is an endless string of delta potentials - aka it looks like a string of delta potentials from - infinity to infinity because the wave function will continue to traverse the circle without 'realizing' it has already gone 2pi.

Anyways, part B was simply algebra. I made it work eventually, but everything I said about how to do the problem was right -for anyone who sees this and is curious

 

[marcusl]

I don't get it. Shouldn't the wavefunctions be propagating Bloch waves? My quantum books are at work so I can't look anything up--but isn't this potential a first approximation for the ion cores in certain solids? (or have I mis-remembered...)

 

[badphysicist]

Yes, you can find this in quantum books (griffiths) and in solid state books (kittel). It is called the Dirac comb.