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VortexLattice

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From what I can tell, this seems to be the general formalism Shankar uses:

1. Expand f (the scattering amplitude) in terms of Legendre polynomials:

[tex]f(\theta,k) = \sum _{l = 0} ^\infty (2l + 1)a_l(k)P_l(cos\theta)[/tex]

(l is angular momentum, k is wave number, a is coefficient)

2. We expand the incoming wave like this too:

[tex]e^{ikz} = e^{ikr cos(\theta)} = \sum _{l = 0} ^\infty i^l (2l + 1)j_l(kr)P_l (cos\theta)[/tex]

3. We take the limit of the incoming wave as r → ∞ (using limit of j_l):

[tex]\frac{1}{2ik}\sum _{l = 0}^{\infty} (2l + 1)(\frac{e^{ikr}}{r} - \frac{e^{-(ikr - l\pi)}}{r})P_l(cos(\theta))[/tex]

4. We know that as r → ∞, the radial wave function has to become the same as the one for a free particle, so we take that limit but note that there can be a "phase shift" in it:

[tex]R_l(r) = \frac{A_l sin[kr - l\pi/2 + \delta_l(k)]}{r}[/tex]

5. Then, we basically combine these two, compare some coefficients, and get the form that the whole wave function should have as r → ∞:

[tex]\psi_k(\vec{r}) = e^{ikz} + [\sum _{l = 0} ^\infty (2l + 1) (\frac{e^{2i \delta_l} - 1}{2ik}) P_l(cos\theta)]\frac{e^{ikr}}{r}[/tex]

So this much makes sense to me, minus a few sketchy little things. But I can buy it. This is the formalism. Now, here's as much as I've gleamed about actually SOLVING a real problem:

1. We're given the potential the electron is going to scatter off of.

2. We solve the Schrodinger equation for that potential.

3. We look at boundary conditions to get more info about some coefficients

4. We take the limit as r → ∞ of this solution, and put the solution in the form of [itex]f(\theta,k)[/itex] like above.

5. This gives us [itex]\delta_l[/itex].

6. Now we have f, and we can solve for stuff like the cross section.

But in the problems I've done, they don't explain it around step 4 very well, which is where it connects to the formalism. Here's a simple example from Shankar. It's a hard sphere such that [itex]V(r) = \infty[/itex] if r<r_0, and V = 0 otherwise (step 1).

So, outside the potential, the particle is basically a free particle and has the form:

[tex]R_l (r) = A_l j_l(kr) + B_l n_l(kr)[/tex]

(step 2)

But we need R = 0 at r = r_0, so that boundary condition demands that:

[tex]\frac{B_l}{A_l} = -\frac{j_l(kr_0)}{n_l(kr_0)}[/tex]

(step 3)

As r → ∞, R becomes

[tex]R_l(r) = \frac{1}{kr}[A_l sin(kr - l\pi/2) - B_l cos(kr - l\pi/2)] = \frac{\sqrt{A_l^2 + B_l^2}}{kr}[sin(kr - l\pi/2 + \delta_l)][/tex]

(step 4)

With [itex]\delta_l = tan^{-1}(-B_l/A_l)[/itex] (step 5?)

From here, I'm pretty confused. I see how he did all this, but I'm not exactly sure what the equivalence between R_l and f is, which seems to be the point of all the formalism he did. He never explicitly says it. Also, in his final expression for f (basically taken from the r → ∞ expression above), he has

[tex]f(\theta) = (1/k)\sum_{l = 0}^{\infty} (2l + 1)e^{i\delta_l} sin(\delta_l) P_l(cos\theta)[/tex]

So it seems like each R_l is one of these terms...but then where are the (2l + 1) and [itex]e^{i\delta_l}[/itex] terms? I guess maybe we don't care about the exponent because you can always shift a function by an arbitrary (complex) phase shift?

Can anyone help me out??

Thanks!