(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

consider the scattering matrix for the potential

2m/hbar^{2}V(x) = λ/a δ(x-b)

show that it has the form

(2ika/(2ika-λ) , (e^{-2kib}) λ/(2ika-λ)

(e^{2kib}) λ/(2ika-λ) , 2ika/(2ika-λ)

(I've used commas just to separate terms in the matrix)

prove that it is unitary and that it will yield the condition for bound states when the elements of that matrix becoe infinite (this will only occur for λ < 0)

2. Relevant equations

suppose the matrix is expressed as

S11 S12

S21 S22

where S11 = (2ika/(2ika-λ)

S12 = (e^{-2kib}) λ/(2ika-λ)

S21 = (e^{2kib}) λ/(2ika-λ)

S22 = 2ika/(2ika-λ)

3. The attempt at a solution

I see that this is a delta potential well at x=b

ok so I know that S11 = T S21= R S22 = T and S12 = R where T and R are the reflection and transmission coefficients so I figure that if I can find those then I show the s-matrix in the above form so here it goes...

take

u(x) = A_{r}e^{kx}+B_{r}e^{-kx}x < b

= A_{l}e^{-kx}+B_{l}e^{kx}x > b

the boundary condition is (du/dx at x = b+) - (du/dx at x = b-) = λ/a u(b)

so

k(A_{r}e^{kb}-B_{r}e^{-kb}+A_{l}e^{-kb}-B_{l}e^{kb})= λ/a u(b)

then for an incoming particle that can be either reflected or transmitted I make A_{r}= 1 A_{l}= r B_{l}=0 and B_{r}=t

where r^{2}= R (reflection coefficient and t^{2}= t (transmission coefficient)

to get

e^{kb}-te^{-kb}+ re^{-kb}= λ/a u(b)

so how do I solve for r and t separately and how do I get rid of the u(b)?

Thank you

Felicity

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# Homework Help: Delta potential well

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