Solve Delta Potential Well for Scattering Matrix & Prove Unitarity

[Felicity]

Homework Statement

consider the scattering matrix for the potential

2m/hbar² 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)

Homework 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)

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_re^kx +B_re^-kx x < b

= A_le^-kx +B_le^kx x > b

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

so

k(A_re^kb -B_re^-kb+A_le^-kb -B_le^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 r² = R (reflection coefficient and t²= 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

 

[Gokul43201]

Aren't you missing a bunch of "i"s in your wave-function exponents?

What is the potential for [itex]x \neq b[/itex]? What are the wave-functions in such a region of space?

PS: These forums support [itex]\LaTeX[/itex]. See this thread: https://www.physicsforums.com/showthread.php?t=8997 or click on my tex to see the code.

 

[Felicity]

Thank you for replying to my question,

About the i's

I realize now that I was assuming that λ was negative which is the condition for bound states but I realize that this was not a good assumption (turns out this only a well when λ <0) and that the i's should in fact be there.

About the potential

I believe the potential where x does not equal b is 0 however in class my professor indicated that the particle/wave in such a potential does not actually live at b but around it and yet it is still bound (assuming a well)

Even with the i's however, I still end up with one equation featuring r, t and u(b). I'm not sure how to get the scattering amplitudes from this or if I am even on the right track.

Thanks for the heads up about Latex, I will familiarize myself with the system

Thanks again,

Felicity

 

[Gokul43201]

So combining all of that and starting over... you should have, for an incoming particle from the left:

[tex]u_l(x<b)=e^{ikx}+re^{-ikx}~~~~~~~~~~~~~~~~~~~~~(1)[/tex]

[tex]u_r(x>b)=te^{ikx}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(2)[/tex]

and

[tex]\left[\frac{du_l}{dx}\right]_{x=b-}-~\left[\frac{du_r}{dx}\right]_{x=b+}=\frac{2m\lambda}{\hbar ^2} \int_{b-}^{b+} \delta (x-b)u(x)dx~~~(3)[/tex]

with (3) evaluated in the limit at [itex]b- \rightarrow b[/itex] from the left and [itex]b+ \rightarrow b[/itex] from the right.

Also keep in mind that while there is a discontinuity in [itex]du/dx[/itex], there is none in [itex]u(x)[/itex] itself.

[tex]u_l(x=b)=u_r(x=b)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(4)[/tex]

This is the second boundary condition that you need to solve for the coefficients.