Physics 4301: Homework 6. Problem 1 (2 points) In class we considered the scattering by a 1D delta-function potential feature V (x) =
~2 γ δ(x), m
(1)
where γ characterizes both the sign and the strength of potential (1). Here you are asked to elaborate on this. (a) Find both the scattering matrix S and transfer matrix M for potential (1). (In what follows you will also need to use modifications corresponding to the arbitrary position x0 of the potential feature (1).) (b) Show that the scattering matrix you derived for this localized feature (with the potential vanishing at both infinities) satisfies the following relation: S† = S−1 ,
(2)
where S−1 stands for the inverse matrix and S† for the hermitian conjugate (transpose conjugate, adjoint) matrix. The latter matrix has its elements both transposed and complex conjugated: S†ij = S∗ji . Operators (matrices) satisfying condition (2) are called unitary. (They are important as they preserve the norm of vectors on which they operate.) (c) Use the transfer matrix method to derive the transfer matrix for the double delta-function potential ~2 V (x) = γ (δ(x − a) + δ(x + a)) . (3) 2m (d) From your result in item (c), derive the transmission coefficient T for potential (3). Does it exhibit transmission resonances, where T = 1? If yes, what is the equation determining the resonance energies? How would you be solving it? (e) With a single delta-function potential (1), transmission coefficient for a given energy was independent of the sign of γ (barrier vs well). Is this also true for potential (3)? Problem 2 In this problem, we explore some general properties of the transfer matrix ( ) a b M= c d
1
(4)
for 1D scattering by a potential feature such that on its left and right sides the potential is constant. In the piece-wise solution of the stationary Schr¨odinger equation b = Eψ, Hψ
(5)
ψ1 = A exp(ik1 x) + B exp(−ik1 x)
(6)
we relate the solution in the left part
to the solution in the right part ψ2 = C exp(ik2 x) + D exp(−ik2 x) with the transfer matrix:
(
C D
)
( =M
A B
(7)
) .
(8)
(a) It is evident that if Eq. (5) holds for ψ, it also holds for ψ ∗ : b ∗ = Eψ ∗ . Hψ Use this fact with functions (6) and (7) to establish that elements of matrix M in (1) actually satisfy the following restrictions: d = a∗ and c = b∗ .
(9)
That’s nice, isn’t it? Make sure that transfer matrices from problem 1 obey this general relationship! (b) Use now the fact that the probability current density is the same on the left and the right sides to establish that determinant of the transfer matrix det(M) = |a|2 − |b|2 = k1 /k2 .
(10)
(c) In terms of the elements of transfer matrix, find the reflection R and transmission T coefficients for the particles incident from the left (use (10) as well – and you may want to exploit these relationships in problem 1). Prove that the reflection coefficients for particles incident from the left and for particles incident from the right are exactly the same independently of the shape of the potential “in the middle” (the difference can be only in phase factors).