Topics covered: Green s function, Lippman-Schwinger Eq., T-matrix, Born Series.

Transcription

1 PHYS85 Quantum Mechanics II, Sprin 00 HOMEWORK ASSIGNMENT 0 Topics covered: Green s function, Lippman-Schwiner Eq., T-matrix, Born Series.. T-matrix approach to one-dimensional scatterin: In this problem, you will use the Lippman- Schwiner equation ψ ψ 0 + GV ψ, () to solve the one-dimensional problem of tunnelin throuh delta potentials. Take ψ 0 (z) e ikz, and let V (z) δ(z) + δ(z L). () (a) Express Eq. () as an interal equation for ψ(z), and then use the delta-functions to perform the interal. It miht be helpful to introduce the dimensionless parameter α M. To solve k for the two unknown constants, enerate two equations by evaluatin your solution at z 0, and z L. Hit with a z from the left, and insert I dz z z after the G to et the interal equation ψ(z) ψ 0 (z) + dz G 0 (z,z )V (z )ψ(z ). (3) Use V (z ) δ(z ) + δ(z L) to handle the interals, ivin: ψ(z) ψ 0 (z) + G 0 (z,0)ψ(0) + G 0 (z,l)ψ(l). () To find the unknowns, ψ(0) and ψ(l), we set first z 0, and then z L, ivin ψ(0) ψ 0 (0) + G 0 (0,0)ψ(0) + G 0 (0,L)ψ(L) (5) ψ(l) ψ 0 (L) + G 0 (L,0)ψ(0) + G 0 (L,L)ψ(L) (6) Solvin simultaneously for ψ(0) and ψ(l) and takin G 0 (z,z ) G 0 ( z z ) ives ψ(0) ψ(l) + iα( e ikl ) + iα α ( e ikl ) e ikl + iα α ( e ikl ) (7) (8) This ives as the solution: ψ(z) e ikz iα eik(l+ z L + e ik z ( + iα( e ikl ) ) + iα α ( e ikl. (9) ) (b) Compute the transmission probability T t, with t defined via lim ψ(z) z teikz. (0) For z > L, this becomes ψ(z) te ikz ()

2 t So that the transmission probability is + iα α ( e ikl ) () T t ( α sin (kl)) + α ( + αcos(kl)sin(kl)) (3) (c) In the stron-scatterer limit α, at what k-values is the transmission maximized? In the limit α, we can keep only the α term in the denominator, ivin T α sin (kl) () which blows up at k nπ/l, n is any inteer. (d) Consider an infinite square-well of lenth L. What are the k-values for each bound-state? How do these compare with the transmission resonances in the stron-scatterer limit? The bound states correspond to k nπ/l, which matches the transmission resonances of the double-delta potential.

3 . The first Born-approximation: In the first Born-approximation, find the scatterin amplitude, f(θ,φ k), for a Gaussian scatterin potential, V (r) V 0 e ( r/r 0). (5) Still within the first Born-approximation, what is the differential cross-section,, and total crosssection, σ tot? First try the interal in spherical coordinates, then when you reach the peak of frustration, try switchin to Cartesian coordinates. In the first-born approximation, we have f( k, k) (π) M k V k k V k d 3 r k r V ( r) r k (π) 3 d 3 re i( k k ) r V 0 e (r/r 0) x dxe ik xx r 0 V 0 (π) 3 V 0 8π 3/r3 0e (( k k) ) y dye ik yy r 0 z dze i(k z k)z r 0 Now ( k k) ( k k e z ) ( k k e z ) k k zk + k. With k k and k z k cos θ, this ives The differential cross section is then The total cross section is then σ tot πv0 M f(θ k) r 0 e k ( cos θ) f(θ k) πv 0 M e k π 0 dφ 0 π V 0 M r 0 r 0 ( cos θ) d(cos θ) πv 0 M r e k 0 due k ( u) π V0 M e k k ( cos θ) 3

4 3. The Huan-Fermi pseudopotential: First, try to compute the T-matrix in three dimensions for a three-dimensional delta-function scatter, V ( r) δ 3 ( r). What happens? A workable zero-rane potential in three-dimensions is called the Huan-Fermi pseudo-potential, V HF, defined via r V HF ψ δ 3 ( r)ψ re ( r), (6) This potential is also referred to as a reularized delta-function. ψ re ( r) d r ψ( r). (7) dr (a) By expandin ψ( r) in powers of r, startin with r, show that the effect of the reularization d operator, dr r is to remove the /r term in the expansion. Thus ψ re( r), is always non-sinular at r 0. ψ( r) c (θ,φ) r + c 0(θ,φ) + c (θ,φ)r +..., (8) Then we have d dr rψ( r) d [ c (θ,φ) + c 0 (θ,φ)r + c (θ,φ)r +... ] dr c 0 (θ,φ) + c (θ,φ)r +... (9) so we see that the sinular term has been removed. Thus ψ re ( r) is non-sinular at r 0. In fact, we can use the siftin property of the delta function to ive r V HF ψ δ 3 ( r)ψ re (0) (0) (b) Compute the T-matrix for V HF, usin the reularization property to solve the sinularity problem encountered with the simple delta-function. We start from the Born-series expansion r T ψ δ 3 ( r)ψ re (0) + δ 3 ( r) d dr r T V + V G 0 V + V G 0 V G 0 V +... () d 3 r G 0 ( r, r )δ 3 ( r )ψ re (0) + 3 δ 3 ( r) d dr r d 3 r d 3 r G 0 ( r, r )δ 3 ( r ) d dr r G 0 ( r, r )δ 3 ( r )ψ re (0) +... δ 3 ( r)ψ re (0) [ + ddr rg 0( r,0) + ddr r d 3 r G 0 ( r, r )δ 3 ( r ) d ] dr r G 0 ( r,0) +... [ δ 3 ( r)ψ re (0) + G 0,re (0,0) + d ] dr rg 0( r,0)g 0,re (0,0) +... δ 3 ( r)ψ re (0) [ + G 0,re (0,0) + G 0,re(0,0) +... ] δ 3 ( r) G 0,re (0,0) ψ re(0) () Expandin G( r,0) in powers of r ives G 0 ( r,0) M π [ ] k + ik r r +... (3)

5 so that so that finally, we have G 0,re (0,0) ika () a M π (5) T is called the re-normalized couplin constant. V HF + ika + ika (c) Use your answer to part (b) to compute the differential cross-section,, as well as the total cross-section, σ tot, for the Huan-Fermi pseudo-potential. The Fourier transform of T is then T( k, k) (π) 3 d 3 r d 3 r e i k r T( r, r)e i k r (π) 3 ( + ika) from f( k, k) (π) M T( k, k), we find The differential cross-section is then f( k, k) (π) M a + ika a + (ka) As this doesn t depend on θ or φ, we have simply σ tot πa + (ka) (π) 3 ( + ika) (6) (7) 5