Quantum Mechanics II

Transcription

1 Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in perturbation theory. We derived the Lippmann-Schwinger equation for the solution of the Schrödinger equation ψ ± = ϕ + G ± V ψ ± () where ϕ is the free particle eigenstate ϕ = k. We defined the G operator as G ± = E H 0 ± iɛ (2) The plus sign is for outgoing states and the minus sign is for incoming states. In terms of wave functions, the Lippmann-Schwinger equation is an integral equation ψ ± (x) = ϕ(x) 2m 2 dx G ± (x, x )V (x )ψ ± (x ), G ± = e ±ik x x 4π x x Last time we found that in the limit that the observer is very far away, x = r, then we have ψ ± (x) = ] [e ik x (2π) 3/2 + eikr r f(k, k ) Note this expression is the definition of the scattering amplitude f, which only makes sense in the limit r. In our formalism we can write f(k, k ) = 2π 2 2m 2 k V ψ + (5) The differential cross section is just the absolute value square of the scattering amplitude. dσ dω = f(k, k ) 2 (6) Although in this quantity only the magnitude of f matters, as f is a complex number in general its phase also matters in the discussion that follows. (3) (4)

2 We also introduced the Born approximation, which largely simplified our calculation for the scattering amplitude f () (k, k ) = 2m 2 dx e i(k k ) x V (x) (7) 4π so the first order scattering amplitude is just the Fourier transform of the scattering potential. If the potential is spherical symmetric, then the scattering amplitude only depends on θ which is the angle between k and k f(θ) = 2m sin qr 2 rv (r) dr (8) q 0 We introduced the transition operator T in order to write Lippmann-Schwinger equation in operator form. We defined T ϕ = V ψ +, and the equation translates into and it has a nice iterative solution. element of the transition operator.2 Optical Theorem T = V + V G + T (9) The scattering amplitude can be interpreted nicely as the matrix f(k, k ) = 2π 2 2m 2 k T k, f (n) k V (G + V ) n k (0) Last time we introduced the optical theorem but did not prove it. The theorem states that Im f(k, k) = k 4π σ tot () where σ tot is the total cross section, the integral of differential cross section over all solid angle. Now let s try to prove it. For notational simplicity we label A = 4mπ 2 / 2. We use the L-S equation to rewrite k = ψ + G + V ψ + = ψ + G + T k (2) Note that G + = G, so we can take conjugate on the above equation Now we can write the imaginary part of f as k = ψ + k T G (3) Im f(k, k) = Im A k T k = A Im [ k V k + k V G + T k ] [ ψ = A Im + V G + T k ] k T G V G + T k = A Im k T k k G + k 2 k 2 T k k,k 2 (4) Note that the matrix element of the operator G + is k G + k 2 = δ(k k 2 ) E 2 k 2 /2m + iɛ (5) 2

3 substituting this into the above equation we can get Now we can use the principal value to unfold the expression and get Im f(k, k) = A k f(k, k ) 2 Im G + (k ) (6) Im x + iɛ = πδ(x), δ(f (x)) = x 0 F (x 0 ) δ(x x 0) (7) Im f(k, k) = 4πk k 2 f(k, k ) 2 dω = k 4π σ tot (8) This result is useful in checking whether our calculation result of scattering amplitude make sense or not..3 Spherically Symmetric Scattering Let s consider again a central potential, or spherically symmetric potential V (x) = V (r). The free Hamiltonian is H 0 = p 2 /2m and k are eigenstates of the free Hamiltonian. However we know that the angular momentum operators L 2 and L z also commute with H 0 and we can use states with definite angular momentum as the free states E, l, m. These are the spherical wave states. Now if there is a scatterer which is spherically symmetric, then the operator p no longer commutes with the Hamiltonian, but L 2 and L z still commute with H. Therefore it is more convenient to use spherical waves as the free states in this situation. So now we need to learn how to operate the mechanism of scattering theory with these states. To start with, we want to impose a normalization for these states E, l, m E, l, m = δ ll δ mm δ(e E ) (9) We would like to expand every state in our scattering theory in terms of these states. At least schematically this is easy and we have k = E, l, m k E, l, m (20) E,l,m However we need to know the coefficients, which are nothing but some inner products. Before going any further let s clarify our objective for doing this. Remember last semester we discussed WKB approximation and barrier penetration. Let s consider similar things here. If we have a solid wall and some incoming plane wave with wave vector k, then upon incidence all the wave will be reflected and by energy conservation the reflected wave will also have wave vector k. Now suppose we have a potential gradually going to infinity instead of a steep wall, then the reflected wave will still have wave vector k, but what is different is the phase of the reflected wave. Similar things happen in 3D. If all particles come in with definite angular momenta, scatter off the spherical symmetric potential, then the scattering process will not change the angular momentum quantum numbers. The only thing can change is the phase. So our goal is to derive a formula for the phase shift. There are a couple of things that we need. The inner product quoted above is k E, l, m = mk δ (E 2 k 2 3 2m ) Y m l (θ k, φ k ) (2)

4 similarly we have the inner product with the position operator x E, l, m = 2mk il π Y l m (θ x, φ x )j l (kr) (22) where j l (x) are spherical Bessel functions. For our purpose we only need to consider the case where l is an integer, so we have the formula ( ) j l (x) = ( x) l d l sin x x dx x Therefore the plane wave can be expanded using the spherical waves as follows (23) e ik x = 4π l m= l i l j l (kr) [Yl m (θ k, φ k )] Yl m (θ x, φ x ) (24) Now let s write our beloved scattering amplitude using this spherical wave representation f(k, k ) = A k T k = A de de k E, l, m E, l, m T E, l, m E, l, m k (25) ll,mm The matrix element for the T operator can t depend on angular momentum quantum numbers, so we can write E, l, m T E, l, m = δ(e E )δ ll δ mm T l (E) (26) Therefore the formula for scattering amplitude is written as f(k, k) = A 2 ( T l E = 2 k 2 ) [ Yl m (ˆk) Yl m (ˆk )] (27) mk 2m l,m We can invoke the spherical harmonics addition theorem to get rid of the m summation P l (cos θ) = Then we can write the scattering amplitude as f(k, k ) = 4π 2l + l m= l π k (2l + )T l(k)p l (cos θ) = In the limit r we can write the spherical Bessel function as Yl m (ˆk ) Yl m (ˆk) (28) (2l + )f l (k)p l (cos θ) = f(θ) (29) j l (kr) = ( ) l d l kr d(kr) l sin(kr) = [ e i(kr πl/2) e i(kr πl/2)] (30) 2ikr where the first term is identified as outgoing partial wave, and the second is identified as incoming partial wave. Therefore the wave function in this limit is written as e ψ k,l,m (x) (2l + )P l (cos θ) [S ikr l ] r r e ikr e iπl (3) 4

5 where S l = + 2ikf l (k). These are the partial scattering amplitudes. There is one thing we haven t used, which is the conservation of number of particles. This means that the number of incoming particles is the same as number of outgoing particles, and this means that S l 2 = (32) This condition is called unitarity. This condition means that we can write S l = e 2iδ l scattering phases. where δ l are called 5