Wave Packet Representation of Semiclassical Time Evolution in Quantum Mechanics

Transcription

1 Wave Packet Representation of Semiclassical Time Evolution in Quantum Mechanics Panos Karageorge, George Makrakis ACMAC Dept. of Applied Mathematics, University of Crete Semiclassical & Multiscale Aspects of Wave Propagation May 30 th, 2012

2 The Semiclassical Limit in Quantum Mechanics Correspondence Principle: Quantum expectations of quantities of motion converge to the classical ones, as 0 +. Reduction of order in Schrödinger equation; inapplicability of regular perturbative approach. i ψ t = 2 ψ + V ψ Semiclassical solutions display essential singularity.

3 Semiclassical Solutions For a semiclassical quantum state ψ corresponding to the mixed classical state δ (Λ,dµ), ψ, Op(f )ψ f dµ, 0 + Λ

4 The WKBM Approximation Ansatz for singular semiclassical solutions real a, S, a being -analytic. ψ (x, t) = a (x, t) exp i S(x, t) WKB approximation to first order: phase and amplitude obey the dynamical equations S t + qs 2 + V (q) = 0 ρ t + q ( ) 2ρ q S = 0

5 The WKBM Approximation Lagrangian manifold: For a generating function S(q, p) Λ S := {(q, p) : p = q S} reflected geometrically in the Hamiltonian transport of the Lagrangian manifold Λ St = g t Λ S0 for initial phase S 0

6 Semiclassical Initial Data in Configuration Space Initial value problem Relevant initial data are themselves semiclassical. 1. Lagrangian states: ψ 0 (x) β e i S0(x) N k=0 k ϕ k (x) ϕ k C 0 (Rd ), S 0 possesses unique analytic extension, Im S Coherent states: ψ0 (x) = d/4 f ( x x 0 )e i S0(x) f S(R d ).

7 Wave Packets Heisenberg inequality: maximum phase space resolution. Structure on finer scales than q, p cannot pertain to probability interpretation. States of minimum simultaneous uncertainty: Isotropic Gaussian Coherent States ψ(q,p) (x) = a exp i (p (x q) + i ) 2 (x q) A(x q) A T = A, Im A 0.

8 Wave Packet Quantization Physical motivation: Pure classical state (q, p) corresponds to isotropic Gaussian wave packet of minimum Heisenberg uncertainty, possessing a de Broglie wavelength G (q,p) (x) = exp i (p (x q) + i (x q)2) 2 Serves formulation of semiclassics in phase space.

9 Wave Packet Transform Configuration space: Analysis of wavefunction in over-complete basis of Gaussian coherent states. Wave packet representation of wavefunction. Ψ (q, p) = Wψ(q, p) = a R d Ḡ (q,p) (x)ψ(x)dx ψ (x) = W 1 Ψ(x) = a G(q,p) R (x)ψ(q, p)dqdp 2d Connection with Bargmann transform Wψ(q, p) = (2π ) d/2 e p2 /2 z Bf ( ) 2 where ψ(x) = f (x/ ).

10 Wave Packet Transform Quantization of quantities of motion f Op wp (f ) Op wp (f )Ψ(q, p) = a 2 Ḡ (q,p) (x)g (y,ξ) (x)f (y, ξ)ψ(y, ξ)dxdydξ Quantization of Canonical Transformations g : R 2d R 2d (q, p) (y, ξ) generated by S T g Ψ(q, p) a 2 z = q ip, ζ = y iξ. Ḡ (q,p) (x)g g(y,ξ) (x)e i S(y,ξ) det z Ψ(y, ξ)dxdydξ ζ

11 Analytic Structure of Wave Packet Transform Wave packet transform W is not onto. W : L 2 (R d ; dx) F L 2 (R 2d ; dqdp) Image of W identified with the kernel ( ) Op wp ( z) z Ψ(z, z) = 0 a twist of Cauchy-Riemann condition. F isomorphic to Hilbert space of analytic phase space functions, with respect to Gaussian density.

12 Physical Interpretation of Analytic Structure F is the subspace of pure phase space states, invariant under phase space Schrödinger evolution. Initial value problem makes sense only for initial data Ψ 0 F for which W 1 exists.

13 Phase Space Schrödinger Equation Canonical variables Op wp (q) = q, Op wp (p) = p + i q Op wp (H) = H(q, p + i q ) After a gauge transformation, Ψ e ip q/ Ψ. Phase Space Schrödinger equation i Ψ t = 2 q Ψ + V (q + i p )Ψ

14 Phase Space Schrödinger Flow Wave packet quantization of time evolution operator U t = W U t W 1 Phase space Schrödinger flow is defined only over the subspace F.

15 Time Evolution in Phase Space Wave packet propagator if Ψ t = Wψ t and Ψ 0 = Wψ 0 Wave packet evolution operator ψ t = U t ψ 0 Ψ t = W U t W 1 Ψ 0 U t = W U t W 1 Schwartz representation Ψ t (q, p) = L(q, p, y, ξ, t)ψ(y, ξ)dydξ

16 Semiclassical Initial Data in Phase Space Lagrangian States in Phase Space Ψ 0(q, p) α e i S0(q) N k=1 k φ k (q) on the Lagrangian manifold Λ S0, while Ψ 0 (q, p) = O( ) outside Λ S0.

17 Propagation of Semiclassical Initial Data Semiclassically concentrated on transported Lagrangian manifold. Sub-algebraic decay outside g t Λ S in. Semiclassical propagated states do not have a classical limit; Phase space density, or averaged over a Heisenberg scale. Ψ t 2 g (q, p) Liouville density.

18 The Wave Packet Ansantz Maslov-Nazaikinskii-Sternin-Schultze-Robert: Semiclassical initial data beyond WKBM form; wave packet propagation. For initial data isotropic Gaussian wave packet ) d/4e Γ i (q,p) (x, t) := (π ) 3d/4( det ImA (S(q,p,t)+p t (x q t)+ 12 (x qt) A(x qt) ) A T = A, Im A 0 satisfies the matrix Ricatti equation da dt + 2A2 = V (q t ) Closely related to the divergence of initially nearby orbits.

19 The Semiclassical Wave Packet Propagator The wave packet propagator, for any initial data, is L(q, p, y, ξ, t) = a 2 G (q,p) (x)u t G (y,ξ) (x)dx R d Based on the MNSSR wave packet ansatz for short times, U t G (y,ξ) (x) Γ g t (y,ξ)(x) we have := ζ. e i b(y, ξ, t) L(q, p, y, ξ, t) ( 2iπ ) d ( S(y,ξ,t)+ 1 2 (ζt z) yt( ζt) 1 (ζ t z) det y t det ζ t ) Ḡ (q,p) (y t )

20 The Semiclassical Wave Packet Propagator Semiclassics: van Vleck formula K(x, r, t) 1 (πi ) d/2 γ det p e i πi S(γ) 2 νγ q t

21 Phase Space Schrödinger Density as a limit of the Liouville flow Ψ(q, p) has no classical limit, while the density Ψ(q, p) 2 does. The quantum density, alike the Wigner quasi-probability, pass smoothly to the Liouville density in the semiclassical limit. ρ t {H, ρ} = 0

22 Phase Space Schrödinger Density as a limit of the Liouville flow For short times (t T ) quantum propagation is approximated by Liouville propagation of the wavefunction. Quantization of Hamiltonian flow: T g t Ψ(q, p) P F (e i S g t J 1/2 Ψ g t) (q, p) Ψ g t (q, p) For short times no structure is formed on sub-heisenberg scale. For semiclassical times, dispersion effects creates fringes on scales. The projector P F = a 2 Ḡ(y,ξ) (x)g (q,p) (x)( )dx dq dp P F as an integral operator, smooths out structures on scales finer than the Heisenberg scale.

23 Future Perspectives Estimates in time. Rigorous estimates using complex phase oscillatory integral theory. Relation of phase space densities to Hussimi density.

24 Thank you for your attention Special thanks to Roman Schubert, Robert Littlejohn, Frederick Faure, for enlightening conversations on wave packets.