Squeezed Coherent States and a Semiclassical Propagator for

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Squeezed Coherent States and a Semiclassical Propagator for the Schr¨o dinger Equation in Phase Space Maurice A de Gosson Universit¨a t Potsdam,Inst.f.Mathematik Am Neuen Palais 10,D-14415Potsdam E-mail address:maurice.degosson@d3ddcad628ea81c758f57859 Serge M de Gosson V¨a xj¨o Universitet,MSI SE-35195V¨a xj¨o ,Sweden E-mail address:sergedegosson@d3ddcad628ea81c758f57859 February 1,2008Abstract We construct semiclassical solutions of the symplectically covariant Schr¨o dinger phase-space equation rigorously studied in a previous paper;we use for this purpose an adaptation of Littlejohn’s nearby-orbit method.We take the opportunity to discuss in some detail the so fruitful notion of squeezed coherent state and the action of the metaplectic group on these states.1Introduction

Let ψbe a square-integrable solution of Schr¨o dinger’s equation

i ??ψ

2π n/2e i

p ·x ′ψ(x ′)

?t = H ph Ψ(2)1

where the operator H ph is de?ned by the intertwining formula

Uφ H ph=H(x,?i ?x)Uφ;

A straightforward calculation shows that we have

Uφ(xψ)=(1

2

p?i ?x)Uφψ;(3) these relations motivate the notation

H ph=H(12p j?i ?x)

and we may thus rewrite(2)as the phase-space Schr¨o dinger equation

i ?

2

x+i ?p,1

?x1

,...,?

?p1

,...,?

(p0·x?1

The Wigner–Moyal transform of a pair(ψ,φ)of functions in the Schwartz space S(R n x)is de?ned by

W(ψ,φ)(z)= 1 p·yψ(x+1φ(x?1

π n/4e?1

(p0·x?1

2π n

φ z0(x)

2π n (ψ,φ z0)L2φ z0(x)d2n z0(10)

and

||ψ||2L2= 1

2π n

Aφ z0(x)

π n/4(det X)1/4e?1

where X and Y are real symmetric n ×n matrices with X >0;we have ||φ

(X,Y )||L 2=1.We will ?nd it convenient to set

M =i (X +iY ),X =X T >0,Y =Y T .

and to write φ (X,Y )=φ

M ;thus:

φ

M (x )=

12 Mx 2,X =Im M.The Wigner transform of φ

M is given by the formula

W φ

M (z )= 1

Gz 2(14)

where G is the symmetric matrix G = X +Y X ?1Y Y X ?1

X ?1Y X ?1 ;(15)

an essential remark is that G is in addition symplectic (this fact was apparently ?rst observed by Bastiaans [1]).More precisely,we have G =S T S with

S = X 1/20

X ?1/2Y X ?1/2 ∈Sp(n )(16)

as results from a direct calculation;notice that S belongs to the isotropy sub-group of the Lagrangian plane ?p =0×R n in Sp(n ).For z 0∈R 2n we set

φ z 0,M = T (z 0)φ

M ;

the Wigner transform of φ

z 0,M is given by

W φ z 0,M (z )=W φ

M (z ?z 0);(17)

it is thus a Gaussian centered at z 0.

Let us generalize formula (14)by calculating the Wigner-Moyal transform W (φ z 0,M ,φ

z ′0,M ′)of a pair of squeezed coherent states;recall for this purpose

the “Fresnel formula” 1 ξ·x e ?1

2 K ?1ξ2

(18)

valid for ξ∈C n ,K =K T ,Im K >0;here (det K )?1/2=λ?1/21···λ?1/2n where

λ?1/2j is the square root with positive real part of the eigenvalue λ?1j

of K ?

1

(see [4,16]).Proposition 2We have

W (φ M ,φ

M ′)(z )=

1 F z 2(19)

where F is the matrix F = 2i M ′)?1M ?i (M +M ′)?1

?i (M ?M ′)2i (M ?

Proof.Setting W M,M′=W(φ M,φ M′)we have

W M,M′(z)= 1 pyφ M(x+1φ M′(x?1

2 (M+

py e i

π 2n(det XX′)1/4

Φ(x,y)=M(x+1M′(x?1

π n(det XX′)?1/4e?1

M′?(M?M′)?1(M?M′(M+

Noting the following formula,which is an easy consequence of the de?nitions

of the Wigner–Moyal transform and the Heisenberg–Weyl operators:

Lemma3For all z0,z′0in R2n z and f,g in L2(R2n z)we have

W( T(z0)ψ, T(z′0)ψ′)(z)=e i2σ(z0,z′0))×W(ψ,ψ′)(z?1

π n e i2σ(z0,z′0))(det XX′)?1/4e?12(z0+z′0))2 where F is given by(20);hence in particular

W(φ z

0,M ,φ z

0,M′

)(z)= 1 F(z?z0)2.

2.2Phase-space coherent states

For eachφ∈S(R n x)the operator

Uφψ(z)= π 2z) is an isometry of L2(R n x)onto its range Hφ;it follows that:

5

Proposition5LetΦ z

0=Uφ(φ z

).For eachΨ∈Hφwe have

Ψ(z)= 1

2π n |(Ψ,Φ z0)L2|2d2n z0.(22)

Proof.Letψbe de?ned byΨ=Uφψ;In view of part(ii)of Proposition1 (formula(10))we have

ψ= 1

2π n (ψ,φ z0)L2Uφ(φ z0)d2n z0

= 1

3Metaplectic Group and Coherent States

The metaplectic group Mp(n)is a faithful unitary representation of Sp2(n),the double cover of the symplectic group Sp(n).There are several di?erent ways to describe the elements of Mp(n)(see for instance[9,16,23]);for our purposes the most adequate de?nition makes use the notion of generating function for a free symplectic matrix because it is the simplest way to arrive at the Weyl symbol of metaplectic operators(and hence to their extension to phase space).

The interest of the metaplectic representation comes from the fact that it links in a crucial way classical(Hamiltonian)mechanics to quantum mechanics (see for instance[18]or[9]and the references therein).Assume in fact that H is a Hamiltonian function which is a quadratic polynomial in the position and momentum variables(with possibly time-dependent coe?cients):thus

H(z,t)=1

2

H′′(t)z2

where H′′(t)is a symmetric matrix(it is the Hessian matrix of H).The associ-ated Hamilton equations˙z=?z H(z,t)determine a(generally time-dependent)?ow consisting of sym …… 此处隐藏：13328字，全部文档内容请下载后查看。喜欢就下载吧 ……