Exact quasiclassical asymptotics beyond Maslov canonical operator

Unfortunately for real physical problems assuption (i) does not satisfies and consequently Maslov global quasiclassical asymptotic of the Green function (0.1.7) does not valid. We improve this difficulties using Colombeau approach [1]- [12], [42], [43].

Let us consider Colombeau-Schrödinger equation [42], [43]:

where ∀ ∈ 0, 1 operator H  t given via formula

and V 0 x  Vx.

Remark 0.1.2. We note that V  x  ∈ G n   ∀ ∈ 0, 1

Maslov local quasiclassical asymptotic of the Green function of the Colombeau-Schrödinger equation (0.1.8) wih potential V  x  ∈ G n  is :

Here

X  0, t, x, y   y, X  t; t, x, y   x.

By using convolution we obtain

We assume now that : (i) for a fixed  ∈ 0, 1, t and y boundary problem is a system, (0.1.12) has a finite number of solutions X k, ; t, x, y, k  1, . . . , N, (ii) for a given t, x, y : ∀ ∈ 0, 1 the set X k, t; t, x, y|k  1, . . . , N does not contain focal points. Then Maslov global quasiclassical asymptotic of the Green function obtained from Eq.(0.1.13) is:

Here the number  k, is the Morse index of the trajectory X k, ; t, x, y.

The main purpose of this paper is: (I) To calculate exact quasiclassical asimptotic of the complex ℂ-valued generalized quantum averages 〈s, t, x 0 ;

without any reference to the corresponding exact quasiclassical asimptotic of the Schrödinger generalized wave function   x, t, x 0 ;   given from Colombeau-Schrödinger Eq.(0.1.8)-(0.1.9) (II) To calculate exact quasiclassical asimptotic of the real -valued generalized quantum averages 〈s, t, x 0 ;

without any reference to the corresponding exact quasiclassical asimptotic of the Schrödinger wave function   x, t, x 0 ;   given from Colombeau-Schrödinger Eq.(0.1.8)-(0.1.9).

Remark.Note that such quasiclassical asimptotic very important from point of view of the “quantum jumps” problem, well known in modern quantum mechanics.The existence of such jumps was required by Bohr in his theory of the atom. He assumed that an atom remained in an atomic eigenstate until it made an instantaneous jump to another state with the emission or absorption of a photon. Since these jumps do not appear to occur in solutions of the Schrodinger equation, something similar to Bohr’s idea has been added as an extra postulate in modern quantum mechanics. The question arises whether an explanation of these jumps can be found to result from a solution x, t, x 0 ;  of the Schrödinger equation (0.1.1)-(0.1.2) alone without additional postulates. We suggest a new asymptotic representation for the quantum averages (0.2.2) with position variable with well localized initial data,i.e.

〈i, 0, x 0 ;       ≃ x 0i , i  1, . . . , n.

The canonical physical interpretation of these asymptotics shows that the answer is “yes.”

As is well-known,when the potential V is sufficiently regular any solution of the Schrödinger equation (0.

Maslov quasiclassical asymptotic of the wave function x, t;  is [22], [25]:

k1 m  dy 0 y|Dq k ; y, x, t| ℒq ̇k; y, x, t, q k ; y, x, t, d, ℒq ̇, q,   m q ̇2 2 -Vq, , Sq k ; y, x, t, t  0, q k 0; y, x, t  y, q k t; y, x, t  x, Dq k ; y, x, t  det ∂Sq k ; y, x, t, t ∂x∂y ≠ 0, x ≠ y.

and the number  k   k q k ; y, x, t is the Morse index of the trajectory q k ; y, x, t.

Eq.(0.2.5) we obtain x, t, x 0 ;   2i -n 2 ∑ k1 m  dy 0 y -x 0 |Dq k ; y, x, t|

Substitution Eqs.(0.2.9) into Eq.(0.2.5) gives [22]: x, t, x 0 ;   2i n 2  dyx -x 0 |Dq k ; y, x, t|

From Eq.(0.2.10) one obtain x, t, x 0 ;  

and therefore  x s x, t, x 0 ; d n x 

Substitution Eq.(0.2.11) and Eq.(0.2.12) into Eq.(0.2.1) gives 〈x i , t, x 0 ;   x s0 cos t , s  1, . . . , n.

Formula (0.2.5) can be written by using Maslov canonical operator K  n t 1/ .

Let  n t be an n-dimensional Lagrangian manifold of class C  in the phase space  x,p 2n where x ∈  n and let d be the volume element on . A canonical atlas is a locally finite countable covering of  n t by bounded simply-connected domains  j (the charts) in each of which one can take as coordinates either the variables x or p a mixed collection

not containing dual pairs p j , x j . The Maslov canonical operator

The canonical operators are introduced as follows: (1) Let the chart  j be non-degenerate, that is,  j is given by an equation p  px, t and

p, dx  ∑ j1 n p j dx j .

Here   1 is a parameter, r 0 ∈  j is a fixed point and  ∈ C 0  .

(2) Let the local coordinates in the chart  j be p, that is  j ,is given by an equation x  xp, t,and let

p, dx -xp, t, p r , r  xp, t, p.

Here

 is defined analogously in the case when the coordinates in are some collection and one fixes a point r