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Exact quasi-classical asymptotic beyond WKB-theory and beyond Maslov canonical operator to the Colombeau solutions of the n-dimensional Schrodinger equation is presented. Quantum jumps nature is considered successfully. We pointed out that an explanation of quantum jumps can be found to result from Colombeau solutions of the Schrodinger equation alone without additional postulates.

A number of experiments on trapped single ions or atoms have been performed in recent years [

Stochastic quantum jump equations [

The question arises whether an explanation of these jumps can be found to result from a Colombeau solution [

i.e. we found the limiting Colombeau quantum averages (limiting Colombeau quantum trajectories) such that [

and limiting quantum trajectories

if limit in LHS of Equation (1.3) exists.

The physical interpretation of these asymptotic given below, shows that the answer is “yes” for the limiting quantum trajectories with localized initial data.

Note that an axiom of quantum measurement is: if the particle is in some state

We rewrite now Equation (1.4) of the form

We define well localized limiting quantum trajectories

and well localized limiting quantum trajectories

if limit in LHS of Equation (1.7) exists.

Let

Let us consider Schrödinger equation:

Here operator

Theorem 2.1. [

where we have set

where

where we have set 1)

Trotter and Kato well known classical results give a precise meaning to the Feynman integral when the potential

We avoided this difficulty using contemporary Colombeau framework [

Potential

1)

Here

Finally we obtain regularized Schrödinger equation of Colombeau form [

Using the inequality (2.7) Theorem 2.1 asserts again that corresponding solution of the Schrödinger Equations (2.8)-(2.9) exist and can be represented via formulae:

where we have set

where we have set

We rewrite Equation (2.10) for a future application symbolically of the following form

or of the following form

For the limit in RHS of (2.12) and (2.13) we will be used canonical path integral notation

where

Substitution

(2.15)

We rewrite Equation (2.15) for a future application symbolically of the following form

or of the following form

For the limit in RHS of (2.16) and (2.17) we will be used following path integral notation

Let us consider now regularized oscillatory integral

Lemma 2.1. (Localization Principle [

Then

Lemma 2.2. (Generalized Localization Principle) Let

Then there exist infinite sequence

Proof. Equality (2.23) immediately follows from (2.21).

Remark 2.1. From Lemma 2.2 follows that stationary phase approximation is not a valid asymptotic approximation in the limit

Theorem 3.1. Let us consider Cauchy problem (2.8) with initial data

where

1) We assume now that: a)

2) Let

Here

3) Let

where master Lagrangian

Let

4) Let

Assume that: for a given values of the parameters

Thus one can to calculate the limiting quantum trajectory corresponding to potential

Proof. From inequality (A.15) and Theorem A1, using inequalities (A.53.a) and (A.53.b) we obtain

where

We note that

where

and

From Equation (3.18) one obtain

where

Let us calculate now path integral

and

From Equation (3.17) and Equation (3.24) we obtain

Substitution Equation (3.25) into Equation (3.26) gives

Similarly one obtain

Let us calculate now integral

Substitution Equations (3.28)-(3.29) into Equation (3.21) gives

(3.30)

Substitution Equation (3.30) into Equation (3.16) gives

Similarly one obtain

Therefore

Substitution Equation (3.1) into Equation (3.33) gives

Let us calculate now integral (3.34) using Laplace’s approximation. It is easy to see that corresponding stationary point

Substitution Equation (3.35) into inequality (3.13) gives the inequality (3.11). The inequality (3.11) completed the proof.

In this subsection we calculate exact quasi-classical asymptotic for quantum anharmonic oscillator with a cubic potential supplemented by additive sinusoidal driving. Using Theorem 3.1 we obtain corresponding limiting quantum trajectories given via Equation (1.3).

Let us consider quantum anharmonic oscillator with a cubic potential

Supplemented by an additive sinusoidal driving. Thus

The corresponding master Lagrangian given by (3.7), are

We assume now that:

where

The corresponding master action

The linear system of the algebraic Equation (3.9) are

Therefore

The linear system of the algebraic Equation (3.10) are

Therefore the solution of the linear system of the algebraic Equation (3.10) are

Transcendental master Equation (3.11) are

Finally from Equation (4.10) one obtain

where

Example 1 (in

We set now

Note that for corresponding propagator

where

Here the initial-

Let us calculate now integral (5.2) using stationary-point approximation. Denoting an critical points of the discrete-time action (5.3) by

for

From Equation (5.2) in the limit

Here the pre-factor

The Gaussian integral in (5.6) is given via canonical formula

Here

Let us consider now Cauchy problem (2.8) with initial data

Note that for corresponding Colombeau solution

Let us calculate now integrals in RHS of Equation (5.8) using stationary-point approximation. Corresponding critical point conditions are

From (5.8) we obtain

Let as denote

Therefore the time discretized path-integral representation of the Colombeau quantum averages given by Equation (1.1) are

(5.13)

where

Here

From Equations (5.13)-(5.14) one obtain

and

As demonstrated in [

with initial data

from which the pre-factor

In the limit

In the limit

with initial data

By integration Equation (5.22) one obtain the first order linear differential equation

In the limit

We set now in Equation (5.1)

Corresponding differential master equation are

From Equation (5.27) one obtain that corresponding transcend dental master equation are

Comparison of the: 1) classical dynamics calculated by using Equation (5.1) (red curve), 2) limiting quantum trajectory

We pointed out that there existed limiting quantum trajectories given via Equation (1.3) with a jump. Such jump does not depend on any single measurement of particle position

An axiom of quantum mechanics is that we cannot predict the result of any single measurement of an observable of a quantum mechanical system in a superposition of eigenstates. However we can predict the result of any single measurement of particle position

A reviewer provided important clarification.

Jaykov Foukzon,Alex Potapov,Stanislav Podosenov, (2015) Exact Quasi-Classical Asymptotic beyond Maslov Canonical Operator and Quantum Jumps Nature. Journal of Applied Mathematics and Physics,03,584-607. doi: 10.4236/jamp.2015.35072

Let us consider now regularized Feynman-Colombeau propagator

where

Here:1)

3)

Therefore regularized Colombeau solution of the Schrödinger equation corresponding to regularized propagator (A.1) are

Here

Let us consider now regularized quantum average

From (A.5) and (A.12) one obtain

From Equations (A.5)-(A.13) one obtain

Using replacement

Here

And

Let us rewrite a function

where

Let use valuate now path integral

(A.22.a)

(A.22.b)

where

(A.25.a)

(A.25.b)

(A.26.a)

(A.26.b)

Let us evaluate now n-dimensional path integral

(A.27)

From Equation (A.27) one obtain the inequality

From In Equation (A.28) one obtain the inequality

where

Using replacement

(A.32)

From (A.29)-(A.35) one obtain

Proposition A.1. [

Then the iterated limit:

Proposition A.2. Let

uation (A.25) and let

tion (A.26). Then

1)

2)

3)

4)

5)

6)

Here

Proof (I) Let us to choose an sequence

1)

2)

We note that from (ii) follows that: perturbative expansion

vanishes in the limit

Let us to choose now an subsequence

exist and

From (A.39) and Proposition A.1 one obtain

From (A.39), (A.40) and (A.38) one obtain

The inequality (A.41) completed the proof of the statement (1).

(II) Let us estimate now n-dimensional path integral

From Equation (A.42) one obtain the inequality

where

Using replacement

(A.45)

From (A.43)-(A.48) one obtain

Let us to choose an sequence

1)

2)

We note that from 2) follows that: perturbative expansion

Vanishes in the limit

Let us to choose now an subsequence

exist and

From (A.51) and Proposition A.1 one obtain

From (A.50), (A.51) and (A.52) one obtain

Proof of the statements (3)-(6) is similarly to the proof of the statements (1)-(2).

Theorem A.1. Let

where

Here

Proof. We remain that

From Equation (A.56) we obtain

Let us to choose now an sequences

1)

2)

3)

Therefore from inequality (A.57), Equation (A.58) and inequality (A.59) we obtain