Il Nuovo Cimento D

, Volume 11, Issue 1–2, pp 73–99 | Cite as

Langevin formulation of quantum mechanics

  • M. Roncadelli
Article

Summary

We present in a rather pedagogical way a new formulation of quantum mechanics. Our starting point is the path integral representation of the quantum-mechanical propagator analytically continued to imaginary timeW(X″, s″|X′, s′). We view the set of random paths contributing toW(X″, s″|X′, s′) as the manifold of solutions of a Langevin equation with a Gaussian white noise. We thus obtainW(X″, s″|X′, s′) as the noise-average of a suitable functional of the solution of the Langevin equation. The standard quantum-mechanical propagator is finally recovered by analytically continuingW(X″, s″|X′, s′) back to real time. The present approach allows for a straightforward application of standard methods of classical stochastic processes to quantum-mechanical problems and offers a new promising way to perform computer simulations of quantum-dynamical systems.

PACS 03.65

Quantum theories quantum mechanics 

PACS 05.45

Theory and models of chaotic systems 

Riassunto

Si presenta in un modo piuttosto pedagogico una nuova formulazione della meccanica quantistica. Il punto di partenza è costituito dalla rappresentazione, in termini di integrale di cammino, del propagatore quanto-meccanicoW(X″, s″|X′, s′) prolungato analiticamente nel tempo immaginario. Si considera l'insieme dei cammini casuali che contribuiscono aW(X″, s″|X′, s′) come l'insieme delle soluzioni dell'equazione di Langevin in presenza di un rumore bianco gaussiano. Si ottiene alloraW(X″, s″|X′, s′) come media su rumore di un opportuno funzionale della soluzione dell'equazione di Langevin. L'usuale propagatore quantistico è poi riottenuto con un ulteriore prolungamento analitico al tempo reale. Questa formulazione consente una diretta applicazione dei metodi dei processi stocastici classici a problemi quantistici e rappresenta un nuovo promettente modo di effettuare simulazioni numeriche di sistemi quanto-meccanici.

Резюме

С педагогической целью мы предлагаем новую формулировку квантовой механики. Мы исходим из представления, использующего интегрирование по траекториям, кватовомеханического пропагатора, аналитически продолженного в область мнимого времениW(X″, s″|X′, s′). Мы рассиатриваем систему случайных траекторий, вносящих вклад вW(X″, s″|X′, s′), как множество решений уравнения Ланжевена в случае гауссова белого шума. Мы получаемW(X″, s″|X′, s′), как среднее по шуму для соответствующего функционала решения уравнения Ланжевена. Стандартный квантовомеханический пропагатор восстанавливается с помошью аналитического продоженияW(X″, s″|X′, s′) обратно в область вещественного времени. Предложенный подход позволяет непосредственно применять стандартные методы классических стохастических процессов к квантовомеханическим проблемам и представляет новый метод для проведения компьютерного моделирования квантоводинамических систем.

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Copyright information

© Società Italiana di Fisica 1989

Authors and Affiliations

  • M. Roncadelli
    • 1
    • 2
    • 3
  1. 1.Dipartimento di Fisica Nucleare e Teorica dell'UniversitàPavia
  2. 2.Sezione di PaviaIstituto Nazionale di Fisica NucleareItalia
  3. 3.CERNGenevaSwitzerland

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