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.
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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Roncadelli, M. Langevin formulation of quantum mechanics. Il Nuovo Cimento D 11, 73–99 (1989). https://doi.org/10.1007/BF02450234
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DOI: https://doi.org/10.1007/BF02450234