[Paper Review] Feynman Integral Approach to Absorption in Quantum Mechanics

Avi Marchewka, Z. Schuss|ArXiv.org|Jun 1, 1999

Quantum Mechanics and Applications10 references3 citations

TL;DR

This paper introduces a Feynman path integral formulation for quantum absorption using absorbing boundaries that terminate trajectories upon contact, thereby modeling unidirectional absorption. It derives the survival probability of a quantum particle, showing decay with beats in a one-dimensional box with absorbing walls, and applies the formalism to slit experiments and energy-dependent absorption modes.

ABSTRACT

We propose a formulation of an absorbing boundary for a quantum particle. The formulation is based on a Feynman-type integral over trajectories that are confined by the absorbing boundary. Trajectories that reach the absorbing wall are instantaneously terminated and their probability is discounted from the population of the surviving trajectories. This gives rise to a unidirectional absorption current at the boundary. We calculate the survival probability as a function of time. Several modes of absorption are derived from our formalism: total absorption, absorption that depends on energy levels, and absorption of non-interacting particles. Several applications are given: the slit experiment with an absorbing screen and with absorbing lateral walls, and one dimensional particle between two absorbing walls. The survival probability of a particle between absorbing walls exhibits decay with beats.

Motivation & Objective

• To develop a consistent quantum mechanical framework for absorption using path integrals.
• To model unidirectional absorption at boundaries by terminating trajectories upon contact.
• To calculate the survival probability of a quantum particle under various absorption conditions.
• To apply the formalism to physical systems such as the double-slit experiment with absorbing screens and one-dimensional absorbing walls.
• To explore different absorption modes, including total, energy-dependent, and non-interacting particle absorption.

Proposed method

• Formulates a path integral over trajectories confined by an absorbing boundary that terminates any trajectory reaching it.
• Assigns zero contribution to terminated trajectories, effectively discounting their probability from the surviving population.
• Defines a unidirectional absorption current at the boundary via the termination mechanism.
• Derives the survival probability as a time-dependent function using the path integral formalism.
• Applies the method to one-dimensional systems with two absorbing walls and to the double-slit setup with lateral or central absorbing screens.
• Distinguishes between absorption modes: total absorption, energy-level-dependent absorption, and non-interacting particle absorption.

Experimental results

Research questions

• RQ1 How can absorption in quantum mechanics be consistently formulated using the Feynman path integral approach?
• RQ2 What is the time evolution of the survival probability for a quantum particle in a system with absorbing boundaries?
• RQ3 How does the presence of absorbing walls affect interference patterns in a double-slit experiment?
• RQ4 What are the differences in survival probability dynamics between total absorption and energy-dependent absorption?
• RQ5 Can the formalism describe decay with beats in a one-dimensional box with absorbing walls?

Key findings

• The survival probability of a particle between two absorbing walls exhibits oscillatory decay with beats, indicating coherent quantum interference during absorption.
• The formalism successfully models unidirectional absorption current at the boundary due to trajectory termination.
• The method reproduces known results for total absorption and extends to energy-dependent absorption scenarios.
• In the double-slit experiment with an absorbing screen, the formalism accounts for the suppression of interference due to absorption.
• The survival probability decays faster in systems with stronger absorption, and the beat pattern arises from quantum interference between surviving paths.
• The approach provides a physically intuitive path integral picture of absorption, consistent with quantum mechanical probability amplitudes.