Relating Classical Mechanics to Quantum Mechanics

1. Introduction

At the atomic and subatomic levels, the world is governed by quantum mechanics, whereas the macroscopic world we directly perceive follows classical mechanics. Systems composed of vast numbers of atoms appear to obey laws different from those of the quantum domain. Historically, classical mechanics preceded quantum mechanics, but its limitations became evident in the late 19th century when experiments contradicted its predictions. This led to the development of quantum mechanics in the early 20th century, which successfully explained these anomalies and introduced several radical conceptual shifts.

Over time, physicists came to regard classical mechanics as a limiting case of quantum mechanics—for large systems or high energies. However, this view was often based on specific examples rather than a fully established logical derivation.

This stand raises a fundamental question: Can the formal structure of classical mechanics be rigorously derived from quantum mechanics? Attempts using the Schrödinger or Heisenberg formulations did not fully succeed in deriving Newton’s laws or the Lagrangian formalism. Ehrenfest showed that expectation values of quantum observables approximately follow Newton’s laws, but this did not constitute a complete derivation.

A significant advance came with Feynman’s path integral formulation, inspired by Dirac’s insights into the conceptual foundations of quantum theory. The path integral approach not only clarified conceptual issues in earlier formulations but also provided a logically consistent framework showing how classical mechanics emerges from quantum mechanics.

Before examining this derivation, it is useful to briefly contrast the fundamental features of classical and quantum laws.

2. Classical and Quantum Laws in Brief

In classical mechanics, a system is described by precisely defined position and momentum at every instant in a chosen reference frame. Given the forces acting on it, future motion can be predicted with certainty using Newton’s laws or, more generally, the Lagrangian formalism. All other observable properties follow from position and momentum (ref.1). Classical dynamics is therefore deterministic. (We are only considering the domain of regular behaviour here).

In contrast, quantum mechanics does not assign definite position and momentum simultaneously. Instead, a system is described by a state vector in a finite or an infinite dimensional Hilbert space, commonly represented by a wave function or a probability amplitude. This wave function evolves according to the Schrödinger equation and yields only probabilistic predictions for measurement outcomes, as interpreted in the Copenhagen framework (ref.2 and ref.6). The probabilistic nature of results and the complementary wave–particle character of matter distinguish quantum theory fundamentally from classical mechanics.

It is this contrast that prompts and triggers any one to demand a convincing justification regarding how the path integral formalism that describes the quantum behaviour of systems can harbour in itself a formalism that is applicable to classical systems.  More explicitly, how does the formalism that accounts for the probabilistic predictions of quantum mechanics can lead to the precisely predictable results of the classical mechanics. The present article is aimed at illustrating the passage that bridges the two seemingly different approaches through the double-slit experiment with electrons – an experiment which was initially conceptualised to show the dual behaviour of electrons and was experimentally demonstrated decades later after the birth of quantum mechanics (ref.4).