There are many ways to the heart of quantum physics

About Richard Feynman's alternative formalism for time-dependent quantum mechanics - and the classical approximations that make it convenient for implementation in physics and theoretical chemistry

By Yonathan Barkheim, quantums in Hebrew

introduction

Almost a century ago, in 1926, the physicist Erwin Schrödinger wrote the famous equation that now bears his name. This equation yielded unprecedentedly accurate results for quantum systems, and these have since been observed countless times in numerous experiments. However, this formalism has inherent limitations since it is not always possible to solve the quantum systems analytically (for example, particles subject to complicated potentials, systems with many electrons, etc.).

Thus, over the years, quantum mechanics has undergone many upheavals, until today it can be said with full mouth that most particle physicists do not recognize the Schrödinger equation as a particularly interesting entity. This remains mainly the domain of atomic physicists and theoretical or computational chemists.

On the other hand, the particle physicists, whose main work is based on quantum field theory, use a formalism completely different from the Schrödinger equation - the path integral formalism invented by Richard Feynman, winner of the 1965 Nobel Prize in Physics and one of the greatest physicists of the 20th century. This formalism has been proven time and time again to be effective in field theory calculations - for example, many particle systems, and calculations in particle physics - and is the current practice.

In this post I will briefly explain the principle behind semi-classical approximations to the path integral formalism, and how they come to realization.

From the Schrödinger equation to the path integral

One of the fundamental principles in quantum mechanics is the correspondence principle, formulated by Niels Bohr a few years before the Schrödinger equation was written. This principle states that within certain limits - a large amount of particles or high energies - the quantum system will strive to behave classically.

This principle received a mathematical configuration with the formulation of Ehrenfast's theorem. One of the well-known results of this theorem says that the quantum observation values ​​behave according to classical equations of motion. For example, Newton's second law (or Hamilton's equations) can be obtained from the time-dependent Schrödinger equation.

Semi-classical approximations are based on the combination of the classical and the quantum point of view, and are related at least on a conceptual level to the matching principle. They assume that part of the system behaves classically, and part quantumly. Think, for example, of a Gaussian wave packet that moves in space and time, under a certain potential, according to the classical equations of motion: on the one hand, this entity has a probabilistic character, and on the other hand, its dynamics conform to Newton's laws.

The central idea of ​​path integrals also draws its strength from classical mechanics. Instead of looking at the wave function as the central quantum motif, Feynman says, let's look at the propagator - a "machine" that promotes an initial state of a particle from a certain point to another point in space and time. Given that the propagator fully determines the dynamics of the particle, it is actually sufficient to calculate only it, and this is completely equivalent to the calculation of the wave function at every moment and point.

Thus, the propagator is defined as a tool that takes the particle from an initial point to an end point under dynamics dictated by the Hamiltonian of the system. A fairly simple development, which should be understandable to anyone who has studied analytical mechanics, replaces the Hamiltonian with a quantity called the classical action. According to Feynman, the propagator is equal to the sum on different "histories" of the particle, where each history is defined exclusively by the action. Each of them has a weight is equal within the sum, so Feynman claimed that the histories behave democratically. Thus, with the help of the propagator, the amplitude for the transition from one point to another and from one moment to another can be recorded, by summing up the phases of all trajectories, where the phase of a trajectories is actually the classical operation.

A full calculation of the propagator using this method requires complicated mathematical and physical tools, which of course does not detract from its power and has led to many developments in theoretical physics over the years. However, physicists are always looking for approximations, and now our story is joined by another Nobel laureate physicist - John Van Velk. Van Vleck's proposal, developed by the Swiss-American physicist Martin Gutzwiler, was to make a sophisticated approximation to the Feynman integral, by developing the operation to higher orders using a Taylor column, or if you like - developing a stationary phase which assumes that the largest contribution that comes from integration over exponent, occurs when its argument is zeroed.

Given an action S, we can also observe the first variation δS and the second variation δ2S. For classical trajectories, which constitute a significant part of the total of all the trajectories that are summed, δS=0 holds as a result of Hamilton's principle of minimal action - and also of the matching principle. In fact, this is the most fundamental feature that distinguishes classical orbits from any general orbits, and is equivalent to any mechanical formulation known to us, such as Newton's second law or the Euler-Lagrange equations. The second variation on the action, δ2S, is mathematically related to a quantity known as stability.

The stability represents the relationship between the initial and final positions and the initial and final momentums for a certain trajectory, and from it one can learn, for example, how chaotic the system is. It is a whole world in itself, deeply related to a pure mathematical field called symplectic geometry.

in fact

Based on the stability properties, numerical methods have been developed over time that make it possible to turn the "law", i.e. the path integrals, into an "action", i.e. an algorithm that can be implemented through code. One of the fundamental methods in this context, allows the calculation as a problem of initial conditions in a basis of Gaussians: it is a complete basis defined over a phase space, that is, a collection of many possible initial positions and many possible initial motions. Each Gaussian has a weight (analogous to probability) which depends on the stability of the problem, and is accompanied by a phase which is the classical operation. This method is called the Herman-Klock method.

Using this method, a series of results were obtained over the years in the fields of molecular dynamics, spectroscopy, dispersion theory, and more. But the main problem is therefore in the built-in limitation, which makes it difficult to calculate effects that are outside the classical domain, such as tunneling. As a result, it is also difficult to gain insights into dynamic phenomena associated with extreme interaction between light and matter, for example the creation of high harmonics, which I talked about in my previous post.

In order to overcome this problem, the theoretical chemist Prof. David Tenor from the Weizmann Institute of Science (the supervisor of the writer of these words) proposed the use of a classical composite operation, from which a set of composite quantities is derived - space, momentum and normalization of the wave function. The representation of the problem in the basis of the Gaussians, similar to the Herman-Klock method, but as a termination condition problem (FVR), together with the conversion of the Schrödinger equation in the collection of customer equations from the field of quantum hydrodynamics (Bohm-de Bruy mechanics), allows obtaining insights that go beyond the classical field (such as on processes tunneling), and all as a derivative of the propagator that went through a series of approximations. Of course, even this method, known as FINCO, has its own problems arising from mathematical phenomena that are unique to the composite plane, since as said - there are no free meals.

The semi-classical dynamics is a magical jungle that is at a crossroads between physics, theoretical chemistry and non-trivial mathematics. Despite being elementary, and maybe precisely because of that - it is so interesting.

More of the topic in Hayadan:

• Quantum teaching - superposition of experiment and theory
• Erwin Schrödinger. A fierce opponent of the Nazi regime and the proud owner of a quantum cat
• The quantum world is not as discrete as we thought and therefore it is possible to save Schrödinger's cat