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Prequantum Dynamics from Hamiltonian Equations on the Infinite-Dimensional Phase Space

Andrei Yu. Khrennikov

Published Mar 19, 2014• 0 citations
Publisher

Abstract

In Section 1.2.8, we postulated that the prequantum field dynamics is given by the Schro¨dinger equation. In this chapter we present the classical mechanical motivation of this approach. It is based on the more or less well-known coupling between classical Hamiltonian dynamics and Schro¨dinger dynamics.
DOI
10.1201/b15540-7

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In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
According to non-relativistic quantummechanics, the evolution of the wave function of a free particle of mass m and momentum p is described by the Schro¨dinger equation: i ∂ψ ∂t = Hψ , (6.1) with the Hamiltonian operator H = −∇2/2m obtained from the expression for energy E = p2 2m , (6.2) using the substitution E → i ∂ ∂t , p → −i∇ . (6.3) Schro¨dinger himself [6] first suggested a generalisation of equation (6.1) based on the use of the relativistic energy equation: E2 = p2 +m2 . (6.4) (+m2)ψ = 0 , (6.5) Multiplying by ψ∗ and subtracting the product of ψ and the complex conjugate of (6.5) from the result, the continuity equation is obtained ∂ρ ∂t = ∇j , (6.6) with ρ = ψ∗ ∂ψ ∂t − ψ∂ψ ∂t (6.7) and j = −∇ (ψ∗∇ψ − ψ∇ψ∗) . (6.8) However, the ρ which appears in equation (6.6) cannot be identified with the probability density, in analogy with the similar quantity obtained from the Schro¨dinger equation, because it does not have the required property of being always positive-definite. To be convinced of this it is sufficient to substitute i∂/∂t→ E in (6.7). The result, ρ = E|ψ|2 , shows that ρ can be either positive or negative, following from the fact that the Klein-Gordon equation has solutions for the energy with both signs E = ± √ p2 +m2 , Note that in the non-relativistic limit E ≈ m > 0, and the familiar result ρ ∝ |ψ|2 is recovered.
Quantum mechanics is essentially a 20th century development that is based on a number of observations that defied classical explanations. While some of these experiments have semiclassical explanations, the triumph of quantum mechanics is that it gives precise verification for an overwhelming number of experimental observations. Its extensions into relativistic quantum mechanics through quantum electrodynamics (QED), electro-weak theory, and quantum chromodynamics (QCD) have led us to the Standard Model of today. While the number of unanswered questions remains approximately constant at each stage of development, the number of answered questions that relate theory with experiment continues to grow rapidly. In this chapter, some of the historical landmarks in the development of the theory are noted. The resolution of the dilemmas presented by classical theory will be dealt with in later chapters, but these are listed to motivate the break from classical mechanics. Because of the apparently unphysical nature of the postulates upon which quantum mechanics is founded, we supply motivation and justification for this break. In the first part of this chapter, we will review some of the experiments that confounded classical theory, and then introduce a formalism that provides some rationale for the postulates upon which quantum mechanics is based. In the end, these postulates will stand on their own.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.