It was Erwin Schrodinger who, in 1925, discovered an appropriate form of wave equation, making use of some deep formal analogies between optics and classical particle mechanics that had been evolved in the nineteenth century by W. R. Hamilton and others. There is all the accumulated evidence that the Schrodinger equations work; they provide the basis for a correct analysis of all kinds of molecular, atomic, and nuclear systems. The theoretical basis for the existence of states of discrete energies emerges more or less directly from the wave-particle duality; one of the most familiar features of classical wave behavior is that of standing waves. The quantum amplitudes for a particle in a bound state can be normalized. In order to make the statistical interpretation of the wave function more precise, we need to use the quantitative ideas of probability distributions.
It was Erwin Schrodinger who, in 1925, discovered an appropriate form of wave equation, making use of some deep formal analogies between optics and classical particle mechanics that had been evolved in the nineteenth century by W. R. Hamilton and others. There is all the accumulated evidence that the Schrodinger equations work; they provide the basis for a correct analysis of all kinds of molecular, atomic, and nuclear systems. The theoretical basis for the existence of states of discrete energies emerges more or less directly from the wave-particle duality; one of the most familiar features of classical wave behavior is that of standing waves. The quantum amplitudes for a particle in a bound state can be normalized. In order to make the statistical interpretation of the wave function more precise, we need to use the quantitative ideas of probability distributions.
Schrödinger’s wave mechanics, developed in a series of papers published one century ago, offered an exceptionally intuitive route into the new atomic physics. The later axiomatic formulation of quantum mechanics brought indispensable mathematical precision, but it can also conceal the physical images that guided the original discovery. We revisit the optical–mechanical analogy behind Schrödinger’s construction and use it as a visual guide to stationary states, superposition, field modes, and gauge interactions. The analogies are deliberately heuristic and their limitations are stated: a quantum wavefunction is not, in general, a classical wave in ordinary space, and discrete classical modes do not by themselves imply quantized energy. Nevertheless, the wave viewpoint makes it possible to see the operator language of first and second quantization as the compact algebraic expression of an underlying mode structure. The purpose of this article is therefore not to replace the modern formalism, but to recover the physical intuition that makes that formalism intelligible.
Quantum mechanics began in very early twentieth century when Max Planck found the expression for black body thermal radiation in which the emitted light was not a continuum as postulated by classical physics. Erwin Schrodinger applied his equation to solve the quantum harmonic oscillator, the quantum rigid rotor, and a hydrogen-like atom. Pertaining to the solution of the Schrodinger equation to hydrogen atoms are the quantum numbers. C. K. Ingold’s overview of Schrodinger wave mechanics: “What can be known about an electron, considered as a particle, is of a statistical nature, and is summarized in the behavior of a wave. When considering singlet substances, the normal Zeeman Effect occurs, providing the discrete values of the orbital angular momentum. In an attempt to rationalize the anomalous Zeeman effect, W. Pauli proposed the fourth degree of freedom of the electron which he called “two-valuedness not describable classically”.
Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie's wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.
Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
Quantum mechanics is introduced here as a fundamental theory that must treat particles equally often as waves. Here, classical waves are introduced in some detail, with discussions of the important concepts of waves: medium, amplitude, frequency, wavelength, period, harmonics, interference, standing waves, etc. Waves can be oscillatory waves with a repeating pattern, or pulse waves, which are localized in space. All of the elements of waves are important in understanding how the Schrödinger equation describes particles. Classical standing waves must have wave lengths that fit between boundaries giving them discrete allowed frequencies, a harbinger of quantized energies in quantum mechanics. Complicated waves can be analyzed in terms of their component frequencies, the fundamental idea of Fourier analysis. This discussion shows that a classical wave has a limit in how accurately one can specify simultaneously the wave’s position and frequency, in analogy with Heisenberg uncertainty principle of quantum mechanics.
It is argued (Part A) that quantum mechanics can be derived as a principle-based dynamical framework, the basic equation of which is an alternative form of the Hamilton-Jacobi equation. Schrodinger's equation obtains as a result of linearising that equation, and so-called wave functions can be given no straightforward physical interpretation. It is suggested, partly in relation to a theorem by Gromov, that a finite action quantum would make it practically inevitable, for purposes of prediction, to resort to a probabilistic formulation. The structure of the space of square-integrable solutions of the Schrodinger equation happens to lend itself to the introduction of the appropriate kind of predictive scheme. Investigating the nature and scope of such a scheme is the subject of Part B. It is shown that basic features of the formalism of quantum theory, like composition rules for 'amplitudes' or the 'Born' probability rule, can be derived independently of any physical assumptions. A generalisation of the basic formalism using tensor product composition appears to be required if all correlations are to be extracted from locally accessed data. A detailed discussion of quantum teleportation leads to the conclusion that a 'one-shot' account leads to a distorted picture of what is actually achieved. An analogy with classical cryptography is made and the statistical significance of the 'transfer', which does not require introducing any novel form of 'quantum information', is emphasised. Results obtained over the last decade using the extended formalism of positive operator-valued measures are reviewed and discussed. These lend further support to the idea that the set of basic 'quantum' rules functions as a general kind of probabilistic scheme for prediction, the structural features of which are not constrained in any direct way by the underlying physics. On the other hand, the very existence of such a predictive framework hinges on selecting a particular class of solutions of the Schrodinger equation, which selection has been incorrectly interpreted as reflecting a physical necessity.