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Cambridge University Press eBooks

Quantum states

John J. Gilman

Published Jan 1, 2001• 0 citations
Publisher

Abstract

The most simple atom is hydrogen consisting of a positive proton at the center, and one negative electron swarming around it. While the proton is concentrated at the center, the electron swarms in a pattern that depends on its total energy, and its angular momentum. In the lowest energy state of the hydrogen atom, the shape of the pattern is simply a sphere, but in higher energy states the pattern resembles a dumbbell, or a three-dimensional cloverleaf. For atoms with more than one electron, each electron has its own pattern, or state. The fundamental postulate that determines the behavior of atomic electrons is that of de Broglie (1924). This states that an electron of mass, m , and velocity, υ, and therefore with momentum, p = m υ, has a wavelength, λ, associated with it that is given by λ = h / p , where h is Planck's constant. If T = m υ 2 / 2 is the kinetic energy, then the expression for the wavelength can also be written λ = hm / 2 T , so the wavelength depends inversely on the kinetic energy. Thus short wavelengths mean high energies and vice versa. Initially, this seemed to be a rather arbitrary proposal, but it has been verified by many, and differing, experiments, and has led to vast numbers of quantitatively precise interpretations of phenomena.

DOI
10.1017/cbo9780511541247.008

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The most simple atom is hydrogen consisting of a positive proton at the center, and one negative electron swarming around it. While the proton is concentrated at the center, the electron swarms in a pattern that depends on its total energy, and its angular momentum. In the lowest energy state of the hydrogen atom, the shape of the pattern is simply a sphere, but in higher energy states the pattern resembles a dumbbell, or a three-dimensional cloverleaf. For atoms with more than one electron, each electron has its own pattern, or state. The fundamental postulate that determines the behavior of atomic electrons is that of de Broglie (1924). This states that an electron of mass, m , and velocity, υ, and therefore with momentum, p = m υ, has a wavelength, λ, associated with it that is given by λ = h / p , where h is Planck's constant. If T = m υ 2 / 2 is the kinetic energy, then the expression for the wavelength can also be written λ = hm / 2 T , so the wavelength depends inversely on the kinetic energy. Thus short wavelengths mean high energies and vice versa. Initially, this seemed to be a rather arbitrary proposal, but it has been verified by many, and differing, experiments, and has led to vast numbers of quantitatively precise interpretations of phenomena.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
An atom consists of a nucleus, containing protons and neutrons, with various electrons swarming around the outskirts. The electrons are held in the vicinity of the nucleus as they are negatively charged and hence attracted by the positively charged protons. Although atoms are complicated objects, there are certain situations where they can be treated as simple lumps of matter. Their internal structure has no relevance to the details of what is happening. This sort of situation is exemplified by experiments in which atoms are deliberately smashed into one another at comparatively low energy and as such provide an ideal start in a study of the quantum physics of identical particles. When we are dealing with atoms, there is something more to being ‘identical’ than we can appreciate with larger-scale objects. Atoms of the same type are really identical, measurably identical if you like, which can make a considerable difference in some physically important situations.
Toward the end of the nineteenth century, it became clear that some experiments cannot be explained on the basis of known physics, now referred to as classical physics. In 1900, Max Planck was able to correctly describe the distribution of blackbody radiation energy as a function of wavelength λ or frequency ν = c/λ. He had to accept the existence of energy packets (quanta) with energy E = hν. A few years later, Albert Einstein found that this unexpected relation between energy and frequency is also necessary in order to understand photoemission from metallic surfaces. A decade later, Niels Bohr arrived at the correct equation for the electronic spectrum of the hydrogen atom by applying “quantum conditions” on the motion of the electron around the nucleus and the Planck’s relation E = hν on the energy difference between the energy levels (1916). A decade later, Heisenberg, Born, and Jordan introduced “quantum mechanics” and de Broglie and Schrödinger (1925-1926) introduced “wave mechanics.”
This work introduces the Saedi Quantum Acceleration (SQA) framework as a mechanical reinterpretation of the Planck–de Broglie relation. Starting from the classical work-energy expression E = Fd = mad, the model assumes that the effective displacement associated with a quantum-scale matter-wave cycle is proportional to the de Broglie wavelength λ. By defining a wavelength-based acceleration aλ = λ/T², the SQA energy expression becomes ESQA = mλvf. Comparison with Planck’s energy relation E = hf leads to the expression h = mλv, which is consistent with the non-relativistic de Broglie relation. The paper does not claim a new numerical value or variable form of Planck’s constant. Instead, it presents h = mλv as a mechanical re-expression of the Planck–de Broglie structure, where Planck’s constant appears as an invariant product of mass, matter-wave wavelength, and velocity within the proposed SQA framework. The work further discusses quantum acceleration, frequency-induced energy changes, and the interpretation of ESQA as an effective wavelength-cycle energy rather than ordinary classical kinetic energy. This preprint is intended as a conceptual and theoretical physics contribution exploring alternative mechanical interpretations of quantum energy relations, matter-wave dynamics, and frequency-dependent energy exchange.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.