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CRC Press eBooks

Applications of Quantum Theory

Jonathan Allday

Published Mar 3, 2009• 0 citations
Publisher

Abstract

While moving inside the box, the electron is free and unin uenced by any forces, but the walls of the box represent an impenetrable barrier. Consequently the amplitude for nding the electron at the walls must be zero. Figure 21.1 shows a simple one-dimensional (1-D) box in which the electron must be found somewhere between 0 and L, the length of the box.

DOI
10.1201/9781584887041-31

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While moving inside the box, the electron is free and unin uenced by any forces, but the walls of the box represent an impenetrable barrier. Consequently the amplitude for nding the electron at the walls must be zero. Figure 21.1 shows a simple one-dimensional (1-D) box in which the electron must be found somewhere between 0 and L, the length of the box.
One of the classic first problems in quantum theory textbooks is that of the particle trapped in a box. Despite the apparent artificiality of the situation, it has quite a few interesting features that we can quickly apply to some more physically plausible situations. The particle in a box is literally an object, such as an electron, trapped inside a ‘box.’ In a realistic physical example, this ‘box’ would be a volume of space bound by some constraining force acting on the particle. For example, an atom is a ‘box’ as far as the orbiting electrons are concerned, with the electrostatic force between the positive protons in the nucleus and the negative electrons providing the constraint. The first move is to assume that the wave function for any particle in the box will have separate spatial and temporal dependencies. One of the key steps in obtaining our wave function was the quantization of the momentum.
The problems of a free classical particle inside a one-dimensional box: (i) with impenetrable walls and (ii) with penetrable walls, were considered. For each problem, the classical amplitude and mechanical frequency of the T -th harmonic of the motion of the particle were identified from the Fourier series of the position function. After using the Bohr-Sommerfeld-Wilson quantization rule, the respective quantized amplitudes and frequencies (i.e., as a function of the quantum label n ) were obtained. Finally, the classical-quantum results were compared to those obtained from modern quantum mechanics, and a clear correspondence was observed in the limit of n » τ.
We study quantum dynamics of one-electron atom confined in a spherical impenetrable box with time-dependent radius. The behavior of the atomic electron interacting with the moving walls of the box is analyzed by computing average binding energy, average force and pressure. Linearly extending, contracting and harmonically breathing boxes are considered in the non-adiabatic regime i.e. when the wall of the box moves with high velocities. It is shown that linearly extending box leads to de-excitation of the atom, while the rapidly contracting box causes the creation of very high pressure on the atom and transition of the atomic electron into the unbound state. In harmonically breathing box diffusive excitation of atomic electron and ionization may occur in analogy with that for atom in a microwave field.
Quantum mechanics has brought revolutionary changes in the conceptual foundations of physics and continues to shape the modern world. In this book, the author's emphasis is on helping students comprehend the significance of the underlying principles and understand the ways the new concepts were introduced. Including many worked examples and problems, this book will be an invaluable resource for students in physics, chemistry and electrical engineering needing a clear and rigorous introduction to quantum mechanics.