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CRC Press eBooks

Quantum Mechanics

Kumari Mamta, Raj Kumar Singh

Published Mar 10, 2026• 0 citations
Publisher

Abstract

In this chapter, we first discuss the photoelectric effect, its characteristics, and Einstein's approach to this phenomenon. This is followed by the Compton effect and the wave-particle duality. The concept of De Broglie wave, wave packet, phase velocity, and group velocity follows this. Then comes the Heisenberg uncertainty principle with application and energy-time uncertainty. Features of a well-defined wave function and its normalization are discussed. Time-dependent and time-independent Schrödinger equation with an application is described. The problem of a particle in a box has been discussed, and the energy eigenvalue and eigen wave function have been determined under suitable boundary conditions.

DOI
10.1201/9781003710486-4

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In this chapter, we first discuss the photoelectric effect, its characteristics, and Einstein's approach to this phenomenon. This is followed by the Compton effect and the wave-particle duality. The concept of De Broglie wave, wave packet, phase velocity, and group velocity follows this. Then comes the Heisenberg uncertainty principle with application and energy-time uncertainty. Features of a well-defined wave function and its normalization are discussed. Time-dependent and time-independent Schrödinger equation with an application is described. The problem of a particle in a box has been discussed, and the energy eigenvalue and eigen wave function have been determined under suitable boundary conditions.
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This chapter starts with the historical development of quantum mechanics and explanations of key concepts behind fundamental discoveries such as Planck's hypothesis, the photoelectric effect, Bohr's atomic model, Compton scattering, the de Broglie hypothesis, the Pauli exclusion principle, the Schrödinger equation, Heisenberg quantum mechanics and the uncertainty principle, the Born interpretation of wave function, the Davisson and Germer experiment, and Dirac relativistic quantum mechanics, which made important contributions to the foundation of modern quantum mechanics. Concepts like the particle properties of radiation, the wave properties of matter, quantized energy and momentum, and the idea that one can no longer know exactly where a single particle like an electron is at any instant of time are brought out clearly through this chapter. A list of important quantum mechanical parameters along with solved examples and unsolved exercises on the key concepts are included towards the end.
The chapter presents the first application of wave ideas to quantum particle descriptions, including describing the Davisson–Germer experiment, in which electrons were diffracted as waves from crystals. The work of famous early developers of quantum mechanics—Planck, Einstein, and de Broglie—led to relating momentum to wavelength, and energy to frequency. The wave function and its probability interpretation by Born are introduced. The wave functions of a particle in a box potential are exactly analogous to classical standing waves, giving these matter waves quantized energies just as frequencies are quantized in the harmonic series of a stringed instrument. The Heisenberg uncertainty principle follows from the general wave analysis. Zero-point motion and zero-point energy are important concepts resulting from the application of wave ideas to particles.
Abstract This chapter examines some crucial experiments and basic principles of quantum mechanics that have strongly influenced the development of quantum theory in its modern form: starting from Planck’s law of black-body radiation and its derivation using the quantization of energy; Einstein’s photon hypothesis for the puzzle presented by the photoelectric effect and its test via the Millikan experiment; the Compton effect; Bohr’s theses for the structure of the hydrogen atom and the complementarity of waves and particles i.e., the wave-particle duality, culminating in de Broglie’s ideas for matter waves and their confirmation in the Davisson-Germer experiment for electron diffraction from a crystal; Heisenberg’s uncertainty principle for position and momentum as well as for energy an time; the famous double-slit experiment and Born’s probability interpretation, and finally the provocative thought experiment of Schrödinger’s cat
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description
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.