Building Blocks of Quantum Mechanics

Theory and Applications

Tao Xiang

• Verlag: CRC Press
• Jahr: 2022
• ISBN:
  9781000583717
• Auflage: 1
• Thema: Naturwissenschaften
• Thema: Mechanik

This textbook provides a concise yet comprehensive introduction to the principles, concepts, and methods of quantum mechanics. It covers the basic building blocks of quantum mechanics theory and applications, illuminated throughout by physical insights and examples of quantum mechanics, such as the one-dimensional eigen-problem, the harmonic oscillator, the Aharonov-Bohm effect, Landau levels, the hydrogen atom, the Landau-Zener transition and the Berry phase.

This self-contained textbook is suitable for junior and senior undergraduate students, in addition to advanced students who have studied general physics (including classical mechanics, electromagnetics, and atomic physics), calculus, and linear algebra.

Key features:

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• Presents an accessible and concise treatment of quantum mechanics

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• Contains a wealth of case studies and examples to illustrate concepts

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• Based off the author's established course and lecture notes

1Introduction

DOI: 10.1201/​9781003174882-1

1.1. Brief history of quantum mechanics

Quantum mechanics is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. It, together with Einstein's theory of relativity, is the two major discoveries of the 20th century. They are also the pillars of modern science.

Quantum mechanics is inescapable. Nearly all fields of modern physics, from elementary particles to the big bang, from semiconductors to solar energy cells, are related to quantum physics. It is undoubtedly one of the greatest intellectual achievements of the history of humankind, probably the greatest of those that will remain for centuries.

The phrase “quantum mechanics” was coined (in German, Quantenmechanik) by Max Born, Werner Heisenberg, and Wolfgang Pauli at the University of Gottingen in the early 1920s, and was first used in Born's 1924 paper “Zur Quantenmechanik”.

Two experimental observations played central roles in the establishment of quantum mechanics. One is the the black-body radiation first reported by Balfour Stewart in 1858 and then by Gustav Kirchhoff in 1859. It was the solution to this problem by Max Planck in 1900 that led to the birth of quantum theory. Planck is also considered to be the father of the quantum physics. The other is the optical spectrum of hydrogen. The basic theory of quantum mechanics, including the Schrödinger equation, was established based on the solution of this problem.

Major progress in the discovery of quantum mechanics includes:

1. In 1900, Max Planck postulated a quantized energy formula of light to solve the black-body radiation problem. He pointed out that in order to solve the black-body radiation problem, the energy ε of a light oscillator of a given frequency ν should not vary continuously, instead it could only take on some discrete values, which are integer multiplies of ν
   $$\begin{array}{l}\epsilon =h\nu ,\hfill \end{array}$$(1.1)
   where h is a fundamental constant which is called the Planck's constant. Based on this assumption, he found that the spectral density of electromagnetic radiation, emitted by a black body in thermal equilibrium, depends only on the temperature T,
   $$\begin{array}{l}I(\nu )=\frac{8\pi h}{c^3}\frac{{\nu }^3}{e^{h\nu /k_{B}T}-1},\hfill \end{array}$$(1.2)
   where k_B is the Boltzmann's constant and c is the speed of light. This formula is also called Planck's law.
   Planck's discovery of light quantization was totally at variance w...