At the end of the 19th century, the macroscopic world was understood in great detail. Newton's laws described mechanics ranging from the collisions of marbles to the motion of planets, Maxwell's equations explained electromagnetic phenomena and statistical physics was the underlying theory for thermodynamic observations. However, a few effects remained truly unexplained, such as the spectrum of black-body radiation and the photoelectric effect. When Planck first quantized the accessible energies for the modes inside a black body, thereby accurately reproducing the observed black-body spectra, he still regarded it as a dirty trick. Einstein was the first to take this idea more seriously when he used it to explain the photoelectric effect by introducing the quantum of light, nowadays called a photon. It was gradually realized that the microscopic world is governed by a set of rules that is completely different from the rules that we know in our everyday life. This set of rules is called quantum mechanics and its success in explaining the microscopic world has been enormous. An important example is Bohr's explanation for the discrete spectra of light that is absorbed and emitted by atoms. Initially quantum mechanics came in two seemingly different formulations, namely the one by Heisenberg now known as matrix mechanics, and the one by Schrödinger now known as wave mechanics. Soon these two pictures were shown to be equivalent, and the unified formulation of quantum mechanics was given by Dirac and Von Neumann. Later, yet another way of doing quantum mechanics was developed by Feynman with the use of path integrals. The latter formalism, which is the topic of Chap. 5, is generalized to quantum field theory in the second part of this book.
In this chapter we review the elementary concepts from quantum mechanics in the elegant formulation of Dirac, focusing on concepts that we need later on for the description of interacting quantum gases. We introduce various important concepts in a familiar setting, such as the number states and the coherent states, before generalizing them to the more abstract formalisms of second quantization and functional path integrals in later chapters.s
KeywordsState Vector Harmonic Oscillator Coherent State Annihilation Operator Relative Momentum
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