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Quantization and Field Equation


Contents

Short-cut to the Introduction of Quantum Theory
First Quantization and Schrodinger Equation
Klein-Gordon Equation of the Scalar Field
Derivation of the Dirac Equation and the Weyl Spinor
Interpretation of the Dirac Equation
Electromagnetic Wave Equation
Electromagnetic Wave Polarization and Photon Spin
Second Quantization and Feynman Diagram
The Different Perspectives of the 1st and 2nd Quantizations
A Footnote - Mathematics for Double-slit Experiment

Short-cut to the Introduction of Quantum Theory

Road to Qunatum The traditional introduction to quantum theory is usually by tracing its historical development from anomalies such as in the blackbody radiation, to the Heisenberg Uncertainty Principle, and finally arriving at the Schrodinger's Equation. Each stage along the way always involves some unfamiliar concepts such as wave-particle duality, measurement uncertainty, quantization, and wave function making the learning process torturous. In his 2014 book on "Quantum Mechanics", L. Susskind adopts a different approach to introduce the subject in a logical way with minimum number of unfamiliar idea. The following is a recap of such style (a scenic short-cut to the same destination, Figure 01a).

Figure 01a Road to Quantum


Hilbert Space The one particle state in classical physics is specified by six numbers, namely three for the coordinates (x, y, z) and three more for the velocities (vx, vy, vz). Once these six variables are known, its subsequent evolution is determined by the dynamic equation, which is usually related to the Hamiltonian involving different kinds of energies such as the kinetic, magnetic, potential, and other interactions. The one particle state in quantum theory is the quantum state | >, which is related to the time, coordinates, momenta (p = mv), spin, energy, ... The | > can be visualized as the generalization of the three dimensional vector to multi-dimensional called state vector (in the so-called Hilbert space, Figure 01b). A quantum state can involve one or more of the unit (basis) vector |n>, the evolution of | > is again governed by the dynamic equation but related to the quantized version of the Hamiltonian. Thus, the deterministic nature of the system is preserved in both paradigms, however the determination is on different object.

Figure 01b Hilbert Space

(See more about state vector in "Bra-ket Notation")

Here's the easy way to quantum theory step by step :







See "Mathematical Terminology" for more on the definition of the various mathematical objects.



See an introduction to quantization in "Harmonic Oscillators and Quantization of Field", and "Second Quantization and Feynman Diagram", "Different Perspectives of the 1st and 2nd Quantizations" for more detail. Also see "Natural Units" used in particle physics


The small insert above demonstrates the Schwartz's Inequality by two vectors. It shows the projection of a vector |B|cos() is always less (shorter) than or equal to the length of the vector |B| itself.

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