Wave Equations

Short-cut to the Introduction of Quantum Theory

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

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 |k>, 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")

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