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Quantization and Field Equations


The Different Perspectives of the 1st and 2nd Quantizations

Quantization The difference between 1st and 2nd quantization is summarized in Figure 08a. In diagram (a) for the 1st quantization, the classical picture of an electron swivelling around the nucleus (as in the hydrogen atom) is replaced by wave function which is a function of r. In this particular case, there are two maxima in the function corresponding to the darker brown and green colors. The averaged circular path is indicated by the circle in blue. The absolute square of the wave function is interpreted as the probability of finding the electron at that point. As shown in diagram (a), the interacting medium is

Figure 08a The 1st and 2nd Quantization [view large image]

the longitudinal component of the electromagnetic field. It is static and by its nature non-relativistic. Such formulation is more suitable for calculating structures in bound state, e.g., in molecules, atoms, ...
The wave function has two separate parts for the spatial and temporal variables respectively - a characteristics of standing wave (see example in "Square Well Potential"). This paradigm is known as Quantum Mechanics and exemplified by the Schrodinger Equation, which is ubiquitous in the studies of atomic, molecular, and solid state physics. See "QM Basic" for an introduction to the subject.

Quantization Diagram (b) shows the classical spinor field for a free electron. The 2nd quantization associates the field with a particle. In this example of Compton scattering, which describes the interaction of the electron with a photon, both of them are treated as particles changing states (energy-momentum p, k and/or spin p, polarization ) in the process. The quantum vacuum is now simmering with all kinds of virtual particles popping into existence briefly according to the uncertainty principle t E > (Figure 08a). The celebrated Higgs particle was discovered when it was excited from the virtual Higgs amid the debris in energetic collisions (Figure 08b, also see Higgs Discovery in LHC).

Figure 08b Higgs Particle Excited from Virtual Higgs [view large image]

As shown in Figure 08a diagram (b), the interaction is now provided by the transverse component of the electromagnetic field A moving with the speed of light and in the form of a particle.
Such formalism is relativistic and tailored to the collision of particles for probing into the mechanism within. Both the real and virtual particles are represented by harmonic oscillators in the form : e-i(kx-t) = e -ikx , which is most suitable for describing traveling wave (v = dx/dt = /k). The probability of the various transitions is calculated by the S-matrix. The formalism of 2nd quantization is known as Quantum Field Theory. The Standard Model is the ultimate product of this methodology. Because of its many shortcomings, a lot of physicists are trying to replace it with a more elegant formulation. The attempts are not very successful so far as new theories keep falling short on either observational or experimental testing.

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