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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),
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Figure 08a The 1st and 2nd Quantization [view large image] |
the interacting medium is 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, ... (see an updated version of "1st Quantization"). |
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Diagram (b) shows the classical spinor field for a free electron. The 2nd quantization (see 2021 version) 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). As shown in Figure 08a diagram (b),
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Figure 08b Higgs Particle from Virtual Higgs |
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 |
t) = e -ik
x , 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.
