10- Second quantization and origin of particles :
The concept of wave particle duality has two different meanings depending on usage:
1. A continuous field versus countable entities present inside the field as somehow contradicting aspects.
The concept of duality had this meaning when it first emerged with the invention of the concept photon in order to refer to the energy portions the particle systems (atoms) can absorb or emit when exchanging energy with electromagnetic fields. In this context there is no reference to the question if each of these countable entities are pointlike or if they are merely somehow portions of the wave .
2. The question if a single particle (electron or photon) is a wave-like or a pointlike entity. The existence of the countable entities are considered as a given fact here.
This meaning of wave-particle duality emerged in discussions on double slit experiment reflecting the fact that one one hand the particle must go through both slits to produce the interference experiment and on the other hand it appeared as a pointlike entity going either through one or the other slit when the position is measured.
In discussions and written material mostly the second meaning is used. We must mention that the concept of pointlike particle automatically requires the existence of countable entities, and it is obvious that the reverse is not true.
Lets summarize the history of the concept of duality.
At the beginning there was a field and its portions (photon) but no pointlike entity chapter 2
Then came the double slit experiment and the duality began to refer to the complementary properties of being spread in space versus being pointlike. The duality in the sense spread versus pointlike was mathematically expressed as uncertainty relation between momentum and position. By the measurement independent formulation of the uncertainty relation as mathematical relation between the average spread of the wave function in momentum space and the average spread in ordinary space (
chapter 9 ) we realized that momentum eigenstates and position eigenstates are two extreme idealized forms of the wave function both nonexistent exactly in these forms in reality. What we perceive as pointlike particle is only a very localized form of the wave function. Thus the concept of pointlike particle has disappeared. This doesnt mean however that the countable entities have disappeared. Their existence is a fact known since Millikans discovery of the elementary charge.Thus a question remained waiting to be answered:
Why are there cauntable entities and not just continuous fields ?
In the case of material particles we are so much accostumed to think in terms of countable entities as self evident since demokritos that we hardly realize that this is a legitimed nontrivial question. That we could realize the importance of this question is due to the quantization of electromagnetic field. The electromagnetic field was previously assumed to be a continuous field It is the discovery of photons in black body spectrum and in photo electric effect that led us to the following question:
Why are there such indivisable portions in the electromagnetic field? It is only after formulation of this question we could realize that the question is nontrivial not only for photons but also for electrons.
The answer became apparent if we include in the Hamilton opreator not only thhe energy of the particle
But also the energy of the electromagnetic field. If one does this for a free electromagnetic field
(no electric charges just EM field is present) We obtain discrete energy levels of the field for each present frequency mode in the field that one can interprete as photons. Encouraged by this succes the method was applied to other particles like fermions where the wavefunction itself becomes an operator. The result is known as the theory ofquantized fields.
Later it was discovered that the field quantization has similarities with structures in a very different area of physics namely solid state physics.
The crystalls are made by ordered arrangements of atoms and molecules. The atoms in the crystall are held together by electromagnetic forces. Each atom has an equilibrium position in crystal. If it is displaced slightly by for example thermal motion from its equilibrium position , it is pulled back towards its equilibrium positions. Atoms oscillate therefore about their equilibrium positions. The random oscillations are related to the temperature of the crystal . If large number of atoms oscillate coherently we have elastic waves or sound waves. This is however a picture of crystal if it were made of classical particles with definite position. The motion of the a single atom around its equilibrium position is in truth a quantum mechanical problem . We have to use wave functions. The solution of the Shrodinger equation gives bounded energy eigenstates with disrete energy levels for such an oscillating atom. If the forces that pull the atom back to its equilibrium position are proportional to the distance from the equilibrium position the oscillator is called to be an harmonic oscillator. The energy levels of an harmonic oscillator are equidistant. One can formulate the quantum mechanical problem not only for one oscillating atom but for the whole crystal as coupled system of harmonic oscillators. The result is the following. The discretization of the energy levels of a single oscillator leads to discretization of the energy levels for whole waves. Thus the waves are not like classical waves but they loose or gain energy in interactions only in form of indivisable quants of energy. These quants of energy are called phonons. Thus we have waves with quantized energy.
The situation has interesting similarities with electromagnetic waves and its quants namely photons. Photons seem to be excitations of coupled imaginary harmonic oscillators in empty space. 1 The analogy is however limited. There are two fundamental differences in mathematical description. In the case of the electromagnetic field there are continuously many oscillators and the elongation of oscillation is not ordinary space but in a separate dimension that is merely an additional degree of freedom.
Thus field quantization is the currently accepted answer to the follwing question:
Why are there countable entities?
The reader should notice there are no pointlike entities in quantized fields but only discrete energy levels for different oscillation modes. Field is fundamental. The quant emerges as a property of the field . In this picture There is no complementarity relation between field and particle .
1 Davydov 1966 pp 545-569