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The scientific methodology adopts essentially a reductionist's approach, which break down the investigation into many levels mainly defined by size and from small to large (see a few examples in Figure 01). Usually, the higher level does not involve the detail of the lower one, which just provides a few parameters as linkage between the two. These parameters serve as macroscopic description of the more complicated microscopic domain, and often can be  
Figure 01 Reductionist's Systems 
measured by experiment or observation. 
 
Figure 02 Particle, Definition 
characterized by its electric charge, mass, spin, and color charge. The ratio R ~ 10^{3}, which is much larger because the strong interaction in this case is a short range force. Actually, not much is known about this system. 
random directions as shown in Figure 03. The thermal energy becomes work if they can be coaxed to move in an unison direction such as in the "Heat Engine". It is called a wave when the organized motion is propagating through the medium such as the sound wave in Figure 04. Thus, particle and wave seem to be very different. Each particle is a single entity with certain characteristics, while the wave involves coherent motion of a collection of particles at macroscopic level as it is understood until the advent of quantum theory which treats these two entities as the flip side of each others and creates severe conceptual problem.  
Figure 03 Thermal Motion 
Figure 04 Sound Wave 
The wave nature of light (as ElectroMagnetic, EM wave) has been demonstrated conclusively by its diffraction pattern since 1801 in Young's experiment (Figure 05). The particle aspect of light was first revealed by Max Planck's "quanta of light" to account for the discrepancy in blackbody radiation; it was finally confirmed by the discovery of photoelectric effect (Figure 06) in 1905. It was found subsequently that electrons can produce similar diffraction as light. The diffraction pattern is attributed to matter wave (or probability wave, see Figure 07) similar in form to the EM wave but with different origin.  
Figure 05 Diffraction of Light (EM Wave) _{} 
Figure 06 Photoelectric Effect [view large image] 
The EM wave is actually at macroscopic level  one step away from quantum. 
In 1924, de Broglie derived a relation between the wave length of the matter wave and momentum p of a particle. Starting from the relativistic version of the kinetic energy E_{k} = pc, which also can be expressed in term of quanta in quantum theory as E_{k} = hf, where the frequency f ~ c/ if the wave travels at relativistic speed, and h = 6.625x10^{27} ergsec is the Planck constant. By putting the two formulas together, we obtain the de Broglie relation : p = h/.  
Figure 07 Matter Wave (Probability Wave) [view large image] 
In Figure 07, P_{1} and P_{2} are the probabilities of finding the particle (from slit 1 or 2) along the screen , and P_{12} is the result of interference  hence the nomenclature of "probability wave". 
Figure 08a Traveling and Standing Waves [view large image] 
Figure 08b Standing Waves [view large image] 
Figure 08c Superposition 
See "Harmonic Oscillator" and "Infinite Square Well". 
There is a deeper level of quantum theory than the particle and wave as mentioned above. In the Standard Model (SM) of elementary particles, it is the quantum fields that are ubiquitous in the formulation. In contrary to the classical fields such as the electromagnetic fields in Maxwell's equations and the metric tensors in General Relativity, in which there is always a source such as charge and current or massenergy tensor to generate the fields, the quantum fields in SM has no source  it is assumed to be everywhere and at all time in this Universe. That's why there is always a  
Figure 09 Quantum Fields [view large image] 
Figure 10 Virtual Particles [view large image] 
suspicion about an even deeper level to describe the detail of their origin  according to the the reductionist's view point. See "Quantum Fields and Vacuum Energy Density" for more on vacuum state. 
The interaction is expressed in the covariant derivative D_{}. There are altogether 3 coupling constants  g and g' are associated with the W_{} and B_{} boson respectively, G_{e} is the Yukawa coupling constant, which defines the interaction between the Higgs and the lepton. The first three terms in Eq.(41c) are responsible for generating the mass of the gauge bosons, while the last term takes care of the fermion mass. The quarks have a slightly form of the Lagrangians by the title of quantum chromodynamics (QCD).  
Figure 11 SM Largranian for Leptons 
Figure 12 Quarks and Gluons [view large image] 
indicates that the matrix element S_{if} has an inverse, which in turn implies that it is possible to return to the initial state from the final state at least in principle although the probability is almost zero in practice so that the second law of thermodynamics is "almost" never violated. This property is also related to the socalled "conservation of information", which caused so much trouble for Stephen Hawking in the "Black Hole Information Paradox".  
Figure 13 SMatrix 
Note: For example in the electron positron scattering process, there are three possible finally states as illustrated in Figure 13. The sum of the probabilities for each one of these has to be: S*_{11}S_{11} + S*_{21}S_{21} + S*_{31}S_{31} = 1 to insure that the final state is in one of the three possibilities. 
Similar to the superposition of waves in quantum mechanics, the unitary condition _{f} S*_{if}S_{if} = 1 is another kind of superposition. That is, given an initial condition there are many possible outcomes until an observation or measurement is made. It is not clear why people would have problem with such process and relates it to "animated suspension". They seem to avert at seeing such process in black and white or in formula. But the realization of probabilities happens all the time as shown in Figure 14 and the chance of lottery win (Figure 15).  
Figure 14 Possibility in Life [view large image] 
Figure 15 Probability in Lottery [view large image] 
The unitary condition for Lottery can be expressed as : _{f} P_{0f} = 1, where the subscript 0 denotes "no ticket", f stands for the 6 digits out of 49, and P_{0f} = 1/13983816 is the chance of winning for each f in fair play, 13983816 is the number of combination. 
In case the game is rigged such that only 6 balls (digits) would fit the exiting hole, then the unitary condition is still : _{f} P_{0f} = 1, but now all P_{0f} = 0, except P_{0w} = 1 for the winning combination w.  
Figure 16 Bell's Theorem 
This is an extreme example of Bell's Theorem, which was proposed to test if there is any (not yet discovered) hidden elements in quantum theory. Since then, many tests have been performed to show there is no hidden variables in quantum theory (no cheating). 
What irked Einstein and friends even more is the phenomena of entanglement which is the superposition of states between two or more particles (Figure 17). It bothered them to witness the simultaneous collapse of corresponding states even when the detectors are separated beyond the distance allowed by the speed of light (called nonlocality). They blamed the incompleteness of quantum theory and proposed the "Hidden Variables" interpretation.  
Figure 17 Entangled States 
Figure 18 Entanglement, 
Way back in 2004, halfadozen experiments have been designed to determine the boundary between the classical and quantum world. One experiment shown in Figure 19 fires C_{70} fullerene balls (70 carbon atoms in the soccerballlike crystal of about 1 nm across) at 190 m/sec toward two diffraction gratings. The first grating creates the matter wave from the fullerenes. The wave is then diffracted by the second grating and the interference pattern is formed on the detecting screen demonstrating the wave property of the fullerenes. However, the interference pattern fades away if the fullerenes are heated by a laser heater (to about 2700^{o}C) or collide with gas (leaking into the vacuum chamber of the experiment). No one has a definitive answer for how the striking photons or molecules switch the quantum to classical domain. One explanation is that the interaction causes an uncertainty in the position of the fullerenes, blurring the interference pattern. Another argument asserts that the disappearance of the quantum property is caused by entanglement between the photons/molecules, the fullerenes, and the rest of the world (the wall of the chamber).  
Figure 19 MatterWave Experiment and quantumclassical boundary _{} 
See explanation by "Decoherence". 
In an article commemorates the 50th anniversary of the "New Scientist" magazine, Roger Penrose suggests that there are three kinds of reality: the physical, the mental and the mathematical, with something (as yet unknown) profoundly mysterious in the relations between them. According to this view, the various "Quantum Interpretations" are attempts to link the mathematical reality to the physical or mental reality. Figure 20 shows the mathematical reality as the patterns of interference computed from a mathematical formula, while the physical reality is in the form of photographic plate with the darker strip corresponding to the higher value of the curve. The mental reality is the image of dark and white strips formed in the retina and perceived by our consciousness.  
Figure 20 The Reality

Interpretation  Feature  Status 

Copenhagen  Measurement plays a key role in decoherence of quantum states  Healthy 
Hidden Variables  Hidden variables carry missing information about quantum states  Challenged 
Many Worlds  All quantum possibilities play out in a multiplicity of universes  Dubious 
Penrose  Outcome of experiments is a result of gravitational interactions  Under investigation 