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Qunatum theory has its foundation in the uncertainty principle, which states in mathematical form : x p _{x} > , where denotes the uncertainty, x is the position of the point mass m along the x-axis, p_{x} = m v_{x} is the momentum along the x-axis, v_{x} is the velocity along the x-axis, and = h/2 = 1.054x10^{-27} erg-sec (Figure 01). A similar relation exists for the y and z components, and for the time t and energy E (including the rest mass energy mc^{2}), e.g., t E > . The uncertainty principle seems to be applicable up to a size of about 10^{-7} cm - the size of the C_{60} buckyball. In case of macroscopic objects with larger size and heavier mass, the uncertainties in
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## Figure 01 Uncertainty Principle |
position and time becomes very small, classical physics is applicable once more. A derivation of the uncertainty principle is appended in the footnote^{§}. |

For example, the mass of the buckyball is about 10

It can be shown that the expression for the uncertainty principle is equivalent to the commutative relations :

xp

Et - tE = i ---------- (2),

with a similar expressions for the y and z components. These formulas can be represented in terms of matrices, but it is more often to keep x, y, z, and t as ordinary c-numbers but the p

p

E = i

The partial differentiation acts only on the indicating variable while others variables are held constant. By substituting Eqs.(3a), (3b) into (1), (2) respectively and applying the equations to a function (x,t), e.g.,

i [- x

it can be verified that the left hand side leads directly to the right hand side. The function is called wave function. It is interpreted as the probability of finding the particle at (x,t) and thus a wave property to a particle. Dynamic of the wave function is to be determined from the energy equation for a free particle :

E = p

which takes into account only the kinetic energy, the rest mass energy mc

-(

Solution for this partial differential equation can be obtained by the method of separation of variables, e.g., (x,t)=A(x)e

(d

It is similar to the equation of motion for the harmonic oscillator:

(d

with taking the role of

= e

where k = (2mE)

(x,t) = A e

where =E/. The integration constant A is determined by demanding that the probability of finding the particle within the extent of length L is 1, i.e.,

which yields A = 1/L

Eqs.(6) and (7) are linear differential equations because they involve only linear terms in the wave function. Linear equation has the special property that if

---------- (12) |

---------- (13) |

where k is the spread of bell curve in k-space (it is related to the spread in x-space by the uncertainty relation kx 1), the wave packet can be evaluated explicitly :
_{o}) and the weighting function (centering around k_{o}). A measurement on the wave packet will detect the particle at certain value of k. This is the Copenhagen Interpretation, which asserts that the wave function collapses to a single state in response to the measurement. In other word, the superposition is decoherent by the detecting apparatus such that the particle is not governed by the linear
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## Figure 02a Wave Packet |
equation (e.g., by Eqs.(6) and (7)) anymore, and thus the break down (or collapse) of the superposition. See more on superposition in "Mathematical Schrodinger's Cat". |

as soon as a coherent electron beam is allowed to pass through both slits (even down to only one electron at a time). The experiment also shows that the properties of particle and wave cannot be displayed simultaneously. Mathematically, there are four terms in the double slits experiment - two "separate-slit" terms from slit 1 and 2, and two "interference" term between the waves from slit 1 and 2, i.e., P_{12} = _{1}^{*}_{1} + _{2}^{*}_{2} + _{1}^{*}_{2} + _{2}^{*}_{1}.
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## Figure 02b Two Slits Exp. |
## Figure 02c Three Slits Exp. |
It is the latter two terms that produce the interference pattern, This is the Born's rule - a key law in quantum mechanics proposed in 1926. In plain language, it states that interference occurs in pairs of possibilities, higher order terms are ruled out (see more in footnote). |

This version of quantum theory takes into account only the mass of the particle, other physical attributes such as spin do not appear in the formulation. It is also non-relativistic in the sense that the equation is not invariant under the Lorentz transformation. The underlying low energy context also implies that it is not capable of treating high energy phenomena such as pair creation and annihilation, production of new particles, ... Nevertheless by adding an interaction potential into the equation, this formulation has been applied successfully to the physics of molecules, atoms and nuclei. Another virtue is its definition of energy, which would never turns into an undesirable negative value (negative energy is not observed in free particles as E = mc

A simplistic way for the derivation is to consider adding two matter (de Broglie) waves different slightly in angular frequency = 2 and wave number k = 2/, where is the frequency and is the wavelength of the wave :

A more realistic analysis would involve a wave packet consisting an infinite number of wave trains as shown in Eq.(12), from which in Eq.(14) yields a slightly different form of x 1//k giving a slightly different uncertainty relation : x p , where = h/2 is the "Reduced Planck Constant". Besides it is more realistic, also turns out to be the basic unit of angular momentum and used in defining the natural units. On the other hand, the light quanta is defined with h as E = h. Anyway, both forms of the Planck constant are used in scientific literatures, it doesn't matter as long as it is not mistaken from each other. A difference of 2 would introduce serious discrepancy in numerical calculation.

See "The Different Perspectives of the 1st and 2nd Quantizations".

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