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|Qunatum theory has its foundation in the uncertainty principle, which states in mathematical form :|
x px > , where denotes the uncertainty, x is the position of the point mass m along the x-axis, px = m vx is the momentum along the x-axis, vx 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 mc2), e.g., t E > . The uncertainty principle seems to be applicable up to a size of about 10-7 cm - the size of the C60 buckyball. In case of macroscopic objects with larger size and heavier mass, the uncertainties in
|position and time becomes very small, classical physics is applicable once more. A derivation of the uncertainty principle is appended in the footnote§.|
|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 :|
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., P12 = 1*1 + 2*2 + 1*2 + 2*1.|
Figure 02b Two 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).|
These mathematical formulas are shown pictorially in Figure 02d.
|The uncertainty in locating the particle x g/2 = 2/k. According to the relationship between the de Broglie wavelength of a particle and its momentum p, i.e., = h/p, the uncertainty in momentum p = hk/2. These two formulas together yields the uncertainty relation : x p h, where h is the Planck Constant.|