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Quantization and Field Equations


First Quantization and Schrodinger Equation

Uncertainty Principle 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

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 10-21gm; with the velocity of about 104 cm/sec at 400oK, the uncertainty in its position should be greater than 10-10cm, which is many times smaller than its size of 10-7cm. On the other hand a single carbon atom moving at such speed would have an uncertainty in position very close to its size of about 10-8cm.

It can be shown that the expression for the uncertainty principle is equivalent to the commutative relations :
xpx - pxx = i ---------- (1),
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 pk's and E become non-commutative q-numbers in the forms of differential operators (the y and z components will be ignored now to simplify the formulation) :
px = -i ---------- (3a),
E = i ---------- (3b).
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 + (x)] = i ---------- (4)
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 = p2/2m ---------- (5)
which takes into account only the kinetic energy, the rest mass energy mc2 is omitted in this equation. It is positively defined as the square of the momentum p is always positive. Substituting Eqs.(3a), (3b) into (5) we obtain the Schrodinger Equation for a free particle (ignoring the y and z components again):
-(2/2m) = i ---------- (6)
Solution for this partial differential equation can be obtained by the method of separation of variables, e.g., (x,t)=A(x)e-iEt/, where the solution for t has been evaluated as an oscillating exponential function by virtue of the imaginary index, otherwise it will be an exponential function not in agreement with observation. Eq.(6) can then be reduced to an ordinary differential equation of the form :
(d2/dx2) = -(2mE/2) ---------- (7)
It is similar to the equation of motion for the harmonic oscillator:
(d2/dt2)X = -KX ---------- (8)
with taking the role of X, x replacing t, and the force constant K = (2mE/2). Thus the solution of Eq.(7) is another oscillating function :
= eikx ---------- (9)
where k = (2mE)1/2/. The final solution can be written as :
(x,t) = A e-i(t-kx) ---------- (10)
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.,
*dx = A2L = 1 ---------- (11)
which yields A = 1/L1/2. Thus the probability density * = 1/L, which indicates that the chance of finding the particle at any point within becomes smaller as the extension getting larger.

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 1, 2, ... are the solutions with k equals to k1, k2, ... respectively, the combined solution = A11 + A22 + ... is again a solution of the same differential equation, where the Ai's are interpreted in quantum theory as the probability amplitude of finding the particle in state ki's and (A1)*A1+(A2)*A2+...=1. This is known as superposition, which forms the base in explaining interference and the production of wave packets, .... in wave theory. For example, superposition in the following form will generate a wave packet:
---------- (12)
where f(k-ko) is the weighting function specifying the k dependency of the coefficients Ak. Since the value of k runs continuously from - to +, the sum becomes integration over the k-space. In particular, when the weighting function is in the Gaussian form:
---------- (13)
Wave Packet 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 :
---------- (14)
Figure 02a shows pictorially the wave packet as a function of x (centering around xo) and the weighting function (centering around ko). 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

Figure 02a Wave Packet
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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".

The wave-particle duality is usually demonstrated by the double slits experiment as shown in Figure 02b. A pattern consistent with the nature of particles emerges in the screen when the electrons go through only one slit. An interference pattern associated with the nature of wave appears
Two Slits Experiment Three Slits Experiment 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.
[view large image]

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).
In the 1990's, a certain theory of quantum gravity suggested that higher order terms such as 1*23*1 should be included in the three slits experiment (Figure 02c). Accordingly, an experiment was run in the late 2000's to check out the claim by subtracting the three slits patterns with the prediction from the Born's rule. The pattern is nullified within the 1% experimental error. Thus, the Born's rule has been vindicated for now. But the verdict is not final, there is room to improve the experimental uncertainty and the theoretical calculation should be reviewed for magnitude of the correction.

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 = mc2 > 0). It is usually referred to as quantum mechanics in distinguishing from the fully relativistic quantum field theory.

§ Footnote - A Derivation of the Uncertainty Principle and the Difference in the 2 Versions of the Planck Constant :

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 :

Uncertainty Principle, Derivation

These mathematical formulas are shown pictorially in Figure 02d.

Figure 02d Uncertainty Principle, Derivation


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.

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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