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Quantization and Field Equations

A Footnote - Mathematics for Double-slit Experiment

Here's more details to expand the hand-waving statement in the main text. All the assumptions usually omitted in textbooks will also be clarified in the process. We begin with the Huyghens' principle, which is applicable to electromagnetic wave as well as matter wave (such as electrons). It states that if we know the value of the wave function on a given wave front, then we can express its value elsewhere as the sum of contributions
Double-slit Experiment Diffraction Pattern from different elements of the wave front in the form of propagating spherical wave, e(2i r/)/r, where r is the distance from the point in question to the element of surface on the wave front, and is the wavelength. This spread of wave is somewhat similar to the pattern (in the wall behind) generated by firing a machine gun at an iron plate with two slits on it. Most of the bullets through the slits would align with each, some may be scattered sideways by hitting the edges although the chance becomes rarer as deviation getting further. One big difference is that bullets don't interfere with each other (story ascribed to Feynman). Anyway, assuming that the slit is very narrow, the two waves from slit 1 and 2 can be represented as:
1 = Ae(2i r1/)/r1 ---------- (66a)
2 = Ae(2i r2/)/r2 ---------- (66b)

Figure 09 Double-slit Exp.
[view large image]

Figure 10 Probabilities [view large image]

where A is the normalization constant, r1, and r2 are the distance from the slits to the screen as shown in Figure 09, which lays out the x-y plane only, while the z direction is perpendicular to the
computer screen. The expression for finite slit size is more complicated but will produce essentially the same result. Since the spacing between the slits is small comparing to the size of the apparatus, i.e., D >> d, therefore, r1 r2 and they are lumped together as r (see Figure 09). However the tiny difference d sin (in the interference term) is crucial in producing the interference pattern (see Figure 09 again). Thus, the probabilities can be expressed as :
1*1 2*2 (A2/r2) ---------- (67a)
1*2 + 2*1 (2A2/r2) cos[2(dsin/)] ---------- (67b)
The total probability is : P12() = 1*1 + 2*2 + 1*2 + 2*1 = (4A2/r2) cos2[(dsin/)] ---------- (68)
The normalization constant A can be evaluated by integrating Eq.(68) numerically over the angle from -/2 to +/2 :
1 = (8A2/D2)cos2 cos2[(dsin/)] d ---------- (68a)
For the values of d and quoted below, the integral is equal to about 0.390, which gives A = 0.566D. For rapid oscillation of the interference term, i.e., when d >> , the integral approaches a value of /8, giving A2 = D2/ or A = D/1/2 = 0.564D. Then it shows clearly that P12() can be interpreted as the probability per unit radian.

It is impossible to produce a graph with realistic data for light wave using home computer and Basic programming. Figure 10 is produced for some sort of matter wave with = 0.001 cm, d = 0.005 cm, and D = 10 cm. The "separate terms" together is plotted in green, the "interference terms" in blue, and the total probability is in red. The only running variable in the numerical computation is sin, which varies from -0.8 to +0.8 in steps of 0.001. Since cos2 = 1 - sin2, tan = sin/cos, from which we can compute r = D/cos, x = D tan. It is rather obvious from Eq.(68) that maxima occur at :
d sin = n , where n = 0, 1, 2, 3, ...
and minima occur at :
d sin = (n/2) , where n = 1, 2, 3, ...

The avid readers may have already noticed that the probability goes negative in the interference terms every half oscillating cycle (see Eq.(67b) or the blue curve in Figure 10). One explanation assures us that only the total probability P12 is observed, we should not worry about how the various terms added up as long as P12 can never be negative as shown in Eq.(68). It's somewhat like borrowing money from the bank, the account may become negative for awhile, but that's alright as long as payment is made on time. Or as explained by Feynman : no one objects to use negative numbers in calculations, although "minus three apples" is not a valid concept in real life.

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