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Double-slit Experiment, Gateway to Quantum Theory (2020 Edition)

According to the Huyghens' principle proposed in 1678 (some 342 years ago) for wave motion, each point on the wavefront act as a fresh
Double-slit Experiment Diffraction Pattern source of distribution of wave; in other words, 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 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. Assuming that the slit is very narrow, the two waves from slit 1 and 2 can be represented as:

1 = Ae(2i r1/)/r1 ---------- Eq.(01a)
2 = Ae(2i r2/)/r2 ---------- Eq.(01b)

where A is the normalization constant, r1, and r2 are the distance from the slits to the screen as shown in Figure 01, which lays out the x-y plane only, while the z direction is perpendicular to the computer screen.

Figure 01 Double-slit Exp.
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Figure 02 Wave Magnitude

Figure 02 is produced for some sort of 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 magnitude 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 impossible to produce a graph with realistic data for light wave using home computer and Basic programming.

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 01). However the tiny difference d sin (in the interference term) is crucial in producing the interference pattern (see Figure 01 again). Thus, the various contributions to the wave magnitude can be expressed as :
Slit Experiments 1*1 2*2 (A2/r2) ---------- Eq.(02a)
1*2 + 2*1 (2A2/r2) cos[2(dsin/)] ---------- Eq.(02b)
The combined magnitude is :
P12() = 1*1 + 2*2 + 1*2 + 2*1 (4A2/r2) cos2[(dsin/)] ---------- Eq.(03)
The normalization constant A can be evaluated by integrating Eq.(03) numerically over the angle from -/2 to +/2 :

Figure 03 Slit Experiments
[view large image]

1 = (8A2/D2)cos2 cos2[(dsin/)] d ---------- Eq.(04)

Figure 03,b is a schematic diagram of the double-slit experiment showing the wave-like characteristic as described by Eq.(3). Figure 03,a is another version of the experiment in which light sources are in place to pinpoint the location (the diffraction fringes have been ignored in the digarm). In this case, the combined pattern is just the sum of the single sources without the interference term, i.e.,
P12() = 1*1 + 2*2 ---------- Eq.(05),
which is the characteristic of particles. Note that there is no slit in this case; the experiment is performed with just two light sources. The interference disappears because the two light sources are not in coherent, the averaged value of the cosine term in Eq.(02b) vanishes.

Double-Slit Experiment 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 magnitude per unit radian. The experiment was first performed with light by Thomas Young in 1801; thus it is sometimes referred to as Young's experiment.

It is rather obvious from Eq.(03) that maxima occur at :
d sin = n , where n = 0, 1, 2, 3, ... ---------- Eq.(06a)
while the minima are at :
d sin = (n/2) , where n = 1, 2, 3, ... ---------- Eq.(06b)

By early 20th century technology had been advanced enough to provide low intensity light and its detection. Observation then revealed that the interference pattern is produced in random specks

Figure 04 Double-Slit Experiment [view large image]

(in the detecting film). The pattern emerges only after long exposure (see Figure 04 for a modern version of the sequence of increasing number of photons). The discovery is now understood in the framework of quantum theory as explained below :