Understanding interference in classical physics

Understanding interference in classical physics 

The theory of diffraction is the foundation for study of the double-slit interference experiment. When first performed in 1800 by Thomas Young (1773-1829), this experiment was considered definitive proof of the wave nature of light. We can modify the single-slit apparatus of Fig. 2.1 to suit Young's experiment by simply cutting a second slit in the diaphragm.

As shown in Fig. 2.3, the width of the second slit is the same as that of the first, w. We must position the second slit so that the two slits are close together-but not too close: for observable interference to occur, the slit separation s must be greater than w. Again, we shine a beam of monochromatic light of frequency v on the diaphragm and see what happens at the detector, which is far to the right of the diaphragm. Figure 2.3

This time, our detector shows an interference pattern like the one in Fig. 2.4. At first glance, this figure may look similar to the diffraction pattern of Fig. 2.2, but on closer examination we see striking differences. For one thing, the interference pattern exhibits more bright bands than the diffraction pattern. This means that the energy of the radiation scattered by the double-slit diaphragm is more evenly distributed than that scattered by a single slit. (In the diffraction pattern about 90% of the energy appears in the central band.) Finally, the individual bands in the interference pattern, which are called interference fringes, are narrower than those of the diffraction pattern.

If we study how the interference pattern changes as we fiddle with the incident wavelength ., the slit separation s, and the slit width w, we discover that the separation between the bright bands increases with increasing just as it did in the diffraction pattern, for which the separation is proportional to  /w. But also we find a difference: the separation in Fig. 2.4 is independent of the slit width w but is inversely proportional to the slit separation s.

The key to understanding interference is superposition. When the incident plane wave encounters the double-slit diaphragm it "splits," and a diffracted wave emerges from each slit. (In a sense, each slit becomes a source of radiation that travels to the detector.) These waves add in the region between the slit and the detector, and this device measures the intensity of their superposition . Now, at the detector the amplitudes of the electric fields of the diffracted waves are equal, but their phases are not. Consequently, their superposition manifests regions of constructive and destructive interference, as seen in Fig. 2.4.

The trick to deriving an equation for the intensity measured in the double-slit experiment-which we'll call is therefore to add the electric fields of the waves diffracted by each slit. (This step is legitimate because Maxwell's equation for these fields is linear.) Having done so, we could calculate the aforementioned phase difference and would find the field from the lower slit lags the field from the upper slit by an amount . The last step is to average the squared modulus of the total electric field at a point  on the detector over one period, which yields 

The source:

Michael A. Morrison - Understanding Quantum Physics.

By. Fady Tarek

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Understanding diffraction in classical physics

Understanding diffraction in classical physics

Diffraction, which was first observed by Leonardo da Vinci, is often considered to be the signature of a wave. Diffraction occurs when ripples in a pond encounter a pair of logs that are close together, when light passes through a narrow slit in a window shade, or when x-rays scatter from a crystal. In each case, we can explain the distinctive pattern that forms using classical wave theory. Understand what happens to a wave when it passes through a small aperture-that is, understand diffraction-and you are well on your way to understanding the behavior of waves in more complicated situation.

A schematic of a single-slit diffraction experiment with light is shown in Fig. 2.1. Monochromatic light of frequency v is incident on a diaphragm in which there is a single slit of width w.

 We'll assume that the light source is far to the left of the diaphragm, so the incident radiation can be represented by a plane wave. The width of the slit must be small compared to the wavelength  = c/v of the radiation if the slit is to appreciably diffract the light; for example, to diffract visible light enough to observe this phenomenon, we require a slit of width w=10^-4 cm. Light scattered by the diaphragm falls on a detector, such as a photographic plate or a photocell, located at a distance D far to the right of the slit. The detector measures the energy delivered by the scattered wave as a function of the distance x in Fig. 2.1.

Light scattered by the single-slit diaphragm forms a beautiful diffraction pattern at the detector. This pattern is characterized by a very bright central band located directly opposite to the slit, surrounded by a series of alternating light and dark regions. The light regions on either side of the central band are called secondary hands, because they are much less intense than the central band. Indeed, the intensity of the secondary bands drops off so dramatically on either side of the central band that only one pair of secondary bands is visible. Additional weak secondary bands exist, though, as you can see in Fig. 2.2, which is a graph of the intensity measured by the detector. If we play around with the frequency control on the light source and study the resulting diffraction patterns, we discover that the separation between adjacent bright bands is proportional to the wavelength of the incident radiation.

A classical physicist would call upon Maxwell's electromagnetic wave theory to explain the pattern in Fig. 2.2. To understand qualitatively what happens when a plane wave passes through a diaphragm we invoke Huygens's Principle, which lets us replace the plane wave and the slit (in our mind's eye) by a large number of closely-spaced , discrete radiating charges that fill the region of space where the slit is located."Scattered waves" radiated by different oscillators-i.e., waves that emerge from different locations in the slit region-have different phases, so the superposition of these scattered waves exhibits regions of high and low intensity, as in Fig. 2.2.

to derive an expression for the electric field at a point (r, ) on the detector (see Fig. 2.1). But the quantity measured in this experiment is not the electric field; it is the intensity the rate at which the scattered radiation delivers energy to the detector. This quantity is proportional to the time-averaged energy flux-i.e., to the average over one period of the square of the modulus of the electric field. I'll denote by  the intensity at a fixed value of r due to radiation scattered by a single slit.

Omitting the details of the derivation of , I'll just quote the result. For convenience, I'll write the intensity in terms of its value  at the principal maximum i.e, at the point r=D, ( = 0, in the central peak in Fig. 2.2-and the handy intermediate quantity

the single-slit intensity at fixed r is

The intensity (2.4) is graphed in Fig. 2.2 as a function of w sin(); plotted this way, exhibits a characteristic pattern of equally-spaced nodes [where  = 0], which occur at

The principal (zeroth-order) maximum of the intensity pattern and higher order maximum occur (approximately) at

Equation (2.4) fully accounts for the properties of patterns such as the one in Fig. 2.2. Thus does classical electromagnetic theory rend the veil of mystery from the phenomenon of diffraction.

The source:

Michael A. Morrison - Understanding Quantum Physics.

By. Fady Tarek

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Basic theorems of limits

Basic theorems of limits

1-The limit of the algebraic sum of a finite number of variables is equal to the sum of their limits.

For the sake of exactness let us take the algebraic sum x-y+z of three simultaneously varying magnitudes. We suppose that x, y and z tend respectively to limits a, b and c. We show that the sum tends to the limit a - b + c.

We have:

where  are infinitesimals. We can write for the sum:

The first bracket on the right-hand side of this equation is a constant, and the second is an infinitesimal. Hence:

2-The limit of the product of a finite number of variables is equal to the product of their limits.

We confine ourselves to the case of the product x*y of two variables. We suppose that x and y vary simultaneously, tending respectively to limits a and b, and we show that x*y tends to the limit a*b.

We have by hypothesis:

where  are infinitesimals; hence:

Using both of the properties of infinitesimals, we see that the sum in the bracket on the right of this equation is an infinitesimal, and hence we have:

3-The limit of a quotient is equal to the quotient of the limits, provided the limit of the denominator is not zero.

We take the quotient x/y, and suppose that x and y tend simultaneously to their respective limits a and b, where  We show that x/y tends to a/b.

To prove the theorem, it is sufficient to show that the difference a/b - x/y is an infinitesimal. By hypothesis:

where  are infinitesimals. Hence:

The denominator of the fraction on the right of this equation is the product of two factors, and tends to b^2. Thus, from some initial moment of its variation, it is greater than b^2/2, the fraction as a whole being included between zero and 2/b^2, i.e. the fraction is bounded. The term  is an infinitesimal. Hence, the difference a/b-x/y is an infinitesimal, and

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek

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The limit of a variable

The limit of a variable

We have called a variable an infinitesimal, if its corresponding point K in the axis X has on displacement the following property: on successive variation the length of the interval OK becomes, and on further variation remains, less than any given positive number . We now suppose that this property is fulfilled, not by the interval OK, but by AK, where A is a definite point on the axis X with abscissa a (Fig. 42).

In this case, the interval S'S of length 2 will have its center at the point A, abscissa x = a, instead of at the origin, and the point K must come within this interval on successive displacement, then remain there on further displacement. We say in this case that the constant a is the limit of variable x, or that x tends to a.

Noting that the length of AK is | a-x |, we can formulate the following definition:

DEFINITION. The constant a is called the limit of the variable x when the difference a-x (or x-a) is an infinitesimal, Having regard to the definition of an infinitesimal, a limit can be thus defined:

DEFINITION. The constant a is called the limit of the variable x, when we have the following property : for any given positive  there exists a value of x such that, for all subsequent values, |a-x| < .

We note some immediately obvious consequences of this definition, without dwelling on their detailed proof.

No variable can tend to two different limits, and not every variable has a limit. for example, the variable sin(a) oscillates between -1 and 1 on successive increase of the angle a, and has no limit. The limit of an infinitesimal is zero.

if x and y vary simultaneously, and each tends to a limit in the course of successive variation, whilst both always satisfy x <= y, their limits a and b satisfy the condition a <= b. We note here, that if the variables satisfy x < y, the sign of equality can be obtained for their limits, i.e. we have a <= b. If x, y, z vary simultaneously and always satisfy the condition x <= y <= z on successive variation, and if x and z tend to the same limit a, y also tends to the limit a.

If a is the limit of x (or x tends to a), we write:

If x tends to a, the difference x-a =  is an infinitesimal, and we can write:

i.e. every variable tending to a limit can be expressed as the sum of two terms: a constant term, equal to the limit, and an infinitesimal. Conversely, if a variable x can be expressed in the form (2), where a is a constant, and  is an infinitesimal, the difference x-a will be an infinitesimal, and hence, a is the limit of x.

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek

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wave-Particle duality(part-3) photons

wave-Particle duality(part-1)

wave-Particle duality(part-2) planck's work

wave-Particle duality(part-3) photons

Complete planck's work

For his second assault on the energy density of a black body, Planck adopted a statistical method based on the concept of entropy as interpreted probabilistically by Ludvig Boltzmann (1844-1906). At one point in his derivation, Planck introduced a simplifying assumption (i .e., a trick to facilitate his mathematics). His ploy was conventional in mathematical analysis , and Planck expected to remove his assumption at the end. Planck was in for a rude shock.

assumption is both simple and radical. He knew that a classical black body could exchange any amount of energy with the walls of the cavity; nevertheless, for purposes of his derivation, Planck assumed that only discrete amounts of energy can be absorbed or emitted by the resonators that comprise the walls of the black body. He called these discrete amounts of energy "quanta." To each quantum Planck assigned an energy equal to an integral multiple of h*v, where h is the constant h =6.63 X 10^-34 J-sec. Having made this assumption, Planck easily derived the radiation law

where c = 3.0 X 10^8 m-sec is the speed of light and  is Boltzmann's constant. Comparing this result with Eq. (2.1) in wave-Particle duality(part-2) planck's work, we see that in his new equation Planck derived expressions for the constants A and B that appeared in his empirical form.

This derivation was the theoretical justification Planck sought for the distribution of radiation in a black body. But in laying the foundation for his theory, Planck paid an enormous price, for try as he might, he could not get rid of his artificial constant h. Setting h = 0 inevitably led to a result that disagreed with a huge body of experimental data. Yet, if h is non-zero, then Planck's theory is seriously at odds with physicists understanding of energy exchange as a continuous process. Planck's rash assumption heralded the strange new physics of the quantum, which dominated the physics community for the next three decades. But physicists, who were then and are today a rather conservative lot, did not take well to being told that their understanding of so basic a process as energy exchange was fundamentally incorrect. Attempts to derive Planck's result without making his drastic assumption failed, and for several years Planck's quanta languished in obscurity, largely ignored. Yet, Planck's assumption was nothing compared to the surprise Einstein had in store.

photons-Particles of Light

Einstein thought he saw an inconsistency in the way Planck used Maxwell's wave theory of electromagnetic radiation in his derivation. With characteristic intellectual courage, Einstein decided that this inconsistency implied a flaw not in Planck's theory but in Maxwell's. This radical contention shifted the emphasis in research on black bodies from the resonators that comprise the walls to the radiation field itself. Ultimately, it completely altered the way we think about light.

In 1905 Einstein proposed that the energy in an electromagnetic field is not spread out over a spherical wave front, as Maxwell would have it, but instead is localized in indivisible clumps-in quanta. Each quantum of frequency v, Einstein averred, travels through space at the speed of light,    C = 3.0 X 10^8 m/sec, carrying a discrete amount of energy h*v and momentum h*v/c. Thus, in Einstein's model, light transports energy in the same way particles do. G. N. Lewis subsequently dubbed Einstein's quantum of radiation energy a photon, the name we use today. The photon model cast Planck's theory of black-body radiation in, so to speak, a new light. Planck thought that energy exchange between the resonators of the cavity and the field occurs in units of h*v because of some strange property of the resonators. To this Einstein said: No, the explanation is that the radiation field itself is quantized. Planck's result is consistent with this extraordinary notion; if the energy in the field is contained in photons-quanta of magnitude h*v-then of course only integral multiples of the photon energy can be exchanged with the walls.

The photon model explained more than just black-body radiation. One of Einstein's greatest achievements was using it to understand the photoelectric effect-the ejection of electrons from a metal, such as sodium, when light impinges on it. (You see the photoelectric effect in action every time an electric eye opens an elevator door for you or closes one on your foot.) Yet, the photon was too radical for physicists of the early 1900's, and Einstein's model of the electromagnetic field encountered strong opposition. Even as late as 1913-years after the publication of Einstein's work on the photoelectric effect-four distinguished German physicists (including Max Planck) wrote in a petition recommending Einstein's appointment to the Prussian Academy of Science:

... That he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta, cannot really be held too much against him, for it is not possible to introduce fundamentally new ideas, even in the most exact sciences, without occasionally taking a risk.

One year later the American experimentalist R. A. Millikan (1868-1953) reported a precise verification of Einstein's equation for the energy of a photon, E = h*v, and the first measurement of Planck's constant. Yet physicists still resisted abandoning their long-cherished idea that light was a wave. Then, in 1923, Arthur H. Compton (1892-1962) published the results of his x-ray scattering experiments, and drove the last nail into the coffin of the wave theory of light. Compton scattered x-rays-electromagnetic radiation with a wavelength around 10^-10 m and a frequency around 10^18 1/sec- from a thin layer of a light element such as carbon and measured the shift in the wavelength of the x-rays due to the scattering. His results were seriously at odds with the predictions of Maxwell's beleaguered theory. Compton's data clearly showed that the wavelength of the scattered radiation is larger than that of the incident radiation. After several foredoomed attempts to explain this result with classical theory, Compton tried the photon idea. If x-rays carry energy in localized clumps, he reasoned, then we can apply classical collision theory to their interaction with the electrons of the target. Compton used the classical laws of conservation of energy and linear momentum-as though he were analyzing a game of billiards-and was able to derive the correct expression for the wavelength shift. This analysis vindicated Einstein's idea. It was too much even for diehards: the photon was accepted.

The source:

Michael A. Morrison - Understanding Quantum Physics.

By. Fady Tarek

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