* The average expectation is then *

*direction*(+ or - ) of each step, the

*length*of each step also varied in some unpredictable way, the only condi- tion being that

*on rhe average*the step length was one unit. This case is more representative of something like the thermal motion of a molecule in a gas. If we call the length of a step S, then S may have any value at all, but most often will be "near" 1. To be specific, we shall let or, equivalently, the "root means square . Our derivation for would proceed as before except that Eq. (6.8) would be changed now to read

*D?*What is, for example, the probability that

*D*= 0 after 30 steps? The answer is zero! The probability is zero that

*D*will be

*any particular*value, since there is no chance at all that the sum of the backward steps (of varying lengths) would exactly equal the sum of forward steps.

We expect that for small ∆

*x*the chance of

*D*landing in the interval is proportional to ∆

*x,*the width of the interval. So we can write

*P(x,*∆

*x)*=

*p(x)*∆

*x*

The function

*p(x)*is called the

*probability density.*

For large

*N,*

*p(x)*is the

*same*for all reasonable distributions in individual step lengths, and depends only on

*N.*We plot

*p(x)*for three values of

*N*in Fig. 6-7. You will notice that the "half-widths" (typical spread from

*x*= 0) of these curves is

*,*as we have shown it should be.

You may notice also that the value of

*p(x)*near zero is inversely proportional to

*.*This comes about because the curves are all of a similar shape and their areas under the curves must all be equal.

The

*normal*or

*gaussian*probability density. It has the mathematical form

where σ is called the *standard deviation* and is given, in our case, by .

We remarked earlier that the motion of a molecule, or of any particle, in a gas is like a random walk. Suppose we open a bottle of an organic compound and let some of its vapor escape into the air. If there are air currents, so that the air is circulating, the currents will also carry the vapor with them. But even in *perfectly still air, *the vapor will gradually spread out-will diffuse-until it has penetrated throughout the room. We might detect it by its color or odor. The individual molecules of the organic vapor spread out in still air because of the molecular motions caused by collisions with other molecules. If we know the average "step" size, and the number of steps taken per second, we can find the probability that one, or several, molecules will be found at some distance from their starting point after any particular passage of time.

The motion of a molecule, or of any particle, in a gas is like a random walk. Suppose we open a bottle of an organic compound and let some of its vapor escape into the air. If there are air currents, so that the air is circulating, the currents will also carry the vapor with them. But even in *perfectly still air, *the vapor will gradually spread out-will diffuse-until it has penetrated throughout the room. We might detect it by its color or odor. The individual molecules of the organic vapor spread out in still air because of the molecular motions caused by collisions with other molecules. If we know the average "step" size, and the number of steps taken per second, we can find the probability that one, or several, molecules will be found at some distance from their starting point after any particular passage of time.

### The uncertainty principle

*,*such that

*is the probability that the particle will be found between*

*x*and

*x*+∆

*x*If the particle is reasonably well localized, say near

*,*the function

*might be given by the graph of Fig. 6-lO(a). Similarly, we must specify the velocity of the particle by means of a probability density*

*,*with

*the probability that the velocity will be found between*

*v*and

*v*+∆

*v*

It is one of the fundamental results of quantum mechanics that the two functions and cannot be chosen independently and, in particular, cannot both be made arbitrarily narrow. If we call the typical "width" of the curve and that of the curve ∆

*v*, nature demands that the

*product*of the two widths be at least as big as the number , where

*m*is the mass of the particle and his a fundamental physical constant called

*Planck's constant.*We may write this basic relationship as

This equation is a statement of the *Heisenberg uncertainty principle.*

This equation says that if we try to "pin down" a particle by forcing it to be at a particular place, it ends up by having a high speed. Or if we try to force it to go very slowly, or at a precise velocity, it "spreads out" so that we do not know very well just where it is.

The necessary uncertainty in our specification of the position of a particle becomes most important when we wish to describe the structure of atoms. In the hydrogen atom, which has a nucleus of one proton with one electron outside of the nucleus, the uncertainty in the position of the electron is as large as the atom itself! Wecannot,therefore,properlyspeakoftheelectronmovinginsome"orbit" around the proton. The most we can say is that there is a certain *chance p(r) *ΔV*, *of observing the electron in an element of volume *V *at the distance *r *from the proton. The probability density *p(r) *is given by quantum mechanics. For an undisturbed hydrogen atom ,* *which is a bell-shaped function like that in Fig. 6-8. The number *a *is the "typical" radius, where the function is decreasing rapidly. Since there is a small probability of finding the electron at distances from the nucleus much greater than *a, *we may think of *a *as "the radius of the atom," about 10^-10 meter.

Our best "picture" of a hydrogen atom is a nucleus surrounded by an "electron cloud" (although we *really *mean a "probability cloud"). The electron is there somewhere, but nature per- mits us to know only the *chance *of finding it at any particular place.

We can define a quantity called the wave number, symbolized as k. This is defined as the rate of change of phase with distance (radians per meter).

The wavelength is the distance occupied by one complete cycle. It is easy to see, then, that the wavelength is .

### Two dipole radiators

In combining the effects of two oscillators to find the net field at a given point. This is very easy in the few cases that we considered in the previous chapter. We shall first describe the effects qualitatively, and then more quantitatively. Let us take the simple case, where the oscillators are situated with their centers in the same horizontal plane as the detector, and the line of vibration is vertical.

We would like to know the intensity of the radiation in various directions. By the intensity we mean the amount of energy that the field carries past us per second, which is proportional to the square of the field, averaged in time. So the thing to look at, when we want to know how bright the light is, is the square of the electric field, not the electric field itself.

Suppose the oscillators are again one-half a wavelength apart, but the phase a of one is set half a period behind the other in its oscillation. In the W direction the intensity is now zero, because one oscillator is "pushing" when the other one is "pulling." But in the N direction the signal from the near one comes at a certain time, and that of the other comes half a period later. But the latter was originally half a period behind in timing, and therefore it is now exactly in time with the first one, and so the intensity in this direction is 4 units.

if we build an antenna system and want to send a radio signal, say, to Hawaii, we set the antennas up and we broadcast with our two antennas in phase, because Hawaii is to the west of us. Then we decide that tomorrow we are going to broadcast toward Alberta, Canada. Since that is north, not west, all we have to do is to reverse the phase of one of our antennas, and we can broadcast to the north. So we can build antenna systems with various arrangements.

## **Diffraction**

**Diffraction**

### The diffraction gratin

Suppose that we had a lot of parallel wires, equally spaced at a spacing d, and a radiofrequency source very far away, practically at infinity, which is generat- ing an electric field which arrives at each one of the wires at the same phase.

Then the external electric field will drive the electrons up and down in each wire. That is, the field which is coming from the original source will shake the electroiJ.s up and down, and in moving, these represent new generators. This phenomenon is called scattering: a light wave from some source can induce a motion of the electrons in a piece of material, and these motions generate their own waves.

A diffraction grating consists of nothing but a plane glass sheet, transparent and colorless, with scratches on it. There are often several hundred scratches to the millimeter, very carefully arranged so as to be equally spaced. The effect of such a grating can be seen by arranging a projector so as to throw a narrow, vertical line of light (the image of a slit) onto a screen. When we put the grating into the beam, with its scratches vertical, we see that the line is still there but, in addition, on each side we have another strong patch of light which is colored. This, of course, is the slit image spread out over a wide angular range, because the angle Bin (30.6) depends upon X, and lights of different colors, as we know, correspond to different frequencies, and therefore different wavelengths. The longest visible wavelength is red, and since d sin B = X, that requires a larger B. And we do, in fact, find that red is at a greater angle out from the central image!

We begin to understand the basic machinery of reflection: the light that comes in generates motions of the atoms in the reflector, and the reflector then regenerates a new wave, and one of the solutions for the direction of scattering, the only solution if the spacing of the scatterers is small compared with one wavelength, is that the angle at which the light comes out is equal to the angle at which it comes in!

### Resolving power of a gratin

Supposing that there were two sources of slightly different frequency, or slightly different wavelength, how close together in wavelength could they be such that the grating would be unable to tell that there were really two different wavelengths there?

Rayleigh's criterion: Two frequency are resolved when the first minimum from one bump sit at the maximum of the other.

The ratio is called the resolving power of a grating;

The parabolic anthena

Now suppose that the radio source is at a slight angle 0 from the vertical. Then the various antennas are receiving signals a little out of phase. The receiver adds all these out-of-phase signals together, and so we get nothing, if the angle is too big.

The smallest angle that can be resolved by an antenna array of length L is .

Colored films crystals

if we look at the reflection of a light source in a thin film, we see the sum of two waves; if the thicknesses are small enough, these two waves will produce an inter- ference, either constructive or destructive, depending on the signs of the phases. It might be, for instance, that for red light, we get an enhanced reflection, but for blue light, which has a different wavelength, perhaps we get a destructively inter- fering reflection, so that we see a bright red reflection. If we change the thickness, i.e., if we look at another place where the film is thicker, it may be reversed, the red interfering and the blue not, so it is bright blue, or green, or yellow, or whatnot. So we see colors when we look at thin films and the colors change if we look at different angles, because we can appreciate that the timings are different at different angles.

We used a grating and we saw the diffracted image on the screen. If we had used monochromatic light, it would have been at a certain specific place. Then there were various higher-order images also. From the positions of the images, we could tell how far apart the lines on the grating were, if we knew the wavelength of the light.

This principle is used to dis· cover the positions of the atoms in a crystal. The only complication is that a crystal is three-dimensional; it is a repeating three-dimensional array of atoms. We cannot use ordinary light, because we must use something whose wavelength is less than the space between the atoms or we get no effect; so we must use radiation of very short wavelength, i.e., x-rays. So, by shining x-rays into a crystal and by noticing how intense is the reflection in the various orders, we can determine the arrangement of the atoms inside without ever being able to see them with the eye!

**The origin of the refractive index**

**The origin of the refractive index**

### The index of diffraction

It is approximately true that light or any electrical wave does appear to travel at the speed cjn through a material whose index of refraction is n, but the fields are still produced by the motions of all the charges-including the charges moving in the material-and with these basic contributions of the field travelling at the ultimate velocity c.

We shall try to understand the effect in a very simple case. A source which we shall call "the external source" is placed a large distance away from a thin plate of transparent material, say glass. We inquire about the field at a large distance on the opposite side of the plate.

According to the principles we have stated earlier, an electric field anywhere that is far from all moving charges is the (vector) sum of the fields produced by the external source (at S) and the fields produced by each of the charges in the plate of glass, every one with its proper retardation at the velocity c.

When the electric field ofthe source acts on these atoms it drives the electrons up and down, because it exerts a force on the electrons. And moving electrons generate a field-they constitute new radiators. These new radiators are related to the source S, because they are driven by the field of the source. The total field is not just the field of the source S, but it is modified by the additional contribution from the other moving charges.

Before we proceed with our study of how the index of refraction comes about, we should understand that all that is required to understand refraction is to under- stand why the apparent wave velocity is different in different materials. The bending of light rays comes about just because the effective speed of the waves is different in the materials. To remind you how that comes about we have drawn in the left figure several successive crests of an electric wave which arrives from a vacuum onto the surface of a block of glass. The arrow perpendicular to the wave crests indicates the direction of travel of the wave. Now all oscillations in the wave must have the same frequency. (We have seen that driven oscillations have the same frequency as the driving source.) This means, also, that the wave crests for the waves on both sides of the surface must have the same spacing along !he surface because they must travel together, so that a charge sitting at the boundary will feel only one frequency. The shortest distance between crests of the wave, however, is the wavelength which is the velocity divided by the frequency. On the vacuum side it is and on the other side it is or , if is the velocity of the wave. From the figure we can see that the only way for the waves to “fit” properly at the boundary is for the waves in the material to be travelling at a different angle with respect to the surface. From the geometry of the figure you can see that for a “fit” we must have or , which is Snell’s law.

The index of refraction is given by

with the number of atoms per unit volume of the plate, , the charge and mass of an electron, w the angular frequency of the radiation, the resonant frequency of an electron bound in the atom.

### Dispersion

For most ordinary gases (for instance, for air, most colorless gases, hydrogen, helium, and so on) the natural frequencies of the electron oscillators correspond to ultraviolet light. These frequencies are higher than the frequencies of visible light, that is, w0 is much larger than w of visible light, and to a first approximation, we can disregard w2 in comparison with Then we find that the index is nearly constant. So for a gas, the index is nearly constant. This is also true for most other transparent substances, like glass. If we look at our expression a little more closely, however, we notice that as w rises, taking a little bit more away from the denominator, the index also rises. So n rises slowly with frequency. The index is higher for blue light than for red light. That is the reason why a prism bends the light more in the blue than in the red.

The phenomenon that the index depends upon the frequency is called the phenomenon of dispersion, because it is the basis of the fact that light is "dispersed" by a prism into a spectrum.

At frequencies very close to the natural frequency the index can get enor- mously large, because the denominator can go to zero.

If we beam x-rays on matter, or radiowaves (or any electric waves) on free electrons the term becomes negative, and we obtain the result that n is less than one. That means that the effective speed of the waves in the substance is faster than c! Can that be correct?

It is correct. In spite of the fact that it is said that you cannot send signals any faster than the speed of light, it is nevertheless true that the index of refraction of materials at a particular frequency can be either greater or less than I.

What the index tell us is the speed at which the nodes (or crests) of the wave travel. The node of a wave is not a signal by itself. In a perfect wave, which has no modulations of any kind, i.e., which is a steady oscillation, you cannot really say when it "starts," so you cannot use it for a timing signal. In order to send a signal you have to change the wave somehow, make a notch in it, make it a little bit fatter or thinner.

We should remark that our analysis of the refractive index gives a result that is somewhat simpler than you would actually find in nature. To be completely accurate we must add some refinements. First, we should expect that our model of the atomic oscillator should have some damping force (otherwise once started it would oscillate forever, and we do not expect that to happen). In presence of a damping with coefficient we replace in the denominator of the refractive index formula by .

### Absortion

As the wave goes through the material, it is weakened. The material is "absorbing" part of the wave. The wave comes out the other side with less energy. We should not be surprised at this, because the damping we put in for the oscillators is indeed a friction force and must be expected to cause a loss of energy. We see that the imaginary part of a complex index of refraction n represents an absorption (or "attenuation") of the wave.

For instance as in glass, the absorption of light is very small. This is to be expected from our Eq. (31.20), because the imaginary part of the denominator, , is much smaller than the term . But if the light fre- quency w is very close to then the index becomes almost completely imaginary. The absorption of the light becomes the dominant effect. It is just this effect that gives the dark lines in the spectrum of light which we receive from the sun. The light from the solar surface has passed through the sun's atmosphere (as well as the earth's), and the light has been strongly absorbed at the resonant frequencies of the atoms in the solar atmosphere.

The observation of such spectral lines in the sunlight allows us to tell the resonant frequencies of the atoms and hence the chemical composition of the sun's atmosphere. The same kind of observations tell us about the materials in the stars. From such measurements we know that the chemical elements in the sun and in the stars are the same as those we find on the earth.

**Polarization **

### The electric vector of light

In ideally monochromatic light, the electric field must oscillate at a definite frequency, but since the x-component and the yy-component can oscillate independently at a definite frequency, we must first consider the resultant effect produced by superposing two independent oscillations at right angles to each other. When the x-vibration and the y-vibration are not in phase, the electric field vector moves around in an ellipse. The motion in a straight line is a particular case corresponding to a phase difference of zero (or an integral multiple of ); motion in a circle corresponds to equal amplitudes with a phase difference of 90 degrees (or any odd integral multiple of ).

Light is *linearly polarized* (sometimes called plane polarized) when the electric field oscillates on a straight line.

**Relativistic effects of radiation **

**Relativistic effects of radiation**

### Moving sources

We recall that the fundamental laws of electrodynamics say that, at large distances from a moving charge, the electric field is given by the formula

The second derivative of the unit vector which points in the apparent direction of the charge, is the determining feature of the electric field.

Associated with the electric field is a magnetic field, always at right angles to the electric field and at right angles to the apparent direction of the source, given by the formula

Let the coordinates of the charge be (x,y,z) with z measured along the direction of observation. Now the direction of the vector depends mainly on x and y, but hardly at all upon z. The transverse components of the unit vector are x/R and y/R with R the distance from the source. One finds

If the time of observation is called t then the time τ to which this corresponds is delayed by the total distance that the light has to go, divided by the speed of light. In the first approximation, this delay is R/c but in the next approximation we must include the effects of the position in the z-direction at the time τ. Thus τ is determined by

In the synchrotron we have electrons which go around in circles in a uniform magnetic field.; they are travelling at very nearly the speed c, and it is possible to see the above radiation as actual *light*! First, let us see why they go in circles. We know that the force on a particle in a magnetic field is given by

and it is at right angles both to the field and to the velocity. As usual, the force is equal to the rate of change of momentum with time. Since the force is at right angles to the velocity, the kinetic energy, and therefore the speed, remains *constant*. All the magnetic field does is to change the *direction of motion*. In a short time , the momentum vector changes at right angles to itself by an amount and therefore p turns through an angle . But in this same time the particle has gone a distance Δs=vΔt =RΔθ. Combining this with the previous expressions, we find that the particle must be moving in a *circle* of radius R, with momentum

If q is expressed in terms of the electronic charge the kinetic energy pc can be measured in units of the *electron volt *

The mks unit of magnetic field is called a *weber per square meter*. Today, electromagnets wound with superconducting wire are able to produce steady fields of over 10 mks units. The field of the earth is weber per meter square at the equator.

We could imagine the synchrotron running at a billion electron volts, then, if we had a B corresponding to, say, 1 kms then we see that R would have to be 3.3 meters. The actual radius of the Caltech synchrotron is 3.7 meters, the field is a little bigger, and the energy is 1.5 billion, but it is the same idea.

We know that the total energy, including the rest energy, is given by and for an electron the rest energy corresponding to , so when we can neglect the rest energy. If , it is easy to show that the speed differs from the speed of light by but one part in eight million!

We turn now to the radiation emitted by such a particle. A particle moving on a circle of radius 3.3 meters, or 20 meters circumference, goes around once in roughly the time it takes light to go 20 meters. So the wavelength that should be emitted by such a particle would be 20 meters—in the shortwave radio region. The effective wavelength is instead much shorter since the time scale is reduced by eight million and the acceleration, which involves a second derivative with respect to time results into the square of that factor, i.e. times smaller than 20 meters, and that corresponds to the x-ray region. Thus, even though a slowly moving electron would have radiated 20-meter radiowaves, the relativistic effect cuts down the wavelength so much that we can *see* it!

To further appreciate what we would observe, suppose that we were to take such light (to simplify things, because these pulses are so far apart in time, we shall just take one pulse) and direct it onto a diffraction grating, which is a lot of scattering wires. The pulse strikes the grating head-on, and all the oscillators in the grating, together, are violently moved up and then back down again, just once. They then produce effects in various directions, The sum of the reflections from all the successive wires is an electric field which is a series of pulses, and it is very like a sine wave whose wavelength is the distance between the pulses, just as it would be for monochromatic light striking the grating! So, we get colored light all right. But, by the same argument, will we not get light from any kind of a “pulse”? No. Suppose that the curve were much smoother; then we would add all the scattered waves together, separated by a small time between them. Then we see that the field would not shake at all, it would be a very smooth curve, because each pulse does not vary much in the time interval between pulses.