Links to specific sections in the text:
2.a. Electromagnetic waves
2.b. Blackbody radiation
2.c. Planck's law
2.d. Photoelectric effect and photons
2.e. Compton effect
Maxwell predicted the existence of electromagnetic waves based on his equations. He found that his equations have solutions that corraspond to an electromagnetic field with no matter present. Since he beleived that his equation described nature correctly, every solution of his equation should correspond to an existing physical phenomenon. Electromagnetic waves propagate in vacuum, such that the electric and magnetic fields, which are perpendicular to each other, are also perpendiculaer to the direction of propagation of the waves.
The observation of electromagnetic waves was waiting for the brilliant German experimental phsysicist, Heinrich Hertz. If on applies a high voltage to a pair of metal spheres with an air gap between them then sparks will jump over the air gap between the two metal spheres. Hertz showed that as a byproduct of these sparks electromagnetic radiation is produced every time when he switched the voltage off or on. The radiation could be detected by a wire loop with another gap, through which sparks jumped through when the loop was exposed to the electromagnetic field generated by sparks between the metal spheres.
Hertz was able to prove that the electromagnetic waves he detected had exactly the same properties as light: They could be reflected, refracted, focussed, polarized, diffracted, and in general made to interfere.
Thomas Wedgwood (founder of the still existing British porcellain factory), who had to work with kilns and look at objects heated to very high temperature, discovered in 1792 that light emitted by heated bodies has a certain universal character. Color seemed to depend only on the temperature of the heated object and not on the material, quality of surface, etc. Kirchhoff refined and quantified Wedgwood's observation. Using thermodynamics, he derived a relationship between the emitted and absorbed power of a surface in thermal equilibrium. Thermal equilibrium means that the system is in a state when the amount of emitted and absorbed power are not depending on time but reach a steady state. His law describes the frequency and temperature dependence of emitted radiation as
ef = J(f,T) af,
where ef is the power emitted by a unit surface of an object in unit frequency range, i.e. efDf is the power emitted by unit surface in the frequency range Dfor alternatively the total energy emitted (at all frequencies) by unit surface is
etotal = Ú ef df = ÚJ(f,T) af df.
As Wedgwood predicted, J(f,T) is an universal function of the temperature, but the intensity of emission also depends on the absorbtion coefficient, af, which is the fraction of the incident power absorbed in unit frequency range and unit surface. It is quite easy to understand that in thermal equilibrium, the more radiation is absorbed, the more is emitted. A black body absorbes all radiation incident to it (nothing is reflected). Mathematically, a black body is defined to have af=1.
Simple physical intuition tells us a lot about the frequency (f) and temperature dependence of universal function J(f,T). First of all, knowing that the total emission is finite, the function must drop off at large frequencies, i.e. J(f,T)Æ0 fairly fast, if fÆ0. Furthermore, a common observation, known since antient times that the hotter an object is the more its color is shifted from red towards violet, i.e. there is a frequency, fmax at which the readiation is maximal. fmax increases with temperature (hence the percieved color change). In fact it has become clear very soon that this frequency increases linearly with temperature.
The next important observation, by Stefan, Austrian physicist, came in 1879. He realized that the dependence of etotal on temperature is very simple
etotal = a s T4.
which is called Stefan's law now. a is an average absorbtion coefficient, i.e. a=1 for black bodies. The value of Stefan's coefficient is s = 5.67¥10-8 Wm-2K-4.
Using Stefan's law we are able to figure out the surface temperature of the sun, using measurements performed on the surface of earth. There is no need to travel to the sun with a thermometer. Strange it may sound, the sun can be regarded as a black body. Remember that the black body is not really defined by color. Its color changes with temperature. The definition hinges on the fact whether the object absorbs all radiation falling on it or not. The sun does, it does not reflect any of it.
The trick to find the surface temperature of the sun is to scale the amount of radiation falling on unit area of the surface of earth to that emitted at the surface of the sun. The basic assumption is that energy is conserved while it travels from the sun outwards and the emission from the sun is isotropic. What is emitted by the sun in our direction will arrive to the surface of earth. Suppose the energy emitted from unit area on the surface of the sun is etotal. Then the total energy emitted by the sun is
E = Ssun etotal = 4 p Rsun2 etotal = 4 p Rsun2 s T4.
Suppose now that the radiation/unit area observed at the surface of earth is eearth. Then the total energy reaching the surface of a sphere the center of which is the sun and intersects the earth is the same,
E = 4 p Dsun-earth2 eearth = 4 p Rsun2 s T4.
Now it is easy to express the temperature of the sun from the above equation
T = [ Dsun-earth2 eearth / Rsun2 s ]1/4 = 5800K.
Research on radiation was soon concentrated to black bodies, because it was understood that the idealized condition af=1 simplified the problem, leaving all the important physics behind radiation intact. Even more important was the change of emphasis during that time, under the influence of Maxwell, from the radiating surface to cavities, that are inside black bodies, in thermal equilibrium with the surface of the cavity. It is easy to understand that the radiation energy density inside the cavity, u(f,T), must be proportional to J(f,T). We define u(f,T) as the total energy of electromagnetic radiation per unit frequency range and unit volume, inside the cavity. Assume for a moment that radiation falls only perpendiculary on the surface. Then the amount of electromagnetic radiation radiation, traveling at velocity c, that will hit unit surface per unit time (the power of absortion, J(f,T)) is equal to the total energy contained in a column of radiation of length c and unit cross section, traveling toward the surface. The volume of that column is c m3, so the total energy per unit frequency range contained in the column is J(f,T)=u(f,T)c. In fact, the exact relation, owing to radiation traveling in other directions (not perpendicularly towards the surface) and complications due to the polarizability of light the exact relation is
J(f,T) = u(f,T) c / 4
A large amount of experimental effort was spent toward the measurement of u(f,T). Using theoretical arguments Lord Rayleigh and Jeans derived an expression from the Maxwell equations which had the form
u(f,T) = Eav 8 p f2 / c3
where Eav, the average energy of oscillators, that emit radiation, in the cavity wall. According to Boltzmann, this average energy is just
Eav = kB T,
where kB is Boltzmann's constant, constant of nature. Now every one new that there was a serious flaw with the Rayleigh-Jeans law. The total energy of radiation in the cavity should be obtained by integrating over frequencies, which should result in Stefan's law. Instead of that integration over the the Rayleigh-Jeans density,
Ú u(f,T) df,
leads to an infinity. This disasterous result is called the ultraviolet catastrophe. Everyone new that at high frequencies this law is got to be modified, but neither the physical reason, nor the mathematical form existed for several years.
Many experiments were performed by heating cavities with black walls and letting radiation leek out through a hole and analysing the radiation. Since reflection from various surfaces is frequency dependent, radiation at various frequencies could be separated by multiple reflectios and analysed at various frequencies independently. The results, obtained at visible frequencies, by Wien, were disasterous for the Rayleigh-Jeans law. Wien wrote down his empirical law in 1893
u(f,T) = A f3 e-bf/T.
This law did not agree with the Rayleigh-Jeans law neither at low nor at high frequencies. This was verified in several experiments until, doing experiments in the far infrared range (small f), a deviation was observed and it was found that the Rayleigh Jeans law indeed works at the smallest of the frequencies. The great beauty of Wien's law was that it explained the relation between the frequency of most intense radiation and Stefan's law as well. Take first the frequency at which radiation is most intense, fmax. The derivative of Wiens energy densiy function with respect to the frequency should vanish at that point. Differentiate Wiens law to find for such a maximum
du(f,T) / df = 0 = A ( 3f2- f3b/T )e-bf/T.
This equation for f can be solved if we define z = bf/T. Then after multiplying by b2/T2 we can write the extremum equation for z as
(3 z2 - z3 ) e-z = 0,
the solution of which is z=3, or equivalently we can write fmax= 3T/b. In other words, the frequency at the maximum is indeed proportional to T. The empirical constant in Wien's law can be written as b=3k/c Then, using l = c/f, where l is the wavelength, we can write
lmax T = k = 2.9 ¥ 10-3 m K.
The temperature of the sun can also be figured out from this relation. The wavelength corrsponding to the maximum intensity of light (on the top of the atmosphere) is around 500nm = 5¥10-7 m. Substituting this value we obtain the same result for the temperature of the sun as before, 5800K.
Stefan's law can also be obtained from Wien's law. Just integrate Wien's density function over the complete frequency range from zero to infinity. We have then
etotal = Ú u(f,T) df = A Ú f3 e-bf/T df .
Now we can substitute z, just like above to get
etotal = A T4 / b4 Ú z3 e-z dz = 6A T4 / b4.
Thus, we obtain Stefan's law with the identification
s = 6A / b4.
Summarizing this section, at the turn of the twentieth century there were three, related, basic problems which waited for a resolution:
Planck set out to solve the discrepancies described in the previous section. First of all he set out to solve the problem of ultraviolet catastrophy. He, just like Rayleigh and Jeans before him, started out from statistical mechanics. The basic postulate of statistical mechanics that the probability that the system is in a given state depends only on the energy of that state and the temperature and has the form
P(e) =A e-e/kBT,
where e is the energy of the state, T is the absolute temperature and kB is a constant of nature, called Boltzmann's constant. A is an overall constant, so called normalization constant that is determined from the condition that probabilities are defined in a way that total probability of all alternate events is 1. In other words,
A = 1 / (¬ e-e/kBT ),
where the summation is performed over the possible energies the system can have.
A crash course in probability: Assume that the probability that a variable x, takes a cerain value P(x). P(x) is called the probability distribution. An example: suppose that 5 people in the class are 19 years old, 6 people 20 years old and 2 are 21 years old. Then if I denote the age of a person by x and the probability that the person is x years old is P(x), then in our case P(19)= A 5,, P(20)=A 6, P(21)=A 2, where A is a normalization constant, A=1/(5+6+2)=1/13. Then the probabilities are: P(19)= 5/13,, P(20)=6/13, P(21)=2/13. The sum of these probabilities is indeed one. Compare these with the probabilityof a state with energy e, you can see that the same principle is used. How would you find out the average age of the class? Well, the most primitive way would be to add all the ages and divide it with the total number of 13. That would be
xav=(5¥19+6¥20 +2¥21) / 13=257 / 13 = 19.77.
Aternatively, I could divide out every term by 13, to get
xav=5/13¥19+6/13¥20 +2/13¥21.=19 P(19)+ 20 P(20)+21 P(21)= ¬ x P(x)
It is easy to see that this is the general method of finding the average of a variable. E.g. if we wish to determine the average of the energy in a Boltzmann distribution then we should calculate the sum
eav = ¬ e P(e) = ¬ e e-e/kBT / ¬ e-e/kBT
What happens if the variable x or for that matter e is continuous, i.e. instead the age we used finding the average age of the class we use the exact age, e.g. someone is 19.65... years old. Also suppose that Clearly then summation over x is out, one rather uses integration over all possible ages. The probability distribution then should have the form P(x) dx is the probability that the variable x is between x and x+dx. We normalize P(x) as ÚP(x)dx=1. If this integral is not one then we can always normalize P(x) dx by dividing with ÚP(x)dx. In other words, the distribution
p(x) = P(x) / ÚP(y)dy
is certainly normalized because its integral
Ú p(x) dx = Ú P(x) dx / ÚP(y)dy = 1.
Again if we wish to find out the average of the variable x then one has to multiply the distribution by x before integration.
xav = Ú x P(x) dx / Ú P(x) dx.
This is completely analogous to the average taken over a variable x distributed discretely. This now completes the crash course in probability.
Counting of modes of radiation. Rayleigh and Jeans argued that the radiation is emitted from the wall is in thermal equilibrium with the electromagnetic radiation inside the cavity. That means that the radiation field inside also has temperature T. The radiation inside the cavity, which is assumed to be a cube, for the sake of simplicity (though this assumption canbe lifted), forms standing waves. Now, the elctric field in standing waves is periodic and has the form (assume polarization along the x axis)
Ex µ A sin(kxx) sin(kyy) sin(kzz)
provided we use the boundary condition that the electric field vanishes at the walls. If the length of an edge of the cube is L, then there are further conditions on the wave numbers ki. The choice of sines for the periodic function insures the the electric field vanishes at the walls at x=0, or y=0, or z=0. We need to insure that the electric field vanishes at the remaining walls, x=L, y=L, and z=L. This can also be insured in a simple manner. Suppose x=L. then the factor sin(kxL) must vanish. This vanishes only if the argument of the sine is a multiple of p. In other words, we have the constraints
kx L = nx p
ky L =
ny
p
kz L =
nz
p,
where nx,ny, nz are arbitrary positive integers. These are the allowed modes of oscillations inside the cavity. The wave numbers are of course closely related to frequencies. The three wave number components form a wave number vector, k. The magnitude of k, k is related to the wave length (the distance of two subsequent crests of the wave k=2p/l.
We can count the number of possible modes if we count the number of possible wave vectors in a range of magnitude. Notice that boundary conditions imply that
k2 = p2 ( nx2 + ny2 + nz2 ) / L2.
or eqivalently
k = (p / L ) [nx2 + ny2 + nz2]1/2
So now, how many wave vectors are smaller than K0? The same number as combinations [nx2 + ny2 + nz2]1/2 are smaller then k0 L/p. The number triple (nx, ny, nz) can be regarded a vector in n-space. There is one vector for every unit volume of the space, so their number in given volume is just the volume. In particular, their number inside the positive one eight of a sphere of radius R=k0 L/p is just one eight of the volume of a sphere of this radius,
n(k0) = v / 8 = (4p / 3) R3 / 8 = (k0 L)3 / 6 p2 = k03 V / 6 p2 ,
where V is the volume of the cavity, V=L3. Do not mix this with v used above.
So now we have the number wave numbers smaller then k0. To get the number wave numbers between k and k+dk we have to take n(k+dk)-n(k) = d n(k)/dk dk = N(k) dk. We obtain
N(k) dk = (V/2p2) k2 dk.
Using the relation of wave numbers to the wave length and frquency of oscillations
k= k=2 p / l = 2 p f / c
we can easily get the number of oscillation modes in the frequency range (f,f+df) per unit volume
N(f) df = 8 p f2 df / c3.
In the above equation an extra factor of 2 was included, to take care of polarization. For every mode there are two possible polarizations are allowed.
Calculations of the average energy by Rayleigh and Jeans. To get the average energy per unit volume of the electromagnetic radiation inside the cavity, one need to multiply number of modes per unite volume by their average energy. Rayleigh and Jeans defined the energy of oscillators as e. Then Boltzmann's law told them how the energy of oscillators is distributed at different temperatures. They had no reason, whatsoever, to assume that the energy of oscillators is related to the the frequency of radiation. Photons (or gamma-quanta) were completely unknown at that time. So, they just went ahead an calculated the average energy of oscillators using the rules for calculating averages, described above. They had no reason to assume that the oscillators had a discrete energy spectrum. They obtained
Ú e P(e) de = Ú e e-e/kBT / Ú e-e/kBT = (kBT)2 / kBT = kBT.
Rayleigh-Jeans law. The average energy of oscillators should be multiplied by the number of oscillators emitting electromagnetic radiation in the range between f and f+df N(f) df, to get the total radiation energy emitted between frequencies f and f+df. The result is
u(f,T) df = kBT 8pf2 df / c3,
a result that, as we saw earlier, had disasterous consequences. Max Planck worked long and hard on the resolution of this problem. Clearly, to solve the problem he needed new ideas. His idea was that the energy of the oscillators in the wall of the cavity is related to the frequency of the electromagnetic radiation emitted. Taking a hint from Wien's law he assumed that the energy was proportional to the frquency. He had no idea why that was true, but he realized that simple assumption allowed him to derive the correct distribution formula, hitherto called Planck's law. Let us follow this idea and using our knowledge of distributions to derive Planck's law.
Planck's law. Planck could not find anything wrong with the Rayleigh-Jeans calculation of the numer of oscillation modes. Thus, he concentrated on the calculation of the average energy of oscillators. He realized that if he wants to get agreement with experiments the average energy should be frequency dependent. Using the simplest of all assumptions, Planck used oscillators that took values that were integral multiples of a fixed multiple of f, nhf, where n is an arbitrary integer and h is a constant. He had every intention to let eventually h go to zero, to go back to a continuous distribution. He thought that the Rayaigh-Jeans calculation was almost good, only some mathematical fine point is missing. Of course he got his cue from Wien's law which showed an exponential drop in frequency. With this asssumption the continuous distribution of Rayleigh and Jeans becomes a discrete distribution. IHe later realized that no matter how he takes the limit he ends up with the incorrect Rayleigh-Jeans distribution. On the other hand, if he kept h finite then at some finite h he obtained complete agreement with experimental observations. This value of h is called lanck's constant, a universal constant of nature. With Planck's assumptions it only takes a few simple manipulations to derive Planck's law.
Boltzmann's factor for oscillators of energy nhf have the form:
P(f) µ e-nhf/kT.
Averaging over all possible energies of this form means summation over n (Remember we donot want to average over f, because we are after the frequency distribution. So the average energy of oscillators, which can have energy 0, hf, 2hf,... is
Eav = ¬ nhf e-nhf/kT / ¬ e-nhf/kT. (2)
Let us calculate first the sum in the denominator. This will be easy. The sum has the form of
¬ xn = 1 / (1-x),
a geometric series, where x= e-hf/kT. Thus, the sum in the denominator of eq. (2) is
¬ e-nhf/kT = 1 / (1- e-hf/kT).
The sum in the numerator of (2) is of the form
hf ¬ n xn = hf x d/dx ¬ xn = hf x d/dx 1/(1-x) = hfx / (1-x)2.
The first relation in this chain is obtained if we exchange the order of summation and differentiation and differentiate the sum term by term. Taking the ratio of the denominator and numerator we obtain for the expectation value of the energy of individual oscillators
(nhf)av = hf x 1/ (1-x)2 ¥ (1-x),
where the last multiplier is the inverse of 1/(1-x),, the normalization factor. Canceling this factor and substituting x= e-hf/kT we obtain
eav = (nhf)av = hf / (ehf/kT - 1),
where both the denominator and numerator were multiplied by ehf/kBT to arrive at this form of the average energy of an oscillator. Note now two important properties of this expression: if we take the (so-called classical) limit hÆ0,, or alternatively the low frequency limit, fÆ0, we obtain after expanding the exponent ehf / (kBT) = 1 + hf / (kBT )+ ... in a Taylor series that the average energy becomes
eav = hf / (hf/kBT)+... = kBT+...,
which is exactly the expression obtained by Rayleigh and Jeans assuming a continuous distribution of the oscillator energy. Now it is easy to get the expression for the frequency and temperature dependence of the energy density inside a cavity, by multiplying the average energy by N(f) df, the number of oscillator modes between frquencies f and f+df. We obtain
u(f,T) df = (8 phf3 / c3) df / (ehf/kT - 1).
This is Planck's law. h is Planck's constant, h=6.6¥10-34J.s = 4.11¥10-15eV.s.
We already investigated the behaviour of the average energy at small frequencies. It gave the Rayleigh-Jeans average energy kT. Since Planck used the same expression for the number of modes as Rayleigh and Jeans, we must get back their expression for the energy density as well.
It is worth while to examine the high frequency behavior of the energy density as well. If hf/kBT >> 1 then its exponential is even much bigger so the -1 in the denominator of the expression for energy density can be neglected. We obtain
u(f,T) df = (8 phf3 / c3) df e-hf/kT,
which agrees Wien's law with the coefficients of Wien idenified and given in terms of universal constants, c, h, kB. It is easy to see that at room temperature and visible light this asymptotic form can be safely used. Boltzmann's constant is kB=8.6¥10-5eV K-1. So at room temperature the energy scale is given by kBT=0.025eV. Now hf = hc/l = 1.24¥103eV.nm / l. The wavelength of visible light is about 500nm. So for visible light the fraction in the exponent is
hf / kBT µ 1.24¥103 / (500¥0.025) = 99.2,
a number that is indeed much larger than one. Indeed, Wien's law can be safely used for visible light at room temperature. If we go to very high temperatures, or to very low frequencies, then one must use the exact, Planck's law.
Finally let us check whether Stefan's law is also recovered from Planck's law. The total energy per unit volume of the cavity is the integral of the energy density over the frequency
E = Ú u(f,T) df = Ú df (8 p h f3 / c3) / ( ehf/kT - 1) = (8 p h / c3) Ú f3 df / ( ehf/kT - 1) ,
To do this integral it is necessary to substitute a new variable. The strategy is to try to eliminate constants from the exponent in the denominator. So we introduce the new integration variable x. x is defined as x=hf/kBT. Then we have dx = df h / kBT. In other words, f3 df = x3 dx ( kBT / h )4. Substituting this lastexpression into the integral we obtain
E = (8 ph / c3) ( kBT / h )4 Ú x3 dx / (ex-1).
The last multiplier, the integral over x, is just a calculable number. In fact, it can be shown that it is equal to p4/15. Thus we obtain the final expression for therotal energy per unit volume as
E = (2p5kB4 / 15 c2 h3 ) T4 = s T4,
exactly Stefan's law, with Stefan's constant identified with a combination of constants of nature,
s = 2p5kB4 / 15 c2 h3 .
As we said earlier Planck did not really understand why the oscillators, of energy e, in the electromagnetic field could take the values of nhe only. It was left to Einstein to resolve this mistery. It was suspected that such a quantization of energy, as this phenomenon was called, is somehow just the consequence of some hitherto unknown properties of matter. Light can only be emitted in such quanta, but still the energy of electromagnetic radiation can take continuous values.
Einstein suspected right away that the hypothesis that light consists of quanta is the consequence of the way it is emitted was wrong. He beleived that the fact that one can only obtain agreement with the black body radiation spectrum if the quantization of electromagnetic radiation is assumed points to the fact that this is the basic nature of electromagnetic radiation. After learning about the results of recently performed photoelectic effect experiments by German-Hungarian physicist Philip Lenard he could clearly proove his hypotesis.
Photoelectric effect was discovered by Hertz, who showed that metals under ultraviolet light emit charges, which were later identified to be electrons by Thomson. Lenard performed a careful study of the photoelectric effect. Lenard constructed a photoelectric vacuum tube cosisting of a cathode irradiated by ultraviolet light and an anode collecting electrons emitted by the cathode. A voltage, V, was applied between the anode and cathode to repel the electrons emitted by the cathode. The number of electrons arriving at the anode was measured by the current in an outside circuit connecting the anode and the cathode. The idea of Lenard was that by changing the voltage he would be able to find the spectrum of emitted electrons. From energy conservation one can find that only electrons that have a kinetic energy larger then e¥V. By varying the voltage he was able to find the number of electrons having kinetic energy larger than a given value at every voltage. He found that there was a lowest negative voltage below which no electron could reach the anode, i.e. the maximal kinetic energy of electrons was independent of the intensity of light. This was completely unexpected. Higher intensity corresponds to more light energy and should result in more energetic electrons according to the general beliefs of the time. The only attribute the this voltage, which equals to the maximal kinetic energy of electrons, depended on was the frequency of light. Lenard found that increasing the frequency also increased the maximal energy of electrons. He was not able to establish a precise relation between the frequency and the maximal kinetic energy. Now we a very precise measurment of this relationship, as shown below
In Einstein's time no such curve existed, just a qualitative relationship between Kmax and f. It was the genius of Einstein that coupled this measurement with Planck's law. In fact, Einstein predicted the relationship shown in the above figure, based on the simultaneous physical interpretation of Lenard's measurment and blackbody radiation. Now this seams to be very easy for us but at the time it was just a jumble of inexact data.
The physical interpretation of this curve is the following: The relationship between Kmax and f is the following:
Kmax = hf - f,
where h is Planck's constant and is a minimum potential, called work function, characteristic to metals. Light is made out of quanta, called photons. These photons are absorbed by the metal individually. Each photon carries energy E=hf, which is can be used to knock out an electron from the metal. The electrons are in a potential well inside the metal. The depth of this well is f. Each photon interacts with a single electron only. Thus, the maximal energy an electron can gain is hf. Part of the energy the electron receives, f, is used to raise it to the outside potential. Thus, the maximal kinetic energy a photo-electron can have in the outside world is Kmax = hf - f. Clearly, this is the same linear dependence found in experiments. The value of f0 isexplained simply, it is the solution of the equation hf0 = f, the value of f where the maximal kinetic energy is zero. In other words, f0 = f / h.
Einstein's hypothesis explains the photoelectric effect and the Planck's law at one strike. Planck's assumption that the possible modes of radiation at frequency f are equal to nhf is explained, because rediation consists of indivisible photons of energy hf. each mode of oscillation can have an integral number, n, photons only.
Photons as particles and waves: Duality. There were, of course, significant problems left open by Einstein's explanation of the photoelectric effect and Planck's law. They both required to treat photons as particles. Morover, to explain the photoelectric effect, one must assume that photons are absorbed in the metal at a point of the "size" of an electron. This completely contradicts to classical electrodynamics, in which electromagnetic waves have large (for plane waves infinite) spacial extent. In other words being a wave and being a particle are not compatible. An equally serious problem is that Maxwell's equation allow the field energy to be arbitrary small amd to be changed continuously.
This problem of continuity of energy can be solved, at list on a
phylosophical level, at least if we are governed by the
correspondence principle. Namely, the field energy represented by a
single photon is exceedingly small. As an example, consider a small,
1W, radar working at 100MHz. One can ask the question, how many
photons are emitted by this radar every second? One can answer this
question if 1J of energy, emitted by the radar every second, is
divided by hf = 6.63¥10
-34J.s ¥10 8s -1
= 6.63¥10 -26J. Thus the
number of photons is
n = 1J / 6.63¥10 -26J =
1.5¥10 25. This is an extremely
large number. The point I want to make is that all of our everyday
experienences with electromagnetism involve trillions of trillions
photons. Knowing the laws of electrodynamics as applied to such an
intense field does not give us the right to predict the laws of
electrodynamics, involving one or a few photons, unless experiments
tell us so. Indeed, Einstein's conclusion was that Maxwell's
electrodynamics cannot be valid for single photons. To have an idea
about the correct theory, humanity had to wait a quarter of a century
until P.A.M.Dirac invented Quantum Electrodynamics.
The theory of Quantum Electrodynamics was only completed in the 40's
by Tomonaga, Schwinger, and Feynman, who also succeeded to show that
it gives numerical predicitons for a large array of physical
quantities which are in an agreement with experiments at a hitherto
unprecedented level, up to ten significant digits.
As we will see in the next section, problems with particle-wave duality have also arisen in the study of particles, as well. In the case of particles one arrives to this puzzle from the opposite extreme. While electromagnetic radiation was thought to be of wave nature, but later was recognized to have a particle structure as well, electrons were assumed to be particles. which turned out to have wave properies. Quantum mechanics explains this apparent contradiction, how an object can be a wave and a particle at the same time.
Let us solve a couple of problems now.
(1) The work function of copper is 4.7V. What is the maximal velocity of photoelectrons produced by ultraviolet light of wavelength 100nm?
Solution: Assume that the electrons are nonrelativistic (do not forget that the rest energy of electrons is 510kV, much larger then the kinetic energy an ultraviolet lightray can provide). Then mv2/2 = hf - f and v = [2(hf-f) / m]1/2. Using the relation f=c/l, we obtain
v = [2 ( hc / l - f) / m]1/2 = [2 ( 1.24¥10 3eVnm / 100nm - 4.7V) / m]1/2
Now use m=510keV/ c2 to get
v = [ c 22 (12.4-4,7)/ 510,000]1/ 2 = 5.5¥ 10 -3c.
This velocity is indeed norelativistic so the assumption of using the nonrelativistic energy is justified.
(2) What should be the wavelength of monochromatic radiation to produce photoelectrons of maximal velocity 10 6m/s from iron?
Solution: 10 6m/s = 3.3¥10 -3c. This corresponds to kinetic energy
Kmax = m (3.3¥10 -3c)2 / 2 = 510,000¥11.1¥10 -6eV = 5.66eV. Now,
hf = hc/l = f + Kmax = 4.5eV+5.66eV=10.16eV, thus we have
l = hc / 10.16eV = 1.24¥10 3eVnm / 10.16eV = 122nm.
The most spectacular proof and direct proof of the existence of photons came about through the investigations of Compton. The compton effect is a scattering of photons by electrons. The scattering a two bbody ineraction which satisfies standard relativistic kinematics, provided we use zero mass for the photon. In other words, in the Compton effect we can see electrons knocked out of their rest position by zero mass particles.
The Compton effect was discovered Debye and Compton independently
in the
early 20's. They studied X-ray scattering and observed that the
scattering from free electrons could not be explained by the
classical
wave theory. Classical theory would have predicted that the frequency
of
scattered X-ray should depend on the length of time and intensity of
the
incident X-ray beam. What they observed was that the scattered
X-ray
was emitted at an angle q and
their frequency only depended on this angle. When Compton figured out
the reason he was amazed why Einstein did not predict this phenomenon
almost 20 years earlier. He soon derived a relationship between these
quantities and got complete agreement with the experimental data. He
assumed that X-ray scattering is a two-body relaivistic scattering
process of massless photons and massive electrons. The derivation of
Compton's formula is very simple. it is based on relativistic
kinematics only.
We know that in the center of mass system the energy of the scattered particles is fixed. In fact, energy and momentum conservation requires that the magnitude of the momenta is unchanged in the process. The scattering happens in the following manner in the Center of Mass System (CMS) and the lab system:
Then the Lorentz transformation transforming the momentum pe to zero takes us to the lab system. Since the configuration in the center of mass is completely determined by the scattering angle, after Lorentz transformation, in the lab system, both the magnitudes and the angles of momentum vectors can only depend on the CMS scattering angle. In other words, fixing the magnitude of the momentum of a scattered particle fixes the angles and the momentum of the other scattered particle. Take e.g. the photon. As we will see below its wavelength is determined uniquely by the its momentum. Then relativistic kinematics will determine the scattering angle of the photon from its wave length.
The photon, being a zero mass particle, its energy and momentum are related to each other as E=pc. We also know, following Einstein that the energy of the photon and its frequency are related as E=hn. Then the momentum is p=hn/c = h/l. Now we will write downequations for momentum conservation and energy conservation.
Transverse momentum conservation: p'g sin f = p'e sin q
The square of this equation: p'g2 sin2 f = p'e2 sin2 q (a)
Longitudinal momentum conservation: pg - p'g cos f = p'e cos q
The square of this equation: pg2 + p'g2cos2 f - 2 pg p'g cos f= p'e2 cos2 q (b)
Energy conservation: c (pg - p'g ) - mc2 = ( m2c4 + p'e2c2)1/2
The square of this equation: c2 pg2 + c2p'g2+
m2c4 - 2 c (pg
- p'g ) mc2 - 2
c2 pg p'g=
m2c4 +
p'e2c2 (c)
Now add equations (a) and (b): pg2 + p'g2 - 2 pg p'g cos f= p'e2
and subtract this sum from (c), divided by c2 to get: 2 c (pg - p'g ) m = 2 pg p'g ( 1 - cos f )
Now the last step of the derivation is dividing this equation by 2 m c pg p'g and multiplying it by h:
l' - l = (h / mc ) ( 1 - cos f ),
which is Compton's formula. It gives the change of the wavelength of the electromagnetic radiation (photon) as a function of the scattering angle of the radiation. These are exactly the quantities measured by Debye and Compton.
Obviously in these measurements Compton needed to measure the wavelength of the primary and scattered X-rays. This can easily be done by using Bragg scattering. As one learned in General Physics, waves reflected from a crystal show characteristic interference pattern. The intensity is maximal if the wave length satisfies the realation
nl = 2 d sin q,
where n is an integer, d is the lattice constant and q is the angle of incidence. The first maximum, n=1, has the maximal intensity. The wave length of X-ray lines from metals is close to that of the lattice constant of most crystals, so the wavelength is well measurable by changing slowly the angle of incidence.
Examples:
(1) Photons in gravitational field. As we see massless particles are not very special as far as relativistic kinematics goes. They have a relaivistic energy E=pc, unlike massive particles having energy, E=( m2c4 + p2c2)1/2. Only energy and not the mass is conserved in collisions. It also turns out that from the point of view of interaction with gravitation it is also the energy and not the mass determines the strength of the interaction. This wouldt change calculations of motion of massive objects through gravitational field very slightly. In Newtons gravitational force one should only change the mass of an object from m to ( m2 + p2/c2)1/2 µ m ( 1+ v2/2c2), which is a negligible change at velocities small compared to the velcity of light. Photons, however, at distance R from a star of mass M, would be affected by the gravitational field as if they had a mass of E / c2 = hf / c2 . Now, in the gravitational field of a star an object of mass m has a potential energy of
V= -mMG/R.
Thus the total energy of the photon will be
E = K +V = hf -mMG/R = hf ( 1- MG/Rc2 ) = hf'.
Suppose that the photon approaches the star very close to the surface and further suppose that the star is so massive that MG/Rc2 < 1. Then the total energy of the photon is negative and the photon can never escape from the gravitational field of the star. In fact, in this case, if the photon gets inside the radius R=MG/c2 , called the horizon, then its energy becomes negative and the photon can never escape. Such a star is called a black hole. Even if the photon is outside of horizon its frequency would shift towards lower values, as shown by the formula above. This is called the gravitational redshift.
(2) Determination of Avogadro's number by X-rays (problem 2.40) X rays of wavelength 0.0626nm are incident on an NaCl crystal which has a cubic lattice with alternating Na and Cl ions. The density of NaCl is 2.17gcm-3. The n=1 diffraction maximum is found at q=6.41o. Compute Avogadro's number. Avogadro's number is the number of atoms in one mole. The molecular number of NaCl is 22.99+35.45=58.44. This many grams correspond to 58.44g/ 2.17gcm-3=26.93cm3.
Now the question is that how many Cl or Na atoms are in this volume. To determine this we need the lattice constant of NaCl first. The rule for Bragg scattering is that nl = 2 d sinq, or
d = n l / 2 sinq = 0.0626nm / 2¥sin 6.41o =0.28nm. With every cell in the lattice we can associate one atom (say the one at its lower, left, front corner). That atom is alternately Cl or Na. Thus, as an average, every cell contains 1/ 2 Na or Cl atom. The number of sodium atoms in one cm3 is then N=1cm3/ 2¥(0.28nm)3= 22.78¥1021. This number should be multiplied by the number of cubic centimeters representing a mole, 26.93, to give finally NA =26.93¥22.78¥1021=6.13¥1023, which is pretty close to the best value NA =6.02¥1023