Another Consequence of Einstein's Energy-Momentum-Mass Revelation. Let's discuss one more of the consequences of the innocuous-looking

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1 Modern Physics (PHY 3305) Lecture Notes Modern Physics (PHY 3305) Lecture Notes Radiation as Particles (Ch ) SteveSekula, 1 February 2010 (created 26 January 2010) Review no tags Today We will begin a discussion of radiation and the "blackbody problem" that kicked off the quantum revolution We will explore the "photoelectric effect", which helped to seal the fate of our classical understanding of radiation and light We will discuss other evidence for the particle properties of radiation and bring the discussion of light full circle Another Consequence of Einstein's Energy-Momentum-Mass Revelation Let's discuss one more of the consequences of the innocuous-looking equation, E 2 2 = ( mc ) ( pc) : What happens when the mass of an object is zero? Is that even allowed? When mass is zero, E = pc. It seems that there is no prior reason to believe such a thing cannot exist; the equation allows it, but what does the equation mean about such an object? If you write this as E = Í u mu, the ONLY way that the energy of such an object can be anything but zero is if u = c so that Í u = 1. While relativity fails to tell you about the properties of this object (e.g. exactly what its energy is!), it doesn't rule out that such objects can exist and if they do they must move at the speed of light. What moves at the speed of light? Light. Light is a massless particle, but we won't learn much more about it from relativity since it's undefinable in relativity. Relativity has little more to say right now regarding massless things that 1 of 11 02/04/ :46 PM

2 move at the speed of light. Let's move forward. Basic Questions WHAT IS A WAVE? WHAT IS A PARTICLE? Parallel Revolution: The Quantization of Energy In parallel with the realizations of Lorentz, Einstein, and others regarding the nature of space and time were developments in energy and matter. These developments were initiated by Max Planck and his studies of the so-called "blackbody problem." His hypothesis, which resolved this problem, had stunning implications for the nature of energy and, in turn, matter. We will discuss today the "blackbody problem" and its resolution. We will then begin exploring the implications of this solution, and the experimental evidence that confirmed those implications. Blackbody Radiation A blackbody is defined as follows: A blackbody is any object whose emitted electromagnetic radiation results solely from thermal motion of its electric charges. The name then becomes clear; we only see things because light scatters off of it and strikes our eyes. If no light can scatter (be emitted from) an object, then it appears black to us. However, this is only one class of such object. Name some common blackbodies in nature. Common blackbodies: coal (tar, oil, etc.) or anything painted black an oven the sun the cosmic microwave background A quantity of interest experimentally when dealing with blackbodies is the 2 of 11 02/04/ :46 PM

3 amount of energy radiated per unit radiation frequency, or the spectral energy density. This is written as du=df. Consider the following blackbody. A cavity is constructed such that the entire volume is enclosed except for a pin-sized hole on one wall. Radiation can enter the cavity through this hole, but once inside the probability of it reflecting off the walls and escaping is small. Thus, incident radiation primarily goes into heating the interior walls of the cavity. Such an object is a blackbody; light entering the hole is converted to heat in the cavity walls, and little or no light escapes. Therefore, the cavity is an excellent blackbody, and was a favorite construct for 19th-century blackbody studies. Now, classical physics lets us calculate the spectral energy density of this blackbody. It's a very details calculation (c.f. Eisberg and Resnick, Chapter 1-3: "Classical Theory of Cavity Radiation". The summary of the calculation is as follows. You can think of the radiation trapped in the cavity as sinusoidal waves resonating in the cavity, with arbitrary amplitudes. Classical Thermodynamics tells us that the average energy of any wave of a given frequency f is k T, where k is Boltzmann's Constant: B B k B = 1:380:::  10 À23 J=K and T is the temperature of the cavity. If we then multiply this average energy by the number of waves for a given range of frequency, df, in the cavity volume V, we arrive at the classical prediction: du = T f df kb  8ÙV 2 c 3 How did this compare to experimental observation of such cavities? Poorly! This function DIVERGES as the frequency increases, meaning that if the waves in the cavity have high frequencies their energy diverges as the square of the frequency. This is the so-called "Ultraviolet Catastrophe" - high-frequency radiation (ultraviolet) present in the cavity would have huge amounts of energy. Data, instead, said the energy spectral density peaked at a value and then fell off for high-frequency radiation: 3 of 11 02/04/ :46 PM

4 (from Eisberg and Resnick, Chapter 1-4) Max Planck wrestled with this problem. He finally made a hypothesis that changed our understanding of the universe. One of the assumptions of the classical calculation was that waves of any energy were allowed for a frequency range df. Instead, Planck tried the hypothesis that the energy of the waves was restricted for a frequency range, and that the energy was discrete, rather than continuously distributed: E = nhf where E is the energy of a wave, n is an integer ( n = 0; 1; 2 ; :::), and h is an undetermined constant. When he changed the assumption from one of "any energy is allowed for waves in a frequency range df " to "only discrete values of energy are allowed to waves in a frequency range df ", he hit upon the formula: du = df hf 8ÙV f 2 e hf=k T B À 1 Â c 3 Rather than blowing up as f! 1, it had the useful property that the first part of the equation forces the spectral energy density to turn over and fall 4 of 11 02/04/ :46 PM

5 as f increases. From the data (e.g. the 1916 data shown above), the function was found to describe experimental results perfectly and the value of the constant, h, (now known as Planck's Constant), was determined to be: h = 6:63 Â 10 À34 J Á s Note that this constant has units of momentum, kg Á m=s. It is currently believed that h is one of the fundamental constants of nature (like the speed of light), and its value can only be determined by extraction from experimental data. Planck called this hypothesis, in a letter to R. W. Wood, "an act of desperation." He wrote, "... a theoretical interpretation [of the blackbody spectrum] had to be found at any cost, no matter how high." Casting aside the classical assumption, long held without contradiction, of the continuity of energy distribution, was indeed seen as a high cost. It was assumed that his choice was a conveniently useful one - a trick - but not a fundamental statement about nature. It was Einstein who sealed the fate of radiation. The Quantization of Radiation: the Photoelectric Effect The blackbody was not the only mystery phenomenon involving radiation near the turn of the 20th century. Another one was called "the photoelectric effect." Ball on a Pond Think about how to knock free a weed-entangled rubber ball. Imagine a small rubber ball in the middle of a pond. It is stuck on some weeds (lilypads) in the middle of the lake. Being clever, you know you can knock it free if you can hit it with waves. So you take a bucket and you shove it into the surface of the pond three times in quick succession (a spacing of 0.5 seconds per push). A wave packet travels from the shore to the boat, momentarily straining the hold the weeds have on the ball. So you do it again, making another wave packet of the same amplitude and frequency. The same thing happens - strain, but not freedom. So you crank the intensity - the time-averaged power delivered by the wave per unit area (in this case, the side of the ball facing you). To increase the intensity, you keep the amplitude and frequency of the wave packets the same (the total 5 of 11 02/04/ :46 PM

6 energy of the wave) but decrease the time between the packets. Another way to crank the intensity is to get a friend to send the same wave packets at nearly the same time as you do it, doubling the intensity. This delivers the same energy many times to the ball before it can snap back into the weeds, and knocks it free. As a note, imagine making wave packets with very long wavelengths. Can they still knock the ball free? The answer is yes: intensity is the key. Make the wavelength very long, so the wave's energy is delivered slowly to the ball, but put a hundred friends on one side of the pond all making these wave packets, and the combined energy of the waves delivered to the ball can knock it free. It was assumed that this was how the photoelectric effect worked: negative charges are bound to positive charges in metal by some force (Coulomb's law, to be specific). Hitting the negative charge with a light wave momentarily stretches the bond between the negative and positive charges, but does not break it. Increasing the intensity of the light - the number of waves of some energy per unit time per unit area - should strain the bond even further until it breaks and the negative charge is free to move on its own. Different light waves will act over the same charge at the same time, like trying to free the ball using multiple people making waves. Intensity is the key! Based on the "Ball on the Pond" discussion, talk about how EM wave theory would predict what happens to electrons in a metal But what did experimentalists find? Let's consider this demonstration of the experiment: Demonstration of the Photoelectric Effect (from /sims.php?sim=photoelectric_effect) Set to: metal: sodium final wavelength: 450nm you will find that the stopping potential is -0.46V We have learned: We can determine the energy required to stop the photoelectrons from crossing the gap by reversing the polarity of the battery and cranking 6 of 11 02/04/ :46 PM

7 the voltage. By determine, I mean we can find out the energy needed to strip the electron from the cathode by finding the voltage needed to bring the current in the circuit just to zero. The kinetic energy of the highest-energy photoelectron is given simply by KE = ev 0, where V 0 is the stopping potential. We can change the intensity for a fixed light frequency, and the stopping potential DOES NOT CHANGE. That means that intensity seems to have nothing to do with the maximum energy that photoelectrons have when they leave the cathode; it only appears to affect the number. We can change the frequency of the light for a fixed intensity, and we are forced to change the voltage to bring the current to zero again. The energy of the electrons depends on the frequency! There is a minimum frequency of light at which no photoelectrons are knocked free of the metal, even under a strong positive potential difference which could pull even weak electrons across to the anode. Translation: more friends making long-wavelength waves will not knock the ball free! This suggests that there is some minimum amount of energy needed to remove an electron from the metal, called the work function. These were curious and stunning observations. They seemed completely counter to the wave theory of light, and yet, so many other properties of light were wave-like! Einstein tried to resolve this mystery with the following hypothesis: Light is transmitted not as a wave (a continuous distribution) but rather in smaller units (quanta), which he called photons. The energy of the photons is given as E = hf, where h is Planck's constant. Instead of the ball being on a pond where you can make waves, imagine instead it's stuck in a rigid tree. You have to knock it down. How do you do it? Well, you could get baseballs and throw them at the rubber ball. The chance of hitting it is small, but when you do the baseball will transfer a lot of energy to the rubber ball and knock it loose, if the energy of that baseball is sufficient to break the hold of the branches. Imagine now having 10 friends helping you, all throwing baseballs. The chance that 2 baseballs will hit the rubber ball at the same time is very slight, so all having more friends (more intensity) does is increase the 7 of 11 02/04/ :46 PM

8 chance of any one baseball striking the rubber ball. However, if all the throwers have weak pitching arms, it won't matter how many people are throwing - the tree will not let go of the ball, even if it is struck. This is more like the photoelectric effect. Let's return to the properties of the metal. There is a minimum amount of energy needed to strip the electron from the metal, which we called the work function,. The condition for removing an electron then is hf >. That means that the kinetic energy of the electron is given by: KE = hf À So, what does this mean experimentally? It means we can determine the work function of a metal by setting the voltage such that the current just drops to zero. This, then, is the condition that: KEmax = ev 0 and defines the stopping potential. We can then determine the work function, knowing Planck's constant and the frequency of the light: = hf À ev 0 Night Vision as an application of the Photo-electric Effect 8 of 11 02/04/ :46 PM

9 This gives you a taste of how quantum physics plays an important role in the modern world - law enforcement, the military, search and rescue, all rely on night vision to aid in difficult situations where relying on visible light is not an option (or not the only option). Quantum physics enables us to have sight beyond sight. Einstein's postulate - that light is transmitted by quanta, particles, called photons - explained the photoelectric effect beautifully. Was there other evidence for the particle nature of light? Compton Scattering Briefly describe the Compton Scattering experiment: Prediction: EM waves (light - specifically X-rays) should cause charges 9 of 11 02/04/ :46 PM

10 in a solid to oscillate. Oscillating charges make EM waves radiate in all directions, with frequency or wavelength approximately equal to the incident waves Experiment: some waves came back toward the source of X-rays with much lower frequencies. Hypothesis: light, a particle and not a wave, collides with an electron and in ejecting the electron scatters backward with much less energy: h (Õ after À Õ before ) = (1 À cos Ò ): m electron Á c scatter So why do X-rays do this but not, say, radio waves or visible light? Consider two different kinds of light being backscattered ( cos Ò scatter = À1 ) Visible light ( Õ = 500nm ), gives us Õafter = 500nm + 2h=(m electron Á c ) = 500nm + ( 5 Â 10 À3 nm) is a SMALL effect - you probably wouldn't even notice such a tiny shift in wavelength. (it's a shift of about 0.001%) X-rays have Õ = 5 Â 10 À2 nm. Now, shifting the wavelength by 5 Â 10 À3 nm is more noticeable - more like a 10% effect! So the answer is deceptively simple: for long wavelengths, the classical description of light is very accurate. But for short wavelengths, the classical description is not as accurate, and discrepancies appear. The Correspondence Principle: the classical behavior of radiation is restored when the wavelength is sufficiently long (this is a guideline, not a law) There is a continuity, therefore, between what is classical physics and what is the more general description of radiation. The Momentum of Radiation Special relativity failed by itself to tell us about the properties of light - only that it is massless, which is how it moves at the speed of light. Thanks to Einstein's work on the photoelectric effect, we know that Elight = hf Relativity tells us that: 10 of 11 02/04/ :46 PM

11 Elight = pc We can combine these two insights - special relativity and quantization of radiation - to finally understand the momentum of light: plight = h f=c = h=õ since c = Õf. Next Time Which is it: wave or particle? If radiation was found to be able to express both "wave-like" and "particle-like" qualities, what about things we've traditionally thought of only as "particles" (e.g. electrons, atoms, people, the earth, the stars). Can they be "wave-like" too, and if so, why haven't we noticed this? 11 of 11 02/04/ :46 PM