Transcript محاضرة 8
Describing a wave phase and group velocities particle diffraction “I think that modern physics has definitely decided in favor of Plato. In fact the smallest units of matter are not physical objects in the ordinary sense; they are forms, ideas which can be expressed unambiguously only in mathematical language.”—Werner Heisenberg Last time I digressed a bit into quantum mechanics (because Beiser does so also). Now back to matter waves. De Broglie says particles have wavelengths. We wrote down an expression for particle wavelength. h λ= . p It’s now time to think about the mathematics of waves. What is this “wave” that is associated with a particle? 3.3 Describing a Wave We can easily calculate a velocity for de Broglie waves. What we calculate had better contain results of classical physics within it! Beiser calls the de Broglie wave velocity "vp," and then shows that vp is the phase velocity of the waves. A wave velocity is given by vp=f. De Broglie postulated that =h/mv for particles as well as waves. De Broglie also postulated that a particle's energy should be given by E=hf, which, as Einstein showed us, is E= mc2. We can solve these two equations to get f= mc2/h.* Don’t do this without supervision! Writing a particle’s frequency as f = mc2/h may lead you to using v=c. Combining our usual formula vp = f for waves, with de Broglie's postulated equation for , and our value for f which we just obtained gives the velocity of the de Broglie waves: v p = fλ and h λ= mv this is wave velocity and mc 2 f= h c2 vp = . v this is particle velocity I don’t consider that last (boxed) equation “fundamental,” but is appears in several problems, so I’ll call it an OSE. This equation gives us no problems when we think about photons as particles. Photons travel with a speed v=c, so that the "wave velocity" is vp=c. But particles with nonzero mass travel at velocities less than the speed of light, so that c2/v must be greater than the speed of light. The de Broglie wave "velocity" of a particle with nonzero mass is greater than c! “Are you trying to tell me that the de Broglie wave has a speed greater than c?” YES! De Broglie's postulate of wave-like properties for particles leads us to conclude that the de Broglie wave "velocity" for a particle with nonzero mass is greater than c. photon: faster than a speeding bullet matter wave: faster than a speeding photon? What does this mean? Right now, it means we must re-think what we mean by wave "velocity." In fact, we’d better rereview our ideas about waves. Beiser uses the vibrating string demonstration to derive a “wave equation.” Take a string, attach one end to an “arm” which can vibrate up and down, and attach the other end to some rigid object. When the vibrating arm moves up, a pulse is sent down the string: vibrating arm attach string here pulse travels down string reflects (& inverts) meets another pulse on the way back You end up with standing waves in the string. You may have done this experiment in your Mechanics lab course. Beiser derives several equivalent forms of the formula for this wave, which gives the displacement y of any point on the string (i.e., at any position x) at any time. x y = A cos 2f (t - ) vp f is the frequency and vp is the wave speed x y = A cos 2 (f t - ) λ Using vp = f , = 2f, and k = 2/, we get y = A cos ( kx - ωt ) , or y = A cos ( k r - ωt ) in 3 dimensions. So far today we have done nothing more than "derive" the equation for a propagating wave. We haven’t yet shown how these waves form “packets” like those illustrated 2 slides back. Waves travel with a "phase velocity," which does not represent the actual velocity of any particle with mass. In the next section we will define the physically meaningful "group velocity." vp y x These are "transverse" waves. They are "polarized" in the y direction. In Chapter 2 we derived the phase velocity in a slightly different, but completely equivalent way. vp y x This wave is infinitely extended in space. The wavelength (and therefore the momentum) of the wave is well defined. Where is the particle represented by the wave? We can’t find it. It could be anywhere along the x-axis. To make a wave represent a particle, we have to modulate it by summing multiple waves of different wavelengths and/or frequencies. Then the wave function will have both an obvious wavelength and spatial "length." 3.4 Phase and Group Velocities Wave groups, which we hinted at above, are a superposition of different individual waves. Waves interfere to produce some shape for the group. http://www.colorado.edu/physics/2000/schroedinger/index.html Because the de Broglie wave velocity varies with , the individual waves move with a different velocity than the group. Beiser calculates the velocity of travel, vg, of a simple group made of two sine waves. y1 = A cos (ωt - kx) y 2 = A cos (ω+dω) t - (k +dk) x Two waves are the minimal set out of which we can build a wave "packet" or "group." With a little trigonometry, and using the fact that d and dk are small compared to and k, Beiser shows that dω dk tx) . y1 + y 2 = 2A cos (ωt - kx) cos ( 2 2 The wave represented by y1+y2 is built upon a wave of angular frequency and wave number k, and is has superposed on it a modulation of frequency d/2 and wave number dk/2. y1 y2 y1+y2 “I thought d << . How come the group seems “wider” than the bumps in the sine waves?” This is a plot of y vs t. Frequency is inversely proportional to t, so frequency features should be “wider.” dω dk tx) . y1 + y 2 = 2A cos (ωt - kx) cos ( 2 2 This picture is mildly unsatisfactory, because it is really a snapshot in time of waves that are moving through space and time. The phase velocity of the moving waves is vp=/k, while the groups (the modulation) move with a velocity vg=(d/2)/(dk/2)=d/dk. ω vp = k dω vg = dk The waves in the figure are y=sin(t) and y=sin(1.2t). ω vp = k dω vg = dk Vg may be greater than vp or less than vp. If the phase velocity vp is the same for all wavelengths, as it is for light in a vacuum, then the phase and group velocities are the same. But what does this have to do with particles? Where in the math is the particle velocity? Could it be this mysterious vg? Wave mechanics had better contain classical physics within it. Is this vg consistent with our idea of particle velocity? Angular frequency: Wave number: 2mc 2 2mc 2 ω = 2f = = 2 h v h 12 2mv 2mv k= = = 2 λ h v h 1- Use the above to calculate ω vp = k dω vg = dk . c2 . c2 Result: vp=c2/v (we already knew that) and vg=v (the particle velocity; we suspected that). Here's a question: we have shown that a wave's phase velocity can be greater than c. Does this imply we can find a way to transmit information faster than the speed of light? Relativity says: we can’t accelerate matter or “energy” to a speed faster than c. Also, we can’t observe the result of an event before the event itself. Relativity doesn’t really address the transmitting of “information,” but in this interpretation, the information is in the modulation, which travels at a speed equal to vg, so we aren’t transmitting information at a speed greater than c. Here’s a picture of a wave packet that “looks” more like it might represent a particle: http://www.phy.duke.edu/Courses/100/lectures/Waves/Wa.html Another equivalent way to write the wave is y=A ej( kx -t ) . Remember Euler's relation says ej is made up of sines and cosines. (You do remember Euler's relation??) Try plotting this wave using Mathcad or some other math program: (x) = exp(-x2/0.2) exp(10jx). modulation oscillates Try plotting the magnitude of vs. x. Also look at the real and imaginary parts. There is no “t” in the above function, so it is non-propagating: it varies in space but not in time. To make it propagating, you have to put in a time dependence. 3.5 Particle Diffraction Diffraction is a wave property. “…impossible, absolutely impossible, to explain in any classical way, and has in it the heart of quantum mechanics. In reality, it contains the only mystery.“ —Richard Feynman http://heasarc.gsfc.nasa.gov/docs/xte/ learning_center/universe/universe.html From lecture 7: “Now we have this equation that says particles have a wavelength. What are we going to do with it?” “Experiment! Find experimental verification!” Beiser presents the experimental results in the “traditional” way. It sounds like “these clever experimentalists set out to prove or disprove the hypothesis. Here’s the experiment they came up with. Sure enough, it worked.” Nonsense! If you write a text, you’ll produce a neatly-wrapped, compact package. I’d do the same. But this time the real story is worth telling. First, let me emphasize again that diffraction is something that only waves do, and only when they interact with regular, periodic arrangements of matter, e.g, a crystalline material. In 1921-1923 Davisson and Kunsman at Bell Labs were investigating secondary electron emission from metal surfaces (bombard a metal with electrons, “secondary” electrons are “kicked out”). Their work showed diffraction of electrons by a platinum surface, surface only they didn’t realize it. In early 1925, 21-year old graduate student Walter Elsasser attended a seminar in which the experiments of Davisson and Kunsman were discussed. One of Elsasser’s professors, the brilliant theoretician Max Born, tried unsuccessfully to explain Davisson and Kunsman’s results in terms of deflection of electrons by atomic electronic shells of different densities. A short time later, Elsasser was reading a paper by Einstein on the particle nature of light, which referred to de Broglie’s thesis and the hypothesis of matter waves. Elsasser read the thesis, did some calculations which showed the experiments of Davisson and Kunsman illustrated electron diffraction, and wrote a half-page paper which was favorably reviewed by Einstein, and then published. Of course, for this brilliant insight, Elsasser never won the Nobel prize for physics At the same time, Walter Davisson, unaware the goings-on in Europe, was continuing his research at Bell labs, now in collaboration with Lester Germer, and this time using a nickel target. On February 5, 1925, Germer discovered a crack in the vacuum trap of their electron scattering apparatus. This is not good. Electron scattering experiments require high vacuums and ultraclean surfaces. Loss of vacuum meant their nickel surface had oxidized. What to do? Clean the surface with hydrogen and heat it in a vacuum to “bake away” impurities. Then back to the experiment. Sign says “do not touch this apparatus.” Really. When they started up again, they noticed their data didn’t look “right.” Kind of like those results of Davisson and Kunsman a few years back. What to do? Better tear the experiment apart and examine the nickel sample. The original nickel target had been polycrystalline. They found the annealing process had partially crystallized their sample. They hypothesized their “strange” data had something to do with the arrangement of atoms in the nickel crystal. What to do now? Spend a year going over the theory of secondary electron emission, and build a new, more sensitive apparatus (after all, the experiments are extremely difficult). A year later, Davisson considered their new experimental data very uninteresting and unconvincing. So what did he do? Headed off to England for a vacation and second honeymoon. While in Oxford, Davisson attended a meeting of the British Association for the Advancement of Science. There, in a lecture by Max Born, he was surprised to hear that his (and Kunsman’s) 1923 work was cited as proof of de Broglie’s matter waves. Unknown to Davisson, a number of Europe’s best physicists had tried, unsuccessfully, to do the extremely difficult electron diffraction measurements he had carried out. His and Germer’s “uninteresting” new data created quite a stir. That was the end of the second honeymoon, because now Davisson now had to work through, and understand, the latest papers on quantum mechanics. So who were the “winners,” in the end? Davisson and G. P. Thomson shared the 1937 Nobel Prize for discovering the wave nature of the electron. G. P. is less well remembered by the historians, probably because of his more famous father, of whom we will hear before long. Germer is the “and” part of the Davisson-Germer experiment. Most of us would die happy if we achieved that kind of fame. Kunsman—I had not heard of him before doing some reading for this lecture. Elsasser was brilliant for his ability to grasp concepts; to see the “big idea.” Maybe he should have shared the Nobel Prize. He had successful careers in physics, geophysics, and biology. References: http://www.nap.edu/html/biomems/welsasser.html http://www.acolyte.co.uk/origins/DandG.html http://rodin.hep.iastate.edu/jc/321-03/ Why did I spend so much lecture time on Davisson and Germer’s experiments? There are Nobel Prizes waiting (not many, but a few) for those who are careful not to rationalize away pesky little quirks in their data. On the next slide is a summary of the Davisson-Germer experiments Unexpected “bump.” Interference produces the “bump.” 1937 Nobel prize for “discovery of electron diffraction” went to Davisson and G. P. Thomson. The Davisson-Germer result would be no big deal for x-rays (in fact, x-ray diffraction was established long before their experiment), but the difference here is that electrons are particles. This experiment demonstrated the wave-like properties of electrons. Figure 3.6, showing experimental results, is (perhaps) somewhat difficult to follow, but the equations in the text are sufficient to solve the problems. Here’s a link to a “typical” diffraction pattern. (Scroll about ¾ down the page.) That's all I'm going to lecture about this experiment. Read the text and be able to identify the experiment in a multiple choice question, and be able to work the problems.