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Matter Models (continued…) • Two puzzles remain at this point: – The wave-particle duality of light (shows both somewhat contradicting properties). – The physical basis for the Bohr model-why the fixed orbits. Actually, we need not speak of particles at all. For many experiments it is more convenient to speak of matter waves . . . The two pictures are of course mutually exclusive, because a certain thing cannot at the same time be a particle . . . and a wave . . . but the two complement each other. By playing with both pictures, by going from the one picture to the other and back again, we finally get the right impression of the strange kind of reality behind our quantum experiments.” ~ Werner Heisenberg PS 110 A Hatch Ch 15 -4 The de Broglie Hypothesis • In 1923 a struggling graduate student named Louis de Broglie proposed that moving matter has a wave property too, defined by PS 110 A Hatch Ch 15 -5 But it turned out to describe what we observe. • De Broglie’s idea explained the Bohr orbitals • The quantized orbits of the Bohr model are predicted perfectly by requiring electrons to exactly wrap 1, 2, 3, etc waves around the nucleus. wavelength = h / (mass x speed) where h = Plank’s constant = 6 x 10-34 Nobel Prize, 1929 originally hailed as the “French Comedy”… PS 110 A Hatch Ch 15 -6 PS 110 A Hatch Ch 15 -7 wavelength = h / (mass×speed) Examples where h = Plank’s constant = 6 x 10-34 Wavelength = (nonsense?) 10-38 m 60 mph 100 mph 2,000 mph Wavelength = 10-34 m (again nonsense?) Wavelength = 10-10 m Diameter of an atom… PS 110 A Hatch Ch 15 -8 If particles behave like waves ... They ought to be able to do some of the unique things waves do - like … Diffract Interfere Must get the right slit width. PS 110 A Hatch Ch 15 -9 Why don’t we observe the wave nature of matter? • Remember, to observe wave effects, your “slits” need to be similar to the wavelength wavelength = h / (mass×speed) h = 6 x 10-34 Why don’t we observe the wave nature of matter? • For all material objects except the very least massive (such as electrons and protons), the wavelength is so immeasurably small that it can be completely ignored. PS 110 A Hatch Ch 15 -10 Diffracting a pig through a doorway. PS 110 A Hatch Ch 15 -11 But what about electrons? - to diffract, need wavelength to be about same size as doorway Wavelength = Planck’s constant (mass of object)(its speed) Pig would have to move so slowly it would take a billion years for it to move through the doorway. PS 110 A Hatch Ch 15 -12 Diffraction limits how small you can see with an optical microscope • Let’s see if they diffract. We need slits about the size of the electron wavelength (10-10 m) to witness it. • How do you make a slit that small? You don’t. You use the space between atoms in a crystal. • Fire an electron beam at a crystal and we DO get diffraction rings! Electrons Video: EP #24 ARE waves! PS 110 A Hatch Ch 15 -13 Electron Microscopes • When objects are about the size of the wavelength of light,~10-6 meters, light diffracts around the object so you can’t get a clear image. • Electron wavelengths are 1/1000th the size of optical wavelengths. So using electron beams we can see 1000 times smaller with the same clarity. PS 110 A Hatch Ch 15 -14 PS 110 A Hatch Ch 15 -15 Electrons as waves boggles my mind! Two-slit experiment with a shaky machine gun: 1. With bullets Let’s do examples with bullets (real particles) water (real waves) electrons (particles or waves?) count the hits in each location Probability of hit for each slit Total probability curve P2: What if I block the right slit? PS 110 A Hatch Ch 15 -16 The concept of a probability distribution wavelength of bullet is very small PS 110 A Hatch Ch 15 -17 For waves, we can use the amplitude as a measure of where the wave “is” Probability doesn’t make sense here; the wave hits a wide area. P3: What if you block 1 hole? left slit right slit both slits PS 110 A Hatch Ch 15 -18 Experimental double slit experiment using electrons • Electrons are detected like particles, a local bright spot, but the places where they are detected show interference patterns These slits must be about as close as the size of an atom PS 110 A Hatch Ch 15 -20 PS 110 A Hatch Ch 15 -19 But what if you send the e’s through 1 at a time and detect them? Each electron chooses a different spot to hit. The cumulative effect of many electrons is the interference pattern (Davisson-Germer 1927). P4: In this experiment, which slit does a single electron go through? PS 110 A Hatch Ch 15 -21 Trying to fool mother nature What do I see if I close one hole? P5: What do I see if I detect the electrons coming through one slit or the other? Electron Detector Probability Curve Questions & Answers (?) Observation Point, behind one slit Look at a null point of the electron pattern. Q. What is seen at the observation point with both slits open? A. No electrons! P6: If I close the slit on the other side (right side in picture above) , what do I now see at this null point? Q. Now open the second slit again, what do I see? A. There are no electrons? How can it be like that?? PS 110 A Hatch Ch 15 -22 Trying to Fool Mother Nature An electron approaches two slits. It has a 25% probability of passing through the left slit, a 25% probability of passing through the right slit, and a 50% probability of hitting the barrier and thus not passing through either slit. P7: If I have a detector which detects this electron passing through the left slit, has the probability wave for this electron changed? What are the probabilities associated with it now? PS 110 A Hatch Ch 15 -23 How do we picture it? “Electrons or photons arrive in lumps, like particles, but the probability of arrival of these lumps is determined as the intensity of waves would be. It is in this sense that the electron behaves sometimes like a particle and sometimes like a wave.” Richard Feynman Nobel Prize, 1965 PS 110 A Hatch Ch 15 -24 When we look at the electron ... it acts like a particle -- otherwise like a wave. How do we “look”? Must use a detector or a probe which interacts with the electron. e.g. light … but a photon is like a bomb to an electron! -- like trying to see the trajectory of a baseball by hitting it with grenades Making the measurement affects the PS 110 A Hatch Ch 15 -26 result! PS 110 A Hatch Ch 15 -25 The results we get depend on how and what we measure. • Don’t measure which hole the electron goes through Æ wave. • Do measure which hole the electron goes through Æ particle. • How the electron behaves depends on whether it is observed. • We have found the truth; and the truth makes no sense. (G. K. Chesterton) PS 110 A Hatch Ch 15 -27 This is a little strange I remember discussions with Bohr which went through many hours till very late at night and ended almost in despair; and when at the end of the discussion I went alone for a walk in the neighboring park I repeated to myself again and again the question: Can nature possibly be as absurd as it seemed to us in these atomic experiments? ~Werner Heisenberg Probability wave • The “wave” is interpreted as being the probability of locating the particle. • It acts like a pure wave with diffraction, interference, refraction, etc. • Somehow “particle” electrons “know” about the existence of both slits even when we cannot prove that they ever go through more than one slit at a time. • Clearly we need another model. PS 110 A Hatch Ch 15 -28 PS 110 A Hatch Ch 15 -29 Probability Curve Interference Curve a graph showing the probability of finding a particle at various locations in space a probability curve from two interfering waves 0.6 0.5 0.4 0.3 0.2 0.1 y 0 PS 110 A Hatch Ch 15 -30 PS 110 A Hatch Ch 15 -31 The electron position is described with a probability wave Moving probability wave • When we measure the position, we find it at a certain position. We now know more about the electron. We refer to this as the collapse of the probability wave. PS 110 A Hatch Ch 15 -32 Schrodinger’s Equation ih ∂Ψ ( x, t ) h 2 2 = ∇ Ψ ( x, t ) + V ( x, t ) Ψ ( x, t ) 2m ∂t • I don't like it and I'm sorry I ever had anything to do with it. Schrodinger, 1926 PS 110 A Hatch Ch 15 -33 The Uncertainty Principle and waves • Back to the Bohr model. To refine this model we need to know how electrons move around atoms. • To find their trajectories we must know their position and velocity at the same time. • But how do you locate the position of a wave/particle electron? Pure sine wave Æ no position but clear frequency. Sharp pulse Æ clear position but unclear frequency. The Uncertainty Principle There is a limit to what can be known about a single electron (or other small particle). The more you know about where it is, the less you know about how fast it is moving (wavelength). The Newtonian view will not work on the small scale. It assumes you can know both position and motion…….. EXACTLY! PS 110 A Hatch Ch 15 -34 Heisenberg Uncertainty Principle • Electrons: fuzzy position and fuzzy wave properties. How fuzzy?.. PS 110 A Hatch Ch 15 -35 An Undetected Electron Where is it? We cannot say exactly. What is it doing? We cannot say exactly. We represent the electron mathematically as a wave. • ..The uncertainty in position times the uncertainty in momentum (mass x velocity) is greater than Planck’s constant. Or ∆x ∆(mv) > h Wave equation yields predictions about likelihood of finding an electron at a given position. High likelihood where the amplitude of the wave is high PS 110 A Hatch Ch 15 -36 PS 110 A Hatch Ch 15 -37 Consequence of WaveWaveParticle Duality Consequences • If we try to find out where an electron is, we know less about where it was going. – Measuring position more accurately makes uncertainty in momentum larger. More waves with different momemtums. This is an alternative explanation for electron diffraction. • Because of the wave nature of electrons, we CANNOT know how they move around atoms. [This is not a measurement problem, but a statement of fundamental limitations in nature.] PS 110 A Hatch Ch 15 -38 Cannot tell precisely what a single, unwatched electron will do. Best we can do is talk about probabilities – then true for many electrons. (Analogous to coin flips--we cannot predict accurately the outcome of one coin flip, but we can predict the outcome of 1,000,000 flips) PS 110 A Hatch Ch 15 -39 In other words… Conclusion • We can predict this interaction perfectly using Newton’s Laws of motion. 8 • We cannot predict the results of this interaction perfectly. We can give probabilities that certain outcomes will happen. ? ? ? PS 110 A Hatch Ch 15 -40 The conclusion to be drawn from the electron experiment is that electrons, like photons, arrive in lumps, like particles, but the probability of arrival of these lumps is determined as the intensity of waves would be. Electrons have both a wave and a particle aspect to their nature. PS 110 A Hatch Ch 15 -41 ? So does the electron “know” where it is and where it is going? • Quantum mechanics experiments demonstrate that there is fundamental uncertainty in nature. It is not a matter of the experimentalist not being clever enough to measure both position and momentum at the same time. A particle simply cannot have an exact position and an exact momentum at the same time. Duality of Matter: Summary 1. Matter has both wave and particle properties. 2. Position and speed, as a pair, are uncertain. 3. Without making an observation there is only a probability of where the particle is. 4. Observation affects the results of the experiment. The wave function “collapses”. 5. The probability of finding a particle is only seen when working with a large number of particles. PS 110 A Hatch Ch 15 -42 Duality of Matter: Summary (cont) 6. Reality is caused by observation. 7. The universe seems to be a universe of chance, and one must work with large numbers. 8. The chance is not a result of some underlying structure (hidden variables). 9. These considerations apply only at atomic level. 10. Quantum mechanics does give Newton’s laws at the larger scale. PS 110 A Hatch Ch 15 -44 PS 110 A Hatch Ch 15 -43