particle in a box the uncertainty principle

1. particle in a box the uncertainty principle "As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.“—A. Einstein Reminder: Exam 1 is one week from today! Do you want Quiz 3 on Wednesday or Friday?

2. There is a homework problem on diffraction (if I remember correctly). The equations you need to work the problems are not “new” ones, but I’ll give you them to you as a reminder.

3. Confession A few years ago, for dramatic emphasis, I libelously suggested a student might have been responsible for the mishap that oxidized Davisson and Germer’s nickel sample. In early 1925, a bottle of liquid air exploded in their lab, shattering the glass container containing their nickel sample and causing it to oxidize. Liquid air is very unstable, and doesn’t need human intervention to explode! Hmm…isn’t liquid nitrogen a lot like liquid air? Yes! Don’t let someone sell you cheap liquid air when you want liquid nitrogen!

4. 3.6 Particle in a Box Now you believe that particles “have” waves, right? Could we digress a minute. Just what does this mean? Is the particle the only reality, and the wave just something physicists invented? —ruled out by experiment—matter waves shown to interfere Is it as Schrödinger first believed—the wave is real, not the particle? —abandoned for many reasons, including the fact that unlike waves, particles do not “spread out”

5. Is the particle real, and the wave something that “guides” it (a “pilot wave”)? —no successful theory was found based on this idea Is there some single reality which sometimes manifests itself as particle and sometimes as wave but is really both at once? —this has come to be the commonly accepted view—but it is possible the whole story has not yet been told Going much deeper into this discussion leads into philosophy and away from physics, so back to the physics… Oh…why study a particle in a box…

6. Anyway, supposing that you believe that particles have a wave nature—and experiment certainly backs this up—let’s pursue some of the consequences. Remember your Physics 23 vibrating string experiment. You couldn't set up arbitrary vibrations of the string. Instead, you saw patterns like this. You get standing waves because the wave pulses traveling down the string reflect (and change phase by 180°) when they reach the ends. The standing wave consists of a series of pulses moving up and down the string. These pulses superpose, and when they are in phase, you see maxima. When they are out of phase, you see nodes.

7. You see standing waves only when the pulse speeds are “just right,” and the standing wave “fits” on the length of the string. Here’s a static picture: (http://www.chem.uci.edu/education/undergrad_pgm/applets/dwell/dwell.htm) dead link January 2005—but I only used it for the picture anyway What does this have to do with a particle in a box? Place a particle, represented by a wave, in a box of length L. The particle wave moves “with” the particle, reflects when it hits the wall of the box, and then again when it hits the other end of the box.

8. If the box is small enough (compared to the particle wavelength), the particle wave "folds up" over and over again every time it reflects off a wall. Here’s the “best” visualization I could find:* http://www.zbp.univie.ac.at/schrodinger/ewellenmechanik/simulation.htm The segments of the particle wave bouncing back and forth interfere. If they interfere constructively, our particle can fit in the box. Otherwise -- no particle fits in the box. *Caution! Intense material! Adult content! Shows death of wave!

9. The folding of the particle wave packet to add in phase works when the length of the box is an integral number of half wavelengths of the particle wave (L=n/2), hence equation 3.17 in Beiser for the de Broglie wavelengths of a trapped particle: Because* KE = mv2/2 and  = h/mv, the restriction on  also places a restriction on the allowed particle energies: *So this is a nonrelativistic calculation.

10. The permitted energies are called "energy levels" and the number n is called a "quantum number." Think of the box as a potential well, inside which a particle is placed. A free particle, outside the box, can have any energy, and any wavelength. When you put the particle in the box, only certain wavelengths and energies are allowed (and note that zero is not one of the allowed energies). You most likely will have to add or remove energy to put a free particle into a box.

11. Comment: “quantum” implies something is quantized (energy, momentum). Quantization of properties of matter is a consequence of the wave nature of matter. Thus, the words “wave” and “quantum” are closely associated in my vocabulary. All of us in this room have discrete energy levels and quantum states. However, as the example on page 107 shows, there are enough energy levels to form, for all practical purposes, the continuum that Newtonian mechanics supposes.

12. 10 gram marble in a box 10 cm wide: Minimum energy and speed are indistinguishable from zero, and a marble of reasonable speed has a quantum number on the order of 1030. In other words, we can't perceive the quantum features of the marble in the box.

13. Electron in a “box” 0.1 nm (10-10 m) wide (the size of an atom): The minimum energy is 38 eV, a significant amount, and the energy levels are far enough apart to be measurable.

14. Is this information useful, or is it just physics trash talk? • Quantum well lasers (183000 pages found in Google search, September 2002): • http://nsr.mij.mrs.org/3/1/ • http://www.iis.ee.ethz.ch/research/tcad/laser_sim.en.html • http://www.chem.wisc.edu/courses/801/Spring00/Ch1_3.html • Quantum well memory (209000 pages found in Google search, September 2002): • http://www.physicstoday.org/pt/vol-54/iss-5/p46.html • http://www.sciencenews.org/sn_arc99/2_27_99/fob5.htm • http://theorie5.physik.unibas.ch/qcomp/qcomp.html OK, so maybe we should pay attention to this wave/quantum stuff. Is there anything else important “hidden” in the wave nature of particles? Quantum well lasers, 388000 pages Sept. 2003; 450000 pages Jan. 2005. Quantum well memory, also 388000 pages Sept. 2003; 1100000 pages jan. 2005.

15. 3.7 Uncertainty Principle I -- derivation based on the wave properties of particles Consider the particle represented by this wave group. Where is the particle? What is its wavelength? The position is well-defined. But the wavelength is poorly defined, and therefore there is a large uncertainty in the particle’s momentum (remember--wavelength and momentum are related). “Life is uncertain. Eat dessert first.” –UMR Prof. Emeritus D. M. Sparlin

16. Now consider the particle represented by this wave group. Where is the particle? What is its wavelength? The wavelength seems to be rather well-defined, but the position is poorly defined. There is a large uncertainty in the particle’s position. To quantify the uncertainties in the wave group's position and momentum, we need to go into much more detail about Fourier transforms and representation of wave groups by summations of individual waves.

17. Beiser does this on pages 108-111. Please read this material. You may be tested or quizzed on major concepts (but not “trivial” details). What I want you to know (backwards and forwards), comes out of this derivation, and is called Heisenberg’s* Uncertainty Principle: It is not possible to simultaneously measure, with arbitrary precision, both the position and momentum of a particle. *1932 Nobel Prize for creation of quantum mechanics.

18. The quantity h/2 appears over and over again in modern physics, so we give it a special symbol: ħ = h /2. The uncertainty principle can then be written There are fundamental limits on how precisely we can simultaneously measure a particle's position and momentum. Because of the wave nature of matter, there are fundamental limits on how precisely we can know things. These limits have nothing to do with our measurement techniques; they are built into nature. “Marvelous what ideas young people have these days. But I don’t believe a word of it.”—A. Einstein, referring to the uncertainty principle

19. Example 3.6 A measurement establishes the position of a proton with an accuracy of 1.00x10-11 m. Find the uncertainty in the proton’s position 1s later. Assume v << c. At the time of measurement, the position uncertainty is x1, and it’s OK to do this for vc

20. A time t later, the position uncertainty x2 is About 1/10 the size of an atom, but much larger than a nucleus. Is vc? To put it bluntly, we have no clue where the proton is 1 second later. The proton didn’t spread out, because it is “somewhere,” but its wave certainly did!

21. 3.8 Uncertainty Principle II -- derivation based on the particle properties of waves* I claimed above that the limits implied by the uncertainty principle are fundamental to nature, and are due to the wave properties of matter. This follows cleanly and logically from the mathematics of waves. As humans, we are left with nagging doubts about the uncertainty principle. How dare nature tell us there are things we cannot know! Surely this is just technical glitch that human cleverness can overcome. Heisenberg (although a theorist first, last, and always) believed he had to specify “definite experiments” for measuring an object’s position in order to validate the uncertainty principle. *Caution: reading this section may be hazardous to your grade!

22. Heisenberg proposed a thought experiment (which can be realized in fact): let’s suppose we want to measure the position of this electron very precisely. How do we do it? The sphere doesn’t represent the size of the electron; it represents the size of the region in which we wish to locate the electron. Visible light? A “real” red light wave would have a much longer wavelength than this! The wavelength of visible light is far too large to allow us to detect the position of the electron. The wavelengthneeds to be comparable to the position precision we seek. You might say the electron is somewhere along the wave, but the wavelength is so long that the imprecision in position is enormous.

23. The last sentence contains a clue: find some kind of radiation that has a much shorter wavelength. Gamma rays have short wavelengths. They should work. But short wavelength gamma radiation carries lots of energy and momentum. http://www.aip.org/history/heisenberg/p08b.htm Our gamma-ray microscope can tell us where the electron was, but it can’t tell us where it came from or where it is heading (its momentum). So we can forget about position (but measure momentum), or forget about momentum (but measure position).

24. I find this an interesting thought experiment, but it implies that the uncertainty principle is really only a measurement difficulty after all. A surprising number of students tell me on tests (through their answers to multiple choice questions) that the uncertainty principle is just a result of experimental difficulties. No!------------------------------------------------------------------------ -------------------------------------No!----------------------------------- ----------------------------------------------------------------------- No! “The indeterminacy (of position and momentum) is inherent in the nature of a moving body.”

25. Let me repeat: Heisenberg’s thought experiment is “bad” because it implies the uncertainty is merely some technical measurement difficulty.* 3.9 Applying the Uncertainty Principle Planck's constant is so small that we never encounter the uncertainty principle in Newtonian mechanics… …but its consequences are manifested in materials we constantly use in everyday life! You’ll hear about it repeatedly in this course. *To placate his critics and get his uncertainty principle paper published, Heisenberg had to include this thought experiment in it. The thought experiment has appeared in (probably) most texts, and confused thousands (millions?) of students. All to get a paper published.

26. Frequency and time are related, and velocity and energy are related, so we can derive an alternate expression of the uncertainty principle: It is not possible to simultaneously measure, with arbitrary precision, both a time for an event and the energy associated with that event. “Prediction is very difficult. Especially about the future.”—Neils Bohr and/or Mark Twain (we are not certain who said this)

27. http://www.nearingzero.net

28. Example 3.7 A typical atomic nucleus is about 5x10-15 m in radius. Use the uncertainty principle to place a lower limit on the energy an electron must have if it is to be part of a nucleus. The problem is asking you something about the energy of an electron confined to a region 5x1015 m in size. Obviously, the starting point is NOT! You are given information about the electron’s x. In fact, you are implicitly told to use x = 5x10-15 m. You must use

29. Proceeding trustingly with the math… You could plug numbers in now. But let’s keep it symbolic and think for a bit. We’ve calculated a momentum uncertainty p. Does it make sense to claim the electron has less momentum than p?

30. Suppose all UMR students measure the diameter of the puck by the University Center. Suppose all UMR students measure the diameter of the puck by the University Center. Suppose Physics 107 students also make the measurement. These measurements all have uncertainties. Suppose we calculate the difference between the average of all UMR student results and the average of the Physics 107 student results, and find that the difference is -0.002  0.007 m. What can we conclude? There is no difference between the “overall” average and the “Physics 107” average.

31. In other words, a result of -0.002  0.007 m is consistent with zero. What would be the minimum difference that you would consider not consistent with zero? A statistician would probably say 3, or maybe 3*0.007 = 0.021. Ignoring the mathematicians,* because they have lots of strange ideas, most of us could probably agree that if the result is x  x, then x had better be at least as big as x; otherwise x is consistent with zero. A statistician would probably say 3, or maybe 3*0.007 = 0.021. *“Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.”—A. Einstein

32. This discussion was designed to get you to agree with the statement that if an electron has a momentum uncertainty given by then it doesn’t make sense to talk about a momentum for the electron which is less than px.

33. If we agree that then the minimum electron momentum is Classically, KE = p2 / 2m, so* What’s this business about calculating KE. Didn’t the problem ask for “energy?” Doesn’t that mean E? Well… in this context the problem was really asking for kinetic energy. Ask me if you are unsure about which energy an exam question is asking for.

34. So the minimum electron (kinetic) energy is Units work out correctly if you keep everything in SI. However, I encourage you to show units in your calculations on exams, to prevent mistakes and help me give you more points if you make a simple math error.

35. So, KE=6.11x10-11 joules. Any comments? Hey! Isn’t that kind of a big energy? I mean for an electron? 6.11x10-11 joules x 1 eV / (1.6x10-19 joules) = 3.82x108 eV. 3.82x108 eV = 3.82x102x106 eV = 382 MeV. Now what’s that electron “rest mass” again… 0.511 MeV/c2. KE = p2 / 2m

36. In case you weren’t in class (physically or otherwise), that big red x on that last slide means that the previous 2 or 3 slides worth of classical energy calculation was wrong.

37. For extremely high speeds and energies, pc >> mc2 so ≈ 0 Remind me again: how’d the = sign slip in? p Here we go again replacing KE by E, except for this large an energy, KE≈E.

38. Four things to notice: • The classical calculation far overestimated the kinetic energy (makes sense—classical calculations can come up with speeds greater than c). • The kinetic energy is about 20 MeV, which is much much greater than 0.511 Mev, so E ≈ pc was a reasonable approximation. • The “p” in the uncertainty principle equation is the relativistic momentum. •  If you want to confine an electron to a nucleus, its wave nature requires that it have at least 20 MeV of energy. This concludes (finally) example 3.7. Example 3.8 is similar except that a nonrelativistic calculation is OK.

39. Example 3.9 An “excited” atom gives up its excess energy by emitting a photon, as described in chapter 4. The average period that elapses between the excitation of the atom and the time it radiates is 10-8 s. Find the inherent uncertainty in the frequency of the photon. We have time, want f, but E and f are related, so Beiser uses the ħ version; you’ll see why I use this in a minute. By not calculating numerical values right away, I simplified the math!

40. If you measure the intensity vs. frequency of the light emitted from this atom, the spectrum will have at least this intrinsic linewidth. Application: it is usually desirable to have laser lines be very “sharp,” i.e., the laser emits only a single color of light. The width of the laser line to the right depends on the design of the laser, but not even the cleverest design can produce a line narrower than that given by the uncertainty principle. intensity frequency