Understanding Particle in a Box & Uncertainty Principle in Modern Physics, Slides of Physics

Download Understanding Particle in a Box & Uncertainty Principle in Modern Physics and more Slides Physics in PDF only on Docsity! Review of Modern Physics Zeroeth Semester Lecture Nine: Docsity.com 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 Docsity.com 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 of 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. Docsity.com 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: 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. Docsity.com 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! Docsity.com 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. 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. Docsity.com 10 gram marble in a box 10 cm wide: 2 2 n 2 n h E = 8mL      22 -34 n 2-3 -1 n 6.63×10 E = 8 10×10 10 -64 2 nE = 5.5×10 n Joules 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. Docsity.com Electron in a ―box‖ 0.1 nm (10-10 m) wide (the size of an atom): 2 2 n 2 n h E = 8mL      22 -34 n 2-31 -10 n 6.63×10 E = 8 9.11×10 10 -18 2 2 nE = 6.0×10 n Joules = 38 n eV The minimum energy is 38 eV, a significant amount, and the energy levels are far enough apart to be measurable. Docsity.com 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. Docsity.com 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: x h ΔxΔp 4   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. Docsity.com 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 .xΔxΔp 2 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 Docsity.com 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! Docsity.com 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? Visible light? The wavelength of visible light is far too large to allow us to detect the position of the electron. The wavelength needs 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. 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. A ―real‖ red light wave would have a much longer wavelength than this! Docsity.com 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). Docsity.com Frequency and time are related, and velocity and energy are related, so we can derive an alternate expression of the uncertainty principle: π h ΔEΔt 4  .ΔEΔt 2  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) Docsity.com Photon self-identity problems. http ://www.nearingzero.net 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 .ΔEΔt 2  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 .xΔxΔp 2  Docsity.com 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. *“Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.”—A. Einstein 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. Docsity.com ,xΔp 2 Δx  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. Docsity.com If we agree that ,x,min xp = Δp 2 Δx then the minimum electron momentum is .xp = 2 Δx Classically, KE = p2 / 2m, so*     = . 2 2 2 x 2 p 2 Δx KE = = 2m 2m 8 m Δx       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. Docsity.com 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. Docsity.com For extremely high speeds and energies, pc >> mc2 so   2 2 2 2 2E = mc +p c ≈ 0 c E = 2 Δx       -34 8 -15 1.055×10 3×10 E = 2 5×10 -12E = 3.17×10 joules = 19.8 MeV Here we go again replacing KE by E, except for this large an energy, KE≈E. Remind me again: how’d the = sign slip in? p Docsity.com 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. Docsity.com