Quantum Mechanics Study Sheet: Understanding the Weird World of Particles and Waves, Papers of Humanities

Download Quantum Mechanics Study Sheet: Understanding the Weird World of Particles and Waves and more Papers Humanities in PDF only on Docsity! IS 151 Quantum Mechanics Study Sheet Generally, classical mechanics adequately describes macroscopic (as opposed to microscopic) objects and phenomena. There are exceptions though, especially superconductivity, in which electrons can pair into “Cooper pairs” and flow with zero resistance, and superfluidity, in which liquid helium flows with zero viscosity – compared with it, water is like syrup! Superfluid helium has other bizarre properties. Water in a lake behind a dam cannot flow downhill into the river below because it would have to flow uphill to get over the dam, and that would require energy it does not have. (It can if a siphon goes over the dam, but only if there is water in the siphon – the potential energy the water at the top of the siphon gives up when it flows downhill is partially used to suck up new water into the siphon.) Superfluid helium can flow over dams even without the siphon. This is an example of the quantum-mechanical effect called “tunneling”. Interestingly, even a neutron star is a kind of superfluid of neutrons. In quantum mechanics, objects act like both particles and waves. Loosely speaking, the probabilities one encounters in quantum mechanics are due to the wave properties. The two-slit experiment is an example: the probability that a particle (photons, electrons, protons, neutrons, and atoms have all been used) will be found at any specific point is given by the diffraction pattern that can be calculated by treating the light (or electrons, etc.) as waves. Some of the weirdest things that happen in quantum mechanics are the result of the superposition of states. The classic example is “Schroedinger’s Cat”. The experiment is as follows: Take a small sample of radioactive material, pre-selected so that in one hour the probability of observing a decay with a Geiger counter is exactly one half. (The nuclear decay that causes radioactivity is a quantum event.) Put a cat you don’t like in an opaque, sound-proof box, so that you have no way of measuring what’s going on inside. Put the radioactive material in the box with a Geiger counter hooked up to an electric hammer that will break a flask of poison gas if a nuclear decay is observed (one “click”). Wait one hour. Is the cat now alive or dead? In classical mechanics, which is how we are taught by experience to think, the cat is either in state A (alive) or D (dead). In quantum mechanics, though, it can be in a superposition of these two states, such as (A + B) or (A - B). When a measurement is made to determine whether the cat is alive or is dead, the state of the cat changes to either state A or D, no matter what it was before the measurement. This is called the collapse of the wave function, because waves are associated with probabilities. Opening the box to see if the cat is alive cannot detect a state like (A + B) or (A - B), but there are other experiments that can. This experiment has never been performed with a cat, of course; it is just a “thought experiment” to help understand the ideas. However, atomic-scale experiments see the same kind of quantum weirdness – for example, see http://www.sciencedaily.com/releases/2000/01/000120073305.htm. Another example of a two-state system is provided by silver (Ag) atoms in the Stern-Gerlach experiment. In this experiment, silver atoms are heated in an oven to a high temperature, so that they have a large kinetic energy, and are shot in a straight line to a configuration of magnets. The silver atom has 47 electrons, but only the outermost gives it a magnetic moment; the motions of the others cancel each other out. The magnets are not the same shape, which is necessary to create a magnetic field that is not the same strength on both sides. This causes a force that depends on which way the magnetic moment points. In classical mechanics, the magnetic moment can be expected to come out of the oven pointing in any direction, and the probability of being in one orientation is the same as in any other orientation. This would lead to a single, broad distribution in the intensity of electrons hitting the screen. In quantum mechanics, however, the electron can only point either “up” (U) or “down” (D) when compared with the orientation of the magnets. Instead of a single, broad distribution, there are two sharp peaks. The “wave function collapses” onto these two values, as with the case of Schroedinger’s cat. 1 IS 151 Quantum Mechanics Study Sheet The really strange thing comes when more than one of these is used. Suppose that we make an initial measurement with the magnets aligned in the z-direction (up and down). Then the wave function will collapse into either the state UZ or DZ. Now if we take the atoms that were measured to be in UZ state and again pass them through a pair of magnets aligned in the z-direction, we will observe them to be in the same UZ state – no big surprise. However, suppose we align the second set of magnets in the y- direction (into and out of the paper). The wave function will now collapse into either state UY or state DY. Since no information had been taken about how much of the moment pointed in the y-direction, even in classical physics it’s not surprising that, of the atoms in the UZ state, half will be measured as UY and half as DY. The bizarre thing is that if you then take the atoms out of the y- measurement and again make a measurement using magnets aligned in the z-direction, only half will collapse into the UZ state, even though they all were in that state before the y-measurement was made!!! Making the y-measurement has destroyed information about the z- measurement. The z-orientation of the atom and the y-orientation are called incompatible observables. Position and velocity are also incompatible observables, and they have a similar problem: the Heisenberg Uncertainty Principle. In this case, it is even more interesting; it can be shown that UY = (UZ + DZ) and DY = (UZ - DZ). Incompatible observables are contrary to our everyday experience. It is as though you sorted eggs by weight, keeping only those that weigh 6 ounces. You could measure them again, and the weight would still be 6 ounces. You then sort the 6-ounce eggs by color, and keep the brown ones. But when you weigh them this time, some of them weigh 8 ounces! These properties can be very useful for quantum cryptography. Cryptography is the science of making secret codes. Current methods of passing secret information depend on the fact that it is easy to multiply two large prime numbers together, but very difficult to take the product and determine which two numbers were multiplied. However, quantum computers, if they can be perfected, will be able to factor numbers very efficiently, and will make current encryption useless. For more details, see “From Quantum Cheating to Quantum Security”, by Daniel Gottesman and Hoi-Kwong Lo, Physics Today, Nov. 2000, pp. 22—27. The only sure-fire way to pass a secret code is to incorporate real random data that both the sender, “Alice”, and the receiver, “Bob”, know. This is called the “one-time cipher pad” method. The problem is not for Alice to get random bits (1’s and 0’s, in which data are encoded) – that part’s easy – but in transmitting them securely to the Bob. The problem is solved by using quantum-mechanical bits: q-bits. The way she does it is this: she randomly chooses both a 1 or a 0 (U or D) and the alignments of magnets (y or z). She transmits the q-bits (which can be carried by atoms, but usually either photons or electrons are used) to Bob without telling him how the magnets are aligned. He randomly chooses his orientation of magnets. Both Alice and Bob randomly determine these random orientations independently for each q-bit. After the transmission is finished, Alice reveals how she had aligned her magnets. Bob can use the 0’s and 1’s only from that set of data in which he has guessed right. If someone else, “Eve”, has been eavesdropping, she will have had to guess the orientations (y and z), just like Bob. Suppose Alice has transmitted a 1 q-bit with a y-orientation, Bob has guessed right about the orientation, and Eve has guessed wrong. Eve measures the orientation with the magnets aligned in the z-direction, and will collapse the wave function onto either UZ or DZ. She then retransmits the particle in the state she 2 IS 151 Quantum Mechanics Study Sheet 12. The state U+D is a) a superposition of the states U and D b) an entanglement of the states U and D c) a collapse of the states U and D d) meaningless, because U and D are incompatible observables 13. The experiment that splits silver atoms into the U and D is called a) the Heisenberg experiment b) the EPR experiment c) the Stern-Gerlach experiment d) the Schroedinger’s Cat experiment 14. Classical mechanics predicts that the silver atoms in the Stern-Gerlach experiment should a) form a single, broad distribution of deflections, with most atoms not being deflected b) form two sharp peaks in the distribution of deflections c) form 5 sharp peaks in the distribution of deflections, making a cross-shaped pattern d) be unaffected by the magnets 15. Quantum mechanics predicts that the silver atoms in the Stern-Gerlach experiment should a) form a single, broad distribution of deflections, with most atoms not being deflected b) form two sharp peaks in the distribution of deflections c) form 5 sharp peaks in the distribution of deflections, making a cross-shaped pattern d) be unaffected by the magnets 16. A silver atom is measured to be in the UZ state, after which it is measured in the DX state. Another measurement with the magnets in the z-direction will reveal a) the atom is definitely still in the UZ state b) the atom is definitely now in the DZ state c) the atom will be found in either the UZ state or the DZ state with 50/50 odds d) the atom will be found in the state (UZ-DZ) 17. The unit of data in quantum information theory is the a) bit b) q-bit c) gambit d) photon 18. Quantum cryptography is most useful for a) breaking the most common type of secret code used today b) directly sending messages without the possibility of error c) directly sending random data without the possibility of error d) directly sending random data with the ability to detect eavesdroppers 19. Einstein did not like quantum mechanics because it violated a) causality b) local realism c) Maxwell’s equations d) the second law of thermodynamics 20. Antimatter a) is purely a product of science fiction, like Godzilla and light sabers b) is the subject of pseudoscience and is not taken seriously by real scientists c) was thought to be observed in the 1930’s, but the observations were later shown to be an example of pathological science d) is real and is in fact used in high-energy particle colliders 21. P.A.M. Dirac proposed that antimatter electrons be thought of as a) ordinary electrons moving backwards in time b) ordinary electrons moving in some extra-dimensional “mirror space” c) corresponding to “holes” in semiconductors d) protons that have lost most of their mass 5 IS 151 Quantum Mechanics Study Sheet 22. Richard Feynman proposed that antimatter electrons be thought of as a) ordinary electrons moving backwards in time b) ordinary electrons moving in some extra-dimensional “mirror space” c) corresponding to “holes” in semiconductors d) protons that have lost most of their mass 23. Modern physics, including the contributions of the Theory of Relativity and Quantum Mechanics, says a) the future is uncertain; we can only calculate probable futures b) the past is uncertain; we have to “sum up” over all possible histories c) the present is uncertain; signals from elsewhere cannot reach us instantaneously d) all of the above 24. A proton is made up primarily of a) two up quarks, a down quark, and gluons b) two top quarks, a bottom quark, and gluons c) two up quarks and a down quark only d) two top quarks and a bottom quark only 25. The charge on an up quark is ___ times the absolute value of the charge on an electron. a) +1 b) +2/3 c) 0 d) -1/3 26. The charge on a bottom quark is ___ times the absolute value of the charge on an electron. a) +1 b) +2/3 c) 0 d) -1/3 For problem 27, note that the textbook is wrong. 27. The top quark was first observed in a) 1974 at SLAC, together with the charm, strange, and bottom quarks b) 1977 at Fermilab, as the partner to the bottom quark c) 1995 at Fermilab; the delay was because it has a much larger mass than any other quark d) It has still not been observed, but is expected to be “seen” by 2010 28. Quarks always combine in combinations that have no net a) charge b) “color” c) angular momentum d) all of the above 29. The Pauli Exclusion Principle, “No two objects can simultaneously occupy the same quantum state,” applies to a) all quantum particles b) groups of identical fermions c) groups of identical bosons d) groups of fermions, regardless of whether or not they are of the same type 30. The unusual particles that do not interact via either the electromagnetic nor the strong nuclear force are known as a) neutrinos b) neutrons c) strange quarks d) muons 31. A muon is a sort of a) heavy proton b) heavy electron c) heavy neutron d) heavy neutrino 6 IS 151 Quantum Mechanics Study Sheet 32. The mass of the neutrino has recently been found to be a) zero b) greater than zero, but much less than the mass of the electron c) between the mass of the electron and the mass of the proton d) greater than the mass of the proton 33. The “force” that keeps a neutron star from collapsing is a) due to the Heisenberg Uncertainty Principle b) due to the Pauli Exclusion Principle c) the strong nuclear force d) pressure from neutrinos 34. The hypothetical particle that gives mass to other particles is called the a) Higgs boson b) isaacnewton c) gluon d) top quark 35. “Supersymmetry” is the idea that a) for every fermion there is a corresponding boson b) for every “ordinary” particle there is a corresponding antiparticle c) there exists a set of “mirror dimensions” that are symmetric to our own and contain “dark matter” d) the proton is the antiparticle of the electron 36. A major paradigm shift in Bohr’s theory of the atom was that a) the accelerating electron (in circular orbit) did not emit energy b) the electron could only orbit the atom at specific distances (energies) c) the electon “jumps” between allowed obits, with no intermediate position, when it absorbs or emits a photon d) all of the above 7