Name Final Exam December 7, 015 This test consists of five parts. Please note that in parts II through V, you can skip one question of those offered. Part I: Multiple Choice (mixed new and review questions) [50 points] For each question, choose the best answer ( points each) 1. Suppose you have a barrier potential with a finite but large width L but height V0 > E, the energy. On the other side of the barrier, what does the wave look like? A) There is no wave at all, because the barrier is too high to get across B) The wave always penetrates the barrier, so it has the same magnitude as it had before C) The wave continues as a wave, but the magnitude is slightly reduced D) The wave damps exponentially, falling off to zero eventually E) The wave has a very small chance of penetrating, coming out as a much smaller amplitude wave. Most of the mass of a typical galaxy is apparently made of A) Stars B) Gas C) Dust D) Dark matter E) Black Holes 3. According to Einstein s equations, the curvature of spacetime is related to the A) Presence of energy, momentum, etc., that makes up the stress-energy-momentum tensor B) Gravitational pull from nearby objects C) Motion of test masses present at that location D) Integral of mass over the nearby space divided by the distance E) Number of geodesics passing through a given point 4. If observer A goes to a distant star and returns at high velocity, while observer B remains on Earth, when they are reunited, how will their relative ages compare? A) Observer A will be younger B) Observer B will be younger C) They will be the same age D) It depends on which one you ask: each one will claim to be the younger E) It depends on which one you ask: each one will claim to be the older 5. What was the nature of the ultraviolet catastrophe that Planck resolved? What was wrong with the classical prediction for black body radiation? A) It had too much energy at low frequencies/long wavelengths B) It had too little energy at low frequencies/long wavelengths C) It had too much energy at high frequencies/short wavelengths D) It had too little energy at high frequencies/short wavelengths E) None of the above
6. If the wave function at a point x is given by, the probability density of finding the particle at x is given by * * * A) B) C) D) E) 7. The best way to spot atomic hydrogen, and to map out the spiral arms of a galaxy, is by A) The X-rays caused by collisions of high temperature atoms B) The radio waves from vibrations of molecules C) The 1 cm line caused by the spin flip of the electron D) The glow of stars that have formed from atomic hydrogen E) The infrared light that can make it through the dust clouds 8. Suppose you took 1 kg of water and heated it from 50 C to 100 C. According to special relativity, would its mass change? A) No B) Yes; it would increase a tiny bit C) Yes; it would increase a lot D) Yes; it would decrease a tiny bit E) Yes; it would decrease a lot 9. Which of the following is true when you are near a mass, such as the Earth? A) Masses become greater B) Masses become smaller C) Time speeds up D) Time slows down E) None of the above 10. Which of the following particles experiences the strong force? A) Protons (only) B) Neutrons (only) C) Electrons (only) D) Protons and neutrons, but not electrons E) Protons, neutrons, and electrons 11. When you place an object in an infinite square well of finite width, the minimum possible energy it can have, according to quantum mechanics, is A) Positive B) Negative C) Zero D) Infinity E) Negative infinity 1. In a galaxy like our own, the most likely place to find new stars being born is in the A) Disk B) Nucleus C) Halo D) Bulge E) None of these 13. Which of the following is not a standard categorization of galaxies? A) Spindle B) Spiral C) Elliptical D) Barred Spiral E) Elliptical
14. Which of the following is the correct four-dimensional distance formula in special relativity? A) s x y z B) s x y z C) s x y z c t D) s x y z c t E) s x y z c t 15. According to quantum mechanics, an object with momentum p will have wavelength A) p B) p C) hp D) p h E) h p 16. At the center of many galaxies, including our own, lies a giant A) Dust cloud B) Star C) Molecular cloud D) Neutron star E) Black hole 17. There are two types of -decay. The visible particle that comes out in the two types is one of which two particles? A) Proton or electron B) Proton or neutron C) Electron or neutron D) Proton or anti-electron (positron) E) Electron or anti-electron (positron) 18. Curvature is a measure of A) How much your coordinates are curved compared to straight coordinates B) How much spacetime is actually curved C) Whether you are using Cartesian or spherical coordinates D) How much matter there is locally E) The amount of gravity present 19. When studying the wave function for the ground state of the Harmonic oscillator, it was Ax found that e satisfies Schrödinger s equation, but ultimately we rejected e because A) It led to infinite energy B) It was not continuous C) It was not normalizable Ax D) It is equivalent to e, and it is easier to work with the minus sign E) It was not computable Ax 0. A geodesic in four dimensions is a path through spacetime that
A) Stays at constant space coordinates while time changes B) The coordinates change at a constant speed with respect to time C) Has the longest proper time between two events D) Is not curved in any coordinate system E) Changes direction only when encountering a mass 1. Which of the following is not necessarily true in special relativity? A) Nothing can go faster than light B) Light in vacuum always goes at the speed of light C) Energy is conserved D) Momentum is conserved E) Mass is conserved. Which type of nuclear decay can only happen when a nucleus starts off in an excited state? A) + B) C) D) E) Electron capture 3. Which of the following objects is not believed to be subject to quantum mechanics? A) Electrons B) Nuclei C) Atoms D) Molecules E) All objects are affected by quantum mechanics 4. For a heavy but fairly stable nucleus, what fraction of the nucleons will typically be protons? A) 30% B) 40% C) 50% D) 60% E) 70% 5. According to the equivalence principle, masses in a gravitational field experience much the same thing as A) Charges in an electric field B) Masses as viewed in an accelerating reference frame C) Objects being pulled by an external force D) Waves interacting with a background potential E) Matter absorbing momentum from impacting photons
Part II: Short answer (review material) [0 points] Choose two of the following three questions and give a short answer (1-3 sentences) (10 points each). 6. In one version of the barn and pole paradox, a runner carrying a pole insists that the pole is too long to fit inside a barn, while a farmer standing nearby insists that he can close the front and back of the pole in the barn at the same moment, proving the pole fit inside. Explain the resolution of this apparent paradox. 7. A hydrogen atom has a nucleus with positive charge and an electron with negative charge. Why doesn t the electron simply go to the center of the atom (x = 0) and stay there (v = 0), according to quantum mechanics? Your answer should have at least one equation or inequality in it. 8. For a hydrogen-like atom with a single electron, the electron is described by four numbers, n, l, m, and ms. Explain what physical quantity each of these numbers describes about the electron. You do not have to give any equations.
Part III: Short answer (new material) [30 points] Choose three of the following four questions and give a short answer (1-3 sentences) (10 points each). 9. According to the modern theory of electromagnetism, electric attractions are caused by the interchange of photons. What particles are interchanged in strong interactions that hold the nucleus together? Explain qualitatively how this results in a force that is short range rather than long range. 30. List the following four types of radioactive decay from most to least dangerous: alpha, beta, gamma, and electron capture. 31. Explain qualitatively or in pictures what causes a blazar, which can vary in intensity quickly and is very bright, to be different from other types of active galactic nuclei. 3. Explain approximately what a gravity wave is. Although they have not been detected directly, explain how studying pulsars has given us indirect evidence that they do exist.
Part IV: Calculation (review material) [40 points] Choose two of the following three questions and perform the indicated calculations (0 points each) 33. An atom of the isotope 64 Cu, when at rest, can be thought of as a sphere of diameter of approximately 90 pm, and has a half-life of 1.70 hours. (a) A group of 64 Cu atoms are moving at such high speed that their actual half-life is instead 18.64 hours. How fast are they moving? You may give the answer in m/s or as a fraction of c. (b) What would be the shape of the atoms? Clarify which directions are which. (c) A 64 Cu atom has approximately a mass of 59.6 GeV/c. What is the total energy of each of these atoms in GeV? (d) What is the momentum of these atoms in GeV/c? 34. It is found that when light of wavelength = 37 nm impacts Mg, electrons of energy up to 1.55 ev are emitted. (a) What is the frequency and energy for = 37 nm? (b) What is the work function for Mg? (c) What is the longest wavelength of light that can extract an electron from Mg? 35. A particle is in a 1D harmonic oscillator with angular frequency = 1.6 10 15 s 1. (a) If the particle is in the n = 6 quantum state, what is the energy of the harmonic oscillator, in ev? (b) Suppose it then falls to the n = 3 state. Would it emit or absorb a photon? What would be the corresponding energy of this photon? (c) Suppose, instead, that it absorbed a photon of energy approximately 3.3 ev. What would be the value of n now?
Part V: Calculation (new material): [60 points] Choose three of the following four questions and perform the calculations (0 points each) 36. Potassium is commonly incorporated into rocks. Argon, by comparison, almost never is present when rocks form. Consider a rock that is formed with 1.60 10 8 atoms of 40 K, an isotope with half-life of t1/ = 1.48 10 9 y. (a) What is the decay constant for 40 K in y -1? (b) Suppose a rock is 3.74 10 9 y old. How many atoms of 40 K would remain today? (c) 40 K has multiple decay modes, but approximately 10.7% of the time, it decays to 40 Ar. Assuming these atoms are trapped in the rock, how many atoms of 40 Ar will there be? (d) Suppose another sample of rock had equal numbers of 40 Ar and 40 K. Would you predict the rock is younger than 3.74 10 9 y, or older, or what? 37. Photocopied with the equations on the next page is a portion of Appendix A from the text. 15 Po is a nucleus that might decay by one of mode Daughter Q (MeV) Possible? the modes listed at right. You may use the table at right to summarize your answers, if you wish. electro (a) For each of these, what is the daughter n isotope? capture (b) What is the Q-value for each decay? + (c) Which of the modes can actually occur? 38. A large central galaxy A has a smaller satellite galaxy B orbiting it at a distance of 1.5 Mly (1 Mly = 9.46 10 1 m). The hydrogen- line, normally at a wavelength of 656.8 nm, is observed to be at 678.35 nm coming from A and 678.0 nm coming from galaxy B. (a) What is the approximate radial velocity of each of these galaxies towards or away from us, in km/s? (b) Assume galaxy B is in circular orbit around galaxy A, such that we are viewing the orbit edge on (see sketch at right). Which direction, and at what velocity, is galaxy B orbiting? (c) Estimate the mass of galaxy A, in solar masses (MSun = 1.989 10 30 kg). A 1.5 Mly 39. A probe is studying a neutron star at a distance of r = 10.7 km. The probe is designed to communicate using a laser of wavelength 437 nm, but distant observers see a wavelength of 53 nm. (a) What is the mass of the neutron star, in solar masses (MSun = 1.98910 30 kg)? (b) After 7.00 days, as measured by a distant observer, a signal is sent to the probe to move to a greater distance. How long will have passed on the probe? (c) To what radius would the neutron star have to be compressed to cause it to fall inside its Schwarzschild radius and become a black hole? To Earth B
Constants: Equations 34 15 h 6.6610 J s 4.13610 ev s 34 16 1.055 10 J s 6.58 10 ev s G 6.67310 m / kg / s N A 6.010 11 3 3 u 931.494 MeV / c 7 u 1.66110 kg mc e 1.0 MeV M He 4.0060 u Energy and momentum in relativity: E mc p mv Orbits: v GM r 1 Harmonic Oscillator Energy: E, 0,1,, n n n GM GM Gravitational time dilation: t 1 Schwarzschild radius: R S cr c Red Shift: 1/ 1/ GM GM 01, f f 01 cr cr Isotope Masses