final - Physics 2D Final Exam Department of Physics UCSD...

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Unformatted text preview: Physics 2D Final Exam Department of Physics, UCSD Summer Session II - 2010 Prof. Pathria 04 September, 2010 Instructions: 1. There are EIGHT questions on the exam — you may attempt ANY SIX. 2. All questions are of EQUAL value. 3. Please write your answers in the blue book and make sure that your three-digit code number is written on a_H pages of the blue book with indelible ink. 1. A train has a proper length of 360m and is traveling at a speed of 0.6c with respect to the station's platform. Its conductor, in the process of checking tickets, walks back and forth at a speed of 0.6c with respect to the train. How long will it take him to get from the rear of the train to its front and how long will it take him to get from the front of the train to its rear (a) according to the observers on the train, (b) according to his own clock, and (c) according to the observers on the station’s platform? 2. On New Year’s Day 1996, an astronaut named Robert set out from Earth at a speed of 0.8c and travelled all the way to our nearest star, a-Centauri, which is at a distance of, say, exactly 4 light-years --- as measured in the Earth’s frame of reference. Having reached the star, he immediately turned around and returned to Earth at the same speed at which he had left. Before leaving for a-Centauri, Robert had promised his twin sister Emily (who decided to stay home) that he will send her a message of greetings on every New Year's Day by a radio signal, and she made a similar promise to him. I am pleased to say that they both kept their promise! Now the question is -------- (a) How many New Year messages, during this trip, did Robert send to Emily (including the one sent on the last day of the trip), and (b) how many such messages did Emily send to Robert (again, including the one sent on the last day of the trip)? Furthermore, how many of these messages did Robert receive on his way out and how many on his way in? 3. (a) An electron, of rest mass me and having kinetic energy equal to its rest energy mecz, collides inelastically with a positron at rest --- forming a positronium atom. What is the rest mass Mo of the positronium atom so formed, and what is its speed V? (b) Subsequently, the electron and the positron forming the positronium atom annihilate one another, producing two y-ray photons. What is the maximum possible energy, E*, of a photon so created? [You may express M0 in terms of me, V in terms of c, and E* in terms of mecz]. 4. In a photoelectric experiment, in which a monochromatic light and a sodium photocathode were used, we found a stopping potential of 1.85V for A = 300nm and a stopping potential of 0.82V for A. = 400nm. Using these data, determine (a) Planck’s constant h [in ].s), (b) the work function ¢ of sodium (in eV), and (c) the cut-off wavelength M for sodium (in nm). 5. In a Compton scattering experiment, the scattered photon had an energy of 150 keV while the recoiling electron (which was originally at rest) had a kinetic energy of 50 keV. [Remember that 1 keV = 103 eV]. (a) Determine the wavelengths )V and N of the incident and the scattered photons. (b) Calculate the angle 6 at which the photon is scattered and the angle (I) at which the electron recoils. 6. A hydrogen atom happens to be in a stationary state in which its total energy E (kinetic plus potential) is -3.4eV. Using this information, determine the kinetic energy K and the potential energy U of this system. Next, (a) using the above value of K, determine the wavelength A of the electron, and (b) using the above value of U, determine the radius r of the electron's orbit. (c) What do you think the relationship between this r and this A. should be? Check if the values you found for r and 7» do indeed satisfy that relationship! 7. The ground state wave function of the simple harmonic oscillator in one dimension is known to be wo(x) = C0 exp (- axz), where on is a constant of the system. (a) Verify that this function does indeed satisfy the Schrodinger wave equation of the oscillator, provided that 0L is chosen to be nmm/h. (b) What is the corresponding energy eigenvalue E0? (c) Next, evaluate the root—mean-square deviation (Ax)rms for the oscillator in this state and compare your result with the "amplitude of a classical oscillator with the same energy E0”. 8. Using the Schrodinger wave function for the hydrogen atom in its ground state, calculate (a) the average distance, <r> , of the electron from the nucleus, (b) the most probable distance, r*, of the electron from the nucleus, and (c) the probability that the electron may be found in the region lying between r* and <r>. Some Useful Numbers, Equations, and Identities Speed of light: 0 = 2.998><108 m/s Planck’s constant: h = 6.626x10—34 J-s h = L- 1 eV = 1.602 x 10-19 J Coulomb’s constant: k = 8.99><109 ng/C2 Electron Charge: 6 = 1.602><10_19 C Electron Mass: me = 9.11x10—31 kg = 0.511MeV/c2 Rydberg Constant: R = 1.097X 107111—1 Atomic Mass Unit: 11 = 1.6606X10“27kg = 931.5 MeV/c2 Proton Mass: mp = 1.673><10_27 kg = 938.3 MeV/c2 = 1.0073 u Neutron Mass: mm = 1,675><10-27 kg = 939.6 MeV/c2 = 1.0087 u Compton wavelength for an electron: h — 0.00243 nrn mac _ Compton scattering formula: X — A0 = mhec (1 —— c050) Photo—electric equation: 8V; = hf — (b = h (f — f0) 1 “VI em Momentum for a relativistic particle: p = vmou, *y = \/ 1 Energy for a particle: E = K + ch = *3/ch Energy-momentum relation (particle): p = %\/E2 — 771304 = %\/2moc§K + K 2 Energy-momentum relation (photon): E 2 pc Relative velocity: u’ = "—132 1‘37 old Doppler Effect (light source approaching observer): fobs 2 f0 :— nld De Broglie wavelength: /\ = h/p Schrodinger Equation: ——h——%:%+ m)=¢ Ew 2m 1—Dimensional Normalization Condition: ff; W‘wdx = ff; |w|2d$ = 1 1mafia? Harmonic Oscillator Potential: U = 2 For a Hydrogen—like atom: — Energy: En = —-§f§§, n: 1,2,3,4,... — Bohr radius: (10: 74% = 0 529x10 10m Volume element in spherical coordinates: dV 2 r2sin0drd0d¢ or 47rr2d7" 3/2 Ground state Wavefunction for Hydrogen: \II (7", 0, (15) = f (i) (fr/“0 00 Root—Mean—Square deviation: Ar = 7‘2 —— F2 Expectation Value for an operator Q: Q 2 fan Space dV\II*[Q]\II sin20 = % [1 —- cos (20)] cos2 =21[1 + cos (20)] fooo e_“””2dx = éfl, a > 0 2 —az2 _l .71.. fooo me dart—4 a3? a>0 fab acne—”‘dQJ— — —(a:" + mo" 1 + n(n —1)9c”_2 + . . . + n!)e_1|2 ...
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