 # Introduction To Quantum Mechanics

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1. Consider a particle in the one-dimensional region O ::S x ::S a. The wave function is given by 'tjJ(x) = N x (a-x) where N is a constant to be determined. Outside the region, the wave function is zero. (a) Find N. (b) Where is the most probable location for the particle? (c) Calculate the expectation value of the position of the particle. ,., The wave function for a simple harmonic oscillator with n = 2 and energy (5/2)nooc is where (a) Verify that this wave function satisfies Schrodinger's equation. (b) Show that ~ =11J2t;Jn Recall the useful integrals in Appendix B of Taylor and Zafiratos, but note that In= \dln_2/d/\.\. (The absolute value sign is missing in the text version.) 4. Recall the one-dimensional semi-infinite box problem we treated in class: ~~ -----------.---------- -------1-.---------.:.,.',. "K. .th R /2m(U~ - E) d /2m£ WI /J= 1/ _ an a="\;- \ 11- ~ t? By solving Schrodinger's equation in the regions O < x < a and x ~ a and invoking continuity of both the wave function and its first derivative at x= a, we derived the following equation relating a, f3 and a: .Btan (a a)= -a (1) . Consider the bound state problem, with E < Uo and Uo = 9Eo, where Eo =1i23t2/(2ma2 ) , the ground state energy for the perfectly rigid box. The allowed energy levels for the semi-rigid box are En = x/Eo. By solving eqn. (1), find all the allowed bound state energies; i.e., find all the allowed Xn's. 6. (a) Use your knowledoe of th l . f h S hr'"'-1 · . . e e o uuon o t e c vumger equation for the h drol!en atom to est1mate the quantum number n for the Moon orbiting the Earth. Ignore ;edu~d mass effects and the Moon's rotation on it axis. (b) _Estimate the quantum numbers / and m for the Moon. (The Moon orbit the :.arth essentially 10 the Earth's equatorial plane. so the lunar angular momentum vector is perpendicular to the plane of its orbit.) (mass of Earth = 5.98 x 1024 kg: mass of Moon = 7.35 x 1011 k~: mean distance from Earth to Moon (centerto center) = 3.84 x 108 m; G = 6.67 x 10·11 N-m /kg2 ) 9. Consider the effect of the rotation of the Sun on the wavelen~s that would be observed in a spectroscopy experiment. The radius of the Sun is 6.96 x 10 km, and its period of rotation at the equator is 25.4 days. For the 486.l run line of the hydrogen spectrum (measured in the lab on the earth). what are the wavelengths that would be emitted from opposite edges of the Sun's equator because of the SW1 's rotation? 11. , -#1 _,-_:/\ 8_ _ ~3c/5 A particle of rest mass Mo is at rest in the laboratory when it decays into three identical particles, each of rest mass 111o. Two of the particles (labeled #1 and #2) have velocities as shown in the figure. ( a) Calculate the direction and speed of particle #3. (b) Find the ratio MJ 111o. (c) Compare your results with the classical (non-relativistic) solution. 2.9 • Muons are subatomic particles that are produced · several miles above the earth's surface as a result of collisions of cosmic rays ( charged particles, such as protons, that enter the earth's atmosphere from space) with atoms in the atmosphere. These muons rain down more-or-less uniformly on the ground, although some of them decay on the way since the muon i~ unstable with a proper half-life of about 1. 5 μs. l 1 μs = 1 o-6 s.) In a certain experiment a muon detector is carried in a balloon to an altitude of 2000 m, and in the course of 1 hour it registers 650 muons traveling at 0.99ctoward the earth. If an identical detector remains at sea level, how many muons would you expect it to register in 1 hour? (Re~ember that after n half-lives the number of muons surviving from an initial sample of N0 is N0 /2n.) ( ,~. • A traveler in a rocket of length 2d sets up a coordiV nate system S' with origin 0' anchored at the exact middle of the rocket and the x' axis along the rocket's length. At t' = 0 she ignites a flashbulb at 0'. (a) Write down the coordinates x~, t~ and x~, t~ for the arrival of the light at the front and back of the rocket. (b) Now consider the same experiment as observed in a frame S relative to which the rocket is traveling at speed v (with Sand S' arranged in the standard· configuration). Use the Lorentz transformation to find the coordinates Xp, tp and xB, tB of the arrival of the two signals. Explain clearly why the two times are not equal in frame S, although they were in S'. (This illustrates how two events that are simultaneous in S' are not necessarily simultaneous in S.)

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