PHYS 212 Final, Spring 1996

Final Exam = Phys. 212 (Intro. Mod. Physics) 24 June, 1996 ° ParLA [10 niarks per question} 1)- Label as true or false these statements involving the atomic quantum numbers: nym =(a) One of these subshelfs cannot exist: n=2,/ =0: n=4,/ =3; n=1,/=1. %, “4 ~ -*"(b) The number of values of #7, that are allowed depends on 1 and on /. ~ “(c) The smallest value of n that can go with a given / is /+ 1. (d) the quantum number s can take the values of +/- 1/2 2)- Angutar momentum is a vector and you might expect that it would take three quantum numbers to describe it, corresponding to the three space components of a vector. Instead, only two quantum numbers characterize the angular momentum in an atom. Explain why. “a 3)- Is there a change in the total energy of a hydrogen atom with n =-4, when it is placed in a magnetic field B? Justify your answer thoroughly, -4)-. Suppose that the distance froma given star to the earth is about 400 light-years and that the lifetime of a person is 70 years; at what speed must he travel to reach the star while he is still alive? (Your answer should be calculated to at least 3 decimal points.) [aT enna nenccen = AMELIOQN CNIvensrmy, : LIBRARY > Part B 120 marks per problem| OF Citas 1)- A relativistic snake of proper length 100 cm is moving at speed v = 0.6c to the right across a table. A mischievious boy. wishing to tease the snake, holds two hatchets [00 cm apart and plans to bounce them simultaneously on the lable so that the left hatchet lands immediately behind the snake's tail. The boy argues as follows: “The snake is moving with 6 = 0.6. Therefore its length is contracted bya - factor of y = 5/4, and its length (as measured in my rest frame) is 80 cm. This implies that the right hatchet will fall 20 cm in front of the snake, and the snake will be unharmed.” (The boy's view of the experiment is shown below.) On the other hand, the snake argues thus: “The hatchets are approaching me with B = 0.6, and the distance between them is contracted to 80cm. Since [ am [00 cm Jong, I shall be cut in pieces when they fall.” Use the Lorentz. transformation to resolve this paradox. t=0 v ——p LIBRARY ne ROOT ” ti ic rm ‘ is 2)- An electron and a proton, each with a kinetic energy E,, = 6 MeV, move perpendicular to a uniform magnetic field of | testa. (a) Find the ratio of the radius of curvature of the electron to that of the proton. (b) Assume that the kinetic energy of the proton is 5 GeV. At what kinetic energy will the electron acquire the same momuntum as that of the-proton. 3)- A beam of thermal neutrons is incident on a crystal in which the spacing between Bragg planes is : *. @=0.304 nm. An intense first-order Bragg diffraction is observed for a scattering angle 0. "..,.(a)-Determine the wavelength of the thermal neutrons (b) Find the scattering angle 6. (Hint: Assume that the neutrons behave like the molecules of an ideal gas at Room-Temperature) 4)- A muon (a negatively charged particle with a mass m, = 206 m,) is captured by a lead nucteus (Zp, = 82). The resulting system will behave like a hydrogen atom. (a) Find the “Bohr radius” for this system. (b) Detrmine the transition energy when the muon descends fromthe n= 2 to the n = f level in this muonic (hydrogen-like) lead ion. : 4 ; | Ws 5)- ()) For the hydrogen atom in the ground state, wy), = 22N(4x4,3)] etldo | Evaluate the probability of finding the clectron within a sphere of radius a, = Bohr radius. (I) A one-dimensional harmonic oscillator wave function is: wy = Ax exp[{-bx?] a) Show that y satisfies the Shrodinger equation. b) Find the constant and the energy E. c) Is this a ground state or a first exited state? 6)- (a) Find the wavelength of the light from a sodium atom emitted in the 3p — 3s transition, knowing that the first exited state of the sodium atom is the 3p level and AE = 2.1 eV. (b) Because of the spin-orbit interaction, the 3p level is actually two levels, 2.1x10-3 eV apart. Determine the difference AA between the two wavelengths produced by 3p —» 3s transitions in Na. (c) Represent the 3p — 3s transitions in an energy-level diagram, and write down the quantum states for n=3 (using the spin-orbit coupling scheme) for the sodium atom. 7)- The measured values for the wavelength of the three emission lines in a normal Zeeman effect are 4226.90 A°, 4226.73 A° and 4226.56 A°. (a) Find the strength of the magnetic field B? (b) What is the energy difference AE = Ef - Ei for the states involved in the transition, when B is switched off?. (c) Is there any information that can be inferred, from the above data, about the orbital quantum numbers of the initial and the final states? ye 1-1 ,01 * #16071 “ELE NSZ6LE Lf .2-01 * 0605, Lf .2-01 * oso's Uf o-01 X £276 PR ty, SWS: e101 * OMELB Dy 0l X Wwe 2 61-01 22091 — OE 01 X OPCO'E SL o_O X 7979'S AVN 11 01 x e199 D/H ,.01 x xp Wd 01 X I9S8R Rif sosip MoE cx-01 * 908E°1 Wome evieg 308/H Ol X 6L66'C. 1-(900-89) 01 x zzz0'9 B01 x LS = Oy Goxudew my0g uonose € TBuspaem ° Bordmc> uansop ? a&req> stone IwISVOD SSRI MsSUOS 5 youRLY WEIsUOS PeCONN AR y.. 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TRIGONOMETRIC IDENTITIES EXPONENTIAL EX PANSION sin(90" ~ @) ~ cos @ entree ge ae. * * c0x(90"—@)— sing ae Oe ee: sin O/cos @ = tan @ : sin? 6 + cos? @ = 4 sec? @— tan? ¢@ = 1 csc? 6— cot? @= | LOGARITHMIC EXPANSION sin 20 = 2 sin @ cos @ ~ © Cot 20 = cos? 8 — sin? 0 = 2 cos? @— 1 1 —~ 2 sin? 9 IU xym ax BF + fx? — hd) sinfer & A) — sin or cos f + cos.ar sin f i coma ££) = cos a cos fF sin asin f TRIGONOMETRIC EXPANSIONS (0 in radians) tan a + tan tal Lr er sin o£ sin P — 2 sin Hox +f) con Hor F f) eg cosO—mt-= 42, a, 4t . > 9 ~ BINOMIAL THEOREM ton 04 4 0, a Waaypere rs oe, He Gten Ot aprn te Ey Mat De, tes tet) DERIVATIVES AND INTEGRALS In what follows, the letters w and » stand for any functions of x, and a and m are constants. To each of the indefinite integrals should be added an arbitrary constant of integration. The ! Xeni 1. [ans 2 ion -at 2. [owdemaf ude . 3 Fut My 3 (4+ de fudet fvde 7 p a até +p) u o . bE. . d ory 5 , 4 —nome fran ee (m4-1) ‘ f ne ee . 5. arms 3. [ &-min ie j 6.4 (may 4 Mt do du * 6 ao "ER 6. [«Ge-w- o> dy e i, Lown % [oane Eg 8. Asin x= cox 8. sin x dx = ~cos x ' 9 4 cot = —sin x 9. | con racm sin ix + 10. F tan x= sect x to. [tenet ee i ' ’ tL 4 cot x= - ese! x ; i. Jf rier x de ~ pe jon 28 $2. f see x= tan x s00 x , [evacn-ten a& a 13, Fs x= cot xese x 13. [xem aee Start nen 14, goree 4, [ pesac~—S ate tract new i AS. Ft w= con wt 15. [reves 16 footw ann 16. [Urrew ae EE no z ——— e