97上量子物理上期ä¸

97上量子物理上期ä¸

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Unformatted text preview: Quantum Physics — Midterm Exam, Nov. 2!]. 21108 Page 1 1- Uning oiasaionl physics! of both thermodynamics and electromagnetism. Rayleigh and Jeans derived that the energy density of the black-body radiation in a cavity at he quency o' and temperature T is given by 87w: Pris) = c3 ifl". where l: is Boltzmann’s constant and c is the velocity of light in the vacuum. (a) How is pflu} related to the spectral radiancy Rfln)? Burr/2 63 and hi” in the above {b} 1What are the physical significanoes {if the two factors formula? (c) Specify the region of applicability of the Rayleigh-Jeans‘ formula (d) 1‘vltl'hat modification must be made in Rayleigh-Jeans‘ derivation in order to get the correct formula for the energir density found by Max Planck. {e} that is the implication of the modification made in {c}? giro-Ci _- Ll . r—snr pic: — i f—“ "'3' ' T" r 1; - E F I first. a? t x / 2. The experiment that provides the most direct evidence for the particle nature of radi- ation is the so—called Connan efi'ect. ' (a) Derive the wavelength shift as a function of the Scattering angle for the Compton scattering. Eb} Compton found that the radiation scattered through a given angle actually onnsists of two components: one whose wavelength is essentially the same as that of the incident radiation, the other of wavelength shifted relative to the incident wavelength by a certain amount. How do you explain these two components using the melt derived in {a}? .r" "__t-. 3. A muonic atom contath a nucleus of chargg 3e and a negative moon. a“, moving about it. The p‘ is an elementary particle with charge WT times as large as an electron mass. [3.) Calculate the moon-nucleus separation¢ D, of the first Bohr orbit of a muonic atom with Z = 1. {b} Calculate the binding energy of a muonic atom with 3 = 1. {c} Evaluate the wavelength of the first line in the Balmer series of such an atom. _\ (d) 1What is the wavelength of the most energetic photon thdt can be emitted from such an atom? Quantum Physics - Midterm Exam, Nov. 2t}, 2003 Page 2 4. flmsidar a very small dust particie, of radius r = Ill—Em and densityr ,s 2 1|)“ kyg‘ma. moving at the very slow veIocitg.r a = Iii—i2 mfia. {a} Estimate the the classically excluded region for this dust particle if the particle irnpinges on a potential step of height equal to five times its kinetic energy in the region to the left of the step. {in} Use the uncertainty principle to show that the wavelilte properties exhibited by an entity in penetrating the classically excluded region are not really in conflict with its particle-like properties. Wen HIE-j start by considering an experiment capable of proving that the particle is located somewhere in the classically.r excluded region] 5. Use the relevant uncertahity relationEs) to {a} discuss the “reality” of orbits in the Bohr atom; (b) estimate the “hmflness” or the “size” of an electron, in, the smallest distance within which one can localize the electron without invoking the production of an electron-positron pair. Note that the electron-positron pair production requires at least twice the rest-mass energy of the electron. Gompare this size with the Bohr radius. It Ii. Consider a particle whose wave function is given as 1;,(x’ t) = Ae—me’flh a-—:'Etffi- {a} Evaluate the energy eigenvalue E and the normalization constant A. {h} Calculate £11: E up“ — {$52. the noisiertaint}I of position in the state specified by the above wave fimction. [c] Calculate flip E u“ — (mg, the uncertainty of momentum in the state specified by the above wave function. {:1} Is the product flea - flip obtained from using your answers to {b} and (c) consistent with Heismberg’s uncertainty principle? Note that { } denotes an expectation value. For instance, {are a E? = m we. sense. a a. m Quantum Phyei — Midterm Exam, Nov. 21], 2098 Page 3 7. A particle of tunes m mom in one dimension under the influence of the potential LEI? Eiufl‘ mej= I}, "admit: Va, who where H3. :- [i and a i} 0. Assume that the energfi of the particle ie amaller than III]. (a) 1i-‘iirite down the Schrfidinger equation for the three regions defined rapeetirely by z < —a. no :1: m «a: a, and a: :e a. Write down aleo the general forms of the solutions to the Schrodinger equation in these regions. {b} What are the appropriate boundary oonditione at a: = —e and r = e? {e} .h.r:tua.il:,-T solve the equationin the limit 141; we no to obtain, in particular, the energy eigenvalues and eigenfunctions of the particle moving in this infinite square-wail potential. “V 5: 08‘? "U - \ (V _,- Egg (W. * “x R r“: 'I /.r {\‘5‘ x“ : 2’ x I A} :. ‘3‘ fl, . 2- I "T _ J. x11 K o, I} 0»? 5 . ._ , ’ i h f/fi {1.3%. h |.-' “X Quantum Physics *- Midterm Eaten]:i Nov. 20,. EDDS Instructions 1. Do any 6 of the neven exam problems given below. Each problem ohoeen counts 1'? points for a total of 113E. You have three hours to do the problems. 2. Make eure you write your calculations and answers for the problems not on the problem sheeta but on the anew sheets. Useful Information and Formiflae Gaussian integrals and Gamma. fimetione: m I ‘11" I e'”|ir=1l;, Rekrjnfl. CH} f E—aza+flmd$ ____ Edgifiafil HERE] :3 [L —oo o . [35' He]: f Ede—rah“. o 'r{z+1)=zr{z), r{u}=1, P{m)=fi. Relationship between 3— and pepeee were fimetione: Meat} = viii—h f dpeiipj) ei‘mfll 7T Planck’s must.an e = nee x 10-34.! - 3; e a g; = 111545 x 10-3413 Boltzmamfe constant: k = 1.33 x _1[I'23Jf K , where K denotes degree in Kelvin. Mane of electron: ma 2 9.1 X ill—Elfin. Manna of proton and neutron: M 2 LE? x lfi”27kg. Proton charge: e : 11302 x Ill—19 (3. e2 _ 1 alereulic fl 13?.fl333 Fine—structure constant: e: = Re = 19133121" rn'm. = 19133 MeV - fm. ...
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This note was uploaded on 02/21/2012 for the course EE 101 taught by Professor 張捷力 during the Spring '07 term at National Taiwan University.

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