Preliminary Examination - Day 1 Thursday, May 12, 2016

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1 UNL - Departmet of Physics ad Astroomy Prelimiary Examiatio - Day Thursday, May, 6 This test covers the topics of Quatum Mechaics (Topic ) ad Electrodyamics (Topic ). Each topic has 4 A questios ad 4 B questios. Work two problems from each group. Thus, you will work o a total of 8 questios today, 4 from each topic. Note: If you do more tha two problems i a group, oly the first two (i the order they appear i this hadout) will be graded. For istace, if you do problems A, A3, ad A4, oly A ad A3 will be graded. WRITE YOUR ANSWERS ON ONE SIDE OF THE PAPER ONLY

2 Prelimiary Examiatio - page Quatum Mechaics Group A - Aswer oly two Group A questios 5 A A proto is cofied to a atomic ucleus of diameter 3 m. Use Heiseberg s ucertaity priciple to estimate the proto s bidig eergy. Give the aswer i electrovolts (ev). A We cosider the Hermitia operator Â ad the o-hermitia operator ˆQ. a. Is exp( A ˆ ) a Hermitia operator? b. Is iqq [ ˆ, ˆ ] a Hermitia operator? I the followig part, ˆx ad ˆp are the positio ad mometum operators, respectively. c. Show that [ xˆ, pˆ] = i xˆ for =,, 3,... A3 At t = the wavefuctio of a particle i a oe-dimesioal, ifiitely-deep box with walls at x = ad x= a is give by where C is a costat. ψ( x,) = Csi( πx/ a)cos( πx/ a), a. Fid ψ ( xt,) for t >. b. What is the eergy expectatio value for t >? Express aswers for parts (a) ad (b) i terms of E ( =,,... ), the eergies of the statioary states. c. Fid the probability P that the measuremet of the particle s eergy at the time t fids the value E. A4 A particle with spi is i the state with the spi projectio o the z axis S =. z a. What are the expectatio values of Sx ad S for this state? y b. Suppose S is measured. What are the possible outcomes of this measuremet ad x what are the correspodig probabilities?

3 Prelimiary Examiatio - page 3 Quatum Mechaics Group B - Aswer oly two Group B questios B A electro i the Coulomb field of a proto is i a state described by the wave fuctio ψ() r = ψ () ψ () r r, where ψ () r is the wave fuctio of a statioary state with pricipal quatum umber, m agular mometum quatum umber, ad its projectio m. a. Is this state a eigestate of the Hamiltoia if =? If =? Explai. b. Fid the expectatio value of the eergy for = ad for =. Express your aswers i terms of the eergy of the groud state, E = R, where R is the Rydberg costat. c. Fid the expectatio value of L. d. Suppose =. Fid the expectatio value of r usig the followig iformatio: z ψ () rψ () r d r = 3 a ( a is the Bohr radius). B Cosider a particle with spi ad gyromagetic ratio γ. a. Fid the eigevalues ad eigestates of S. x b. Suppose the particle is i the state correspodig to the larger eigevalue. Fid the expectatio values of S, S, y z S, ad y S for this state. z c. Usig the results obtaied i part (b), fid S + S + S. Ca you obtai the aswer i x y z a more straightforward way? d. Now place the particle i a magetic field B that is directed alog the x axis. Write the Hamiltoia H. Cosider the followig statemet: The solutios foud i part (a) are eigestates of the Hamiltoia. Is this statemet correct? If yes, fid the correspodig eergy eigevalues. If o, fid the expectatio values H for these states.

4 Prelimiary Examiatio - page 4 B3 Cosider a system whose wavefuctio is ψ =, + A, +,, 7 7 where the,m are agular mometum eigestates. The quatity A is a real-valued, positive costat. a. Calculate A so that ψ is ormalized. b. Calculate the expectatio values of L ˆ, L ˆ, L ˆ, ad x y z ˆL for this state ψ. c. Fid the probability that a measuremet of the z compoet of the agular mometum gives L =. z d. Calculate,m Lˆ ( ) ψ ad,m ( Lˆ ) ψ. + B4 Cosider a harmoic oscillator with Hamiltoia ˆ pˆ H= + mω xˆ = ( aa ˆˆ+ ) ω, ad m statioary states give by Hˆ = E = ( + ) ω. The aihilatio operator â is defied β i by aˆ = xˆ + pˆ (with mω β = mω / ), ad the creatio operator â is its Hermitia adjoit. The aihilatio ad creatio operators act as aˆ = ad aˆ = + +, α respectively. The oscillator is i a ormalized so-called coheret state, ψ = C, where =! C ad α are costats. x k a. Determie the (real-valued) ormalizatio costat C. Note: e = Σ x / k! k b. Show that ψ is a eigestate of the aihilatio operator â with eigevalue α. c. Explai why the creatio operator â caot have eigestates. d. Calculate x = ψ xˆ ψ. Hit: write ˆx i terms of â ad â, ad use part (b).

5 Prelimiary Examiatio - page 5 Electrodyamics Group A - Aswer oly two Group A questios A A 9 Ω resistor, a µf capacitor, ad a.45 H iductor are coected i series to a Vrms, 6 Hz power source. What is the average power cosumptio i this circuit? A A microwave ove heats food by usig electromagetic waves that form a stadig wave i the ove. Whe.5 GHz waves are used this way i a sample of food, it is foud that there are bur marks spaced 5.5 cm apart. What is the idex of refractio of the food sample for these electromagetic waves? A3 Each combiatio of capacitors betwee poits X ad Y i the figure is first coected across a -V battery, chargig the combiatio to V. These combiatios are the coected to make the circuits show. Whe the switch S is throw, a surge of charge from the dischargig capacitors flows to trigger the sigal device. How much charge flows through the sigal device i each case? A4 Four poit charges are arraged at the corers of a square, as show i the figure. a. Fid the electric field E at the ceter of the square. b. Fid the potetial V at the ceter of the square, assumig V( ) =.

6 Prelimiary Examiatio - page 6 Electrodyamics Group B - Aswer oly two Group B questios B Two semi-ifiite lie charges with uiform liear charge desity +λ lie alog the x-axis from to a / ad from to a /, respectively. A charge +q is iitially placed at rest at the origi. If it receives a small impulse alog the x-axis, what is its period of oscillatio alog this axis? B Two coils (A ad B) made out of the same wire are i a uiform magetic field with the coil axes aliged i the field directio. Coil A has turs of radius 5 cm ad coil B has turs with a ukow radius. The field stregth is ow doubled i s ad it is foud that the ratio of the emf iduced i each coil EMF : EMF is :. What is the radius of coil B? A B B3 A log, solid metal cylider with radius a is coaxial with log, thi, ad hollow metal tube with radius b> a. The positive charge per uit legth o the ier cylider is λ, ad there is a equal egative charge per uit legth o the outer cylider. We set Vr ( = b) =. a. Calculate the potetial Vr () for r< a. b. Calculate the potetial Vr () for a< r< b. c. Calculate the potetial Vr () for r > b. d. Calculate the electric field at ay poit betwee the cyliders. e. What would the potetial differece betwee the two cyliders have bee if the outer cylider had had o et charge? B4 I a LRC series a.c. circuit, the voltage amplitude of source is = V, the resistace is R = 8 Ω, ad the reactace of the capacitor is 48 Ω. The voltage amplitude across the capacitor is 35 V. a. What is the curret amplitude i the circuit? b. What is the impedace? c. What two values ca the reactace of the iductor have? d. For which of the two values foud i part (c) is the agular frequecy less tha the resoace agular frequecy?

7 Prelimiary Examiatio - page 7 Physical Costats speed of light... 8 c =.998 m/s electrostatic costat... k = πε = 9 (4 ) m/f 34 3 Plack s costat... h = 6.66 J s electro mass... m el = 9.9 kg 34 Plack s costat / π... =.55 J s electro rest eergy kev 3 Boltzma costat... k =.38 J/ K Compto wavelegth.. λ = h/ mc=.46 pm B C el 9 elemetary charge... e =.6 C proto mass... m = = electric permittivity... ε = F/m bohr... a 6 magetic permeability... µ =.57 H/m hartree (= rydberg)... E molar gas costat... Avogrado costat... N R = 8.34 J / mol K gravitatioal costat... p kg 836 el m = / ke m =.59 Å = / m a = 7. ev h el G = m / kg s 3 = 6. mol hc... hc = 4 ev m A el 3 Equatios That May Be Helpful TRIGONOMETRY siαsi β = cos( α β) cos( α + β) cosαcos β = cos( ) cos( ) α β + α + β siαcos β = si( α β) si( α β) + + cosαsi β = si( α + β) si( α β) QUANTUM MECHANICS Particle i oe-dimesioal, ifiitely-deep box with walls at x = ad x= a: / Statioary states ψ = (/ a) si( πx/ a), eergy levels E = π ma Agular mometum: [ L, L ] = i L et cycl. x y z Ladder operators: L, m = ( + m+ )( m), m+ + L, m = ( + m)( m+ ), m

8 Prelimiary Examiatio - page 8 Pauli matrices: i σ, σ =, x = σ y = i z ELECTROSTATICS q ec E d A = E = V ε S b a r r E d = V( r ) V( r ) V() r = 4πε q( r ) r r Work doe W= qe d = qv( b) V( a) Eergy stored i elec. field: W= Ed = Q C ε τ / V MAGNETOSTATICS Loretz Force: F= qe+ q( v B ) Curret desities: I = J d A, I = K d µ Id Rˆ Biot-Savart Law: Br () = 4π ( R is vector from source poit to field poit r ) R I For straight wire segmet: B = µ siθ siθ 4π s where s is the perpedicular distace from wire. Ifiitely log soleoid: B-field iside is B = µ I ( is umber of turs per uit legth) Ampere s law: B d =µ I eclosed Magetic dipole momet of a curret distributio is give by Force o magetic dipole: F= ( m B) Torque o magetic dipole: τ= m B B-field of magetic dipole: Br µ ( ) = 3( ˆˆ ) 3 4π r m= I da. µ The dipole-dipole iteractio eergy is U = ( m m ) 3( m Rˆ)( m R ˆ), where DD 3 4π R R= r r.

9 Prelimiary Examiatio - page 9 Maxwell s Equatios i vacuum ρ. E = Gauss Law ε. B = o magetic charge 3. B E = t Faraday s Law 4. E B= µ J+ εµ t Ampere s Law with Maxwell s correctio Maxwell s Equatios i liear, isotropic, ad homogeeous (LIH) media. D = ρ Gauss Law f. B = o magetic charge B 3. E = Faraday s Law t D 4. H= J + Ampere s Law with Maxwell s correctio f t Iductio Alterative way of writig Faraday s Law: dφ E d = dt Mutual ad self iductace: Φ = M I, ad M = M ; Φ= LI Eergy stored i magetic field: W = µ B dτ = LI = A I d V

10 Prelimiary Examiatio - page

11 Prelimiary Examiatio - page INTEGRALS + ay dy = π /a / y e ay! dy = a + r dr 3 / = ( r + x ) + 3/ / ( x + r ) ( r + x ) dx ta x ta x π = < a + x a a a xdx = l ( a + x ) a + x dx x = l xa ( + x) a a + x dx ax b = l ax b ab ax + b coth ax = ab b t ax = ah, ax < b ab b x

Preliminary Examination - Day 1 Thursday, August 10, 2017

UNL - Department of Physics and Astronomy Preliminary Examination - Day Thursday, August, 7 This test covers the topics of Quantum Mechanics (Topic ) and Electrodynamics (Topic ). Each topic has 4 A questions