PHYSICS 2D FINAL EXAM WINTER QUARTER 2016 PROF. HIRSCH MARCH 15, 2016 Formulas: Time dilation; Length contraction : Δt = γδt' γ Δt p.

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1 PHYSICS D FINAL EXAM WINTER QUARTER 06 PROF. HIRSCH MARCH 5, 06 Fomuas: Time diation; Lengt contaction : Δt = γδt' γ Δt p ; L = L p /γ ; c = m /s Loentz tansfomation : x'= γ(x vt) ; y'= y ; z'= z ; t'= γ(t vx /c ) ; invese : v -v u Veocity tansfomation : u x '= x v u x v /c ; u '= u y y ; invese : v -v γ( u x v /c ) Spacetime inteva: (Δs) = (cδt) -[Δx +Δy +Δz ] Reativistic Doppe sift : f obs = f souce + v /c / v /c Momentum: p = γ mu ; Enegy : E = γ mc ; Kinetic enegy : K = (γ )mc Rest enegy : E 0 = mc ; E = p c + m c 4 Eecton : m e = 0.5 MeV /c Poton : m p = MeV /c Neuton : m n = MeV /c Atomic mass unit : u = 93.5 MeV /c ; eecton vot : ev = J Stefan's aw : e tot = σt 4, e tot = powe/unit aea ; σ = W /m K 4 c e tot = cu /4, U = enegy density = u(λ,t)dλ ; Wien's aw : λ m T = k B Botzmann distibution : P(E) = Ce -E/(k B T ) Panck's aw : u λ (λ,t) = N λ (λ) E (λ,t) = 8π λ c /λ 8πf ; N( f ) = 4 e c / λkbt c 3 Potons : E = f = pc ; f = c /λ ; c =,400 ev A ; k B = (/,600)eV /K Potoeectic effect : ev s = K max = f φ, φ wok function; Bagg equation : nλ = dsinϑ Compton scatteing : λ'-λ = ( cosθ) ; m e c m e c = 0.043A Couomb foce : F = kq q ; Couomb enegy : U = kq q ; Couomb potentia : V = kq Foce in eectic and magnetic fieds (Loentz foce): F = qe + qv B Rutefod scatteing : Δn = C Z Hydogen spectum : K α sin 4 (φ /) λ mn = R( m n ) ; R = m = ke =4.4 ev A 9.3A Bo atom : E n = ke Z Z = E 0 n n ; E 0 = ke = m e(ke ) =3.6eV ; K = m ev a 0 ; U = ke Z f = E i E f ; n = 0 n ; 0 = a 0 Z ; a 0 = m e ke = 0.59A ; L = m ev = n angua momentum de Bogie : λ = p ; f = E ; ω = πf ; k = π p ; E = ω ; p = k ; E = λ m i(kx -ω(k)t ) Wave packets : y(x,t) = a j cos(k j x ω j t), o y(x,t) = dk a(k) e ; ΔkΔx ~ ; ΔωΔt ~ j goup and pase veocity : v g = dω dk ; v p = ω k ; Heisenbeg : ΔxΔp ~ ; ΔtΔE ~

2 PHYSICS D FINAL EXAM WINTER QUARTER 06 PROF. HIRSCH MARCH 5, 06 Pobabiity: P(x)dx = Ψ(x) dx ; P(a x b) = dxp(x)!c = 973 eva Scodinge equation : - Ψ m x +U(x)Ψ(x,t) = i Ψ t Time independent Scodinge equation : - m squae we: ψ n (x) = L sin(nπx L ) ; E n = π n ml ; b a ; Ψ(x,t) =ψ(x)e -i E t ψ +U(x)ψ(x) = Eψ(x) ; x dx m e = 3.8eVA ψ * ψ = - (eecton) Hamonic osciato : Ψ n (x) = H n (x)e mω x ; E n = (n + p )ω ; E = m + mω x = mω A ; Δn = ± Expectation vaue of[q] :< Q >= ψ * (x)[q]ψ(x) dx ; Momentum opeato : p = i x Eigenvaues and eigenfunctions: [Q] Ψ = q Ψ (q is a constant) ; uncetainty : ΔQ = < Q > < Q > Step potentia: efection coef : R = (k k ) m, T = R ; k = (E U) (k + k ) Tunneing : ψ(x) ~ e -α x ; T = e -αδx ; T = e x - α(x )dx x ; α(x) = m[u(x) - E] Scodinge equation in 3D : - m Ψ +U( )Ψ(,t) = i Ψ ; Ψ(,t) =ψ( )e -i E t t 3D squae we: Ψ(x,y,z) = Ψ (x)ψ (y)ψ 3 (z) ; E = π m (n L + n L + n 3 L ) 3 m Speicay symmetic potentia: Ψ n,,m (,θ,φ) = R n ()Y m (θ,φ) ; Y m (θ,φ) = P (θ)e im φ Angua momentum: L = p ; [L z ] = i φ ; [L m ]Y = ( +) m Y m ; [L z ]Y m = m Y Radia pobabiity density : P() = R n () ; Enegy : E n = ke Z a 0 n Gound state of ydogen and ydogen - ike ions : Ψ,0,0 = π / ( Z a 0 ) 3 / e Z / a 0 Obita magnetic moment : µ = e m e L ; µ z = µ B m ; µ B = e m e = ev /T Spin / : s =, S = s(s +) ; S = m ; m = ±/ ; z s s µ s = e gs m e g= Obita +spin mag moment : µ = e ( L + gs ) ; Enegy in mag. fied : U = µ B m e Two patices : Ψ(, ) = + / Ψ(, ) ; symmetic/antisymmetic Sceening in mutieecton atoms: Z Z eff, < Z eff < Z Obita odeing: s<s<p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<6p<7s<6d~5f

3 PHYSICS D FINAL EXAM WINTER QUARTER 06 PROF. HIRSCH MARCH 5, 06 Pobem Te moving wakway at te aipot moves at speed 0.5c. You ae about to miss you pane, so you un on te wakway at speed 0.5c wit espect to te wakway to get to you gate. Accoding to you watc, you wee 60 seconds on te wakway and got to te gate igt on time, unfotunatey accoding to te agent at te gate you missed te pane by: A: 0 s ; B: 0 s; C: 30 s; D: 40 s; E: not sue (E aways counts 0.3 pts) Pobem A mass m moving at speed 0.8c coides ineasticay wit a mass m at est, te two masses stick togete foming a mass M tat is equa to A: m; B:.4m; C:.3m; D:.66m; E: not sue Pobem 3 A 60W igt bub emits maximum powe at waveengt 5700A. Its suface aea is A:.58mm ; B:.44mm ; C: 3.8mm ; D: 5.3mm ; E: not sue Pobem 4 In a potoeectic expeiment wit a cetain meta it is found tat te stopping votage (votage tat wi stop te potocuent) is.5v wen te incident igt as waveengt 4000A. Wat is te maximum waveengt of igt tat wi give ise to a potocuent fo tis meta? A: 8650A; B: 7750A; C: 6850A ; D:5950A; E: not sue Pobem 5 In a Compton scatteing expeiment wit incident potons of waveengt A, te scatteed eectons at a cetain ange θ ave kinetic enegy 50eV. Te ange θ is appoximatey A: 30 o ; B: 45 o ; C: 60 o ; D: 90 o ; E: not sue Pobem 6 In a Rutefod scatteing expeiment wit atoms of atomic numbe Z=35 and α patices of incident kinetic enegy 7 Mev, it is found tat fo evey 800 α patices scatteed at a 90 o ange tee ae appoximatey 00 α patices scatteed at ange 80 o. Wat can you say about te adius R of tis nuceus? (fm=0-5 A). A: R<4.4 fm; B: R>4.4 fm; C: R<7. fm; D: R>7. fm; E: not sue Pobem 7 Te atio of speeds of an eecton in Bo obits of angua momentum 3! and angua momentum 6! is: A: /4; B: 4; C /; D: ; E: not sue Pobem 8 A poton and an eecton bot ave de Bogie waveengt λ=0.00a. Te atio of te speed of te poton to te speed of te eecton is (Hint: conside wete you need to use eativity) A: ; B: 0.008; C: 0.03; D: 0.; E: not sue

4 PHYSICS D FINAL EXAM WINTER QUARTER 06 PROF. HIRSCH MARCH 5, 06 Pobem 9 Fo an eecton wavepacket wit peak at k=0.a -, te atio of pase veocity to goup veocity is A: ; B: /; C: ; D: >0; E: not sue Pobem 0 An eecton moving in a one-dimensiona potentia U(x) is in a stationay state wit enegy E. Its wavefunction in te inteva 0<x<5A is ψ(x) = Cx 4 wit C a constant. Wat is te diffeence in te vaues of its potentia enegy U(x) at positions x =0.5A and x =A, i.e. wat is U(x )-U(x )? A: 37eV; B: 84eV; C: 3.8eV; D: 7.6eV; E: not sue Hint: use te Scodinge equation Pobem An eecton in a one-dimensiona box of engt L can emit potons of waveengt exacty 000A and 000A wen it makes a tansition fom an excited state to te gound state. Wat is te engt of te box? A: 3A; B: 6A; C: 9A; D: A; E: not sue Pobem Te cassica ampitude fo an eecton in te fist excited state of a one-dimensiona amonic osciato potentia is 3A. Wat is te gound state enegy? A:.7eV; B:.54eV; C: 3.8eV; D: 5.08eV; E: not sue Pobem 3 Wic of te foowing functions is an eigenfunction of te opeato x A: e λx ; B: e λx ; C: e λ/x ; D: e λ/x ; E: not sue Pobem 4 3eV ev x? Fo eectons incident wit kinetic enegy ev, te tansmission pobabiity toug te two baies sown above, of eigt 3eV and ev, is te same. If te 3eV baie as widt A, te widt of te ev baie is A: 0.7A; B:.4A C: A; D: 0.5A; E: not sue

5 PHYSICS D FINAL EXAM WINTER QUARTER 06 PROF. HIRSCH MARCH 5, 06 Pobem 5 Fo an eecton in te gound state of a two-dimensiona box of side engt L=4A, te pobabiity of finding te eecton in a sma egion aound (x=a, y=a) is ow many times age tan te pobabiity of finding te eecton in a sma egion (of te same aea) aound (x=a, y=a)? A: ; B: 4 C: 8; D: 6; E: not sue Pobem 6 Te wavefunction fo an eecton in a ydogen-ike ion wit nucea cage Ze is ψ(,θ,φ) = C( D + F )e /a 0 wee C, D, and F ae constants Te vaues of te quantum numbes n,, m ae A: 3,, ; B: 4, 0, 0; C: 3, 0, 0; D:, 0, 0; E: not sue Pobem 7 Te enegy of te poton emitted wen te eecton in te wavefunction of pobem 6 makes a tansition to te gound state is, in tems of E 0 =3.6eV: A: E 0 B: 4E 0 ; C: 8E 0 ; D: E 0 ; E: not sue Pobem 8 Te adia wavefunction fo an eecton in a ydogen-ike ion is R() = Ce /(a 0 ) Te aveage adius, i.e. < >, is A: a 0 ; B:.5a 0 ; C: 3.5a 0 ; D: 5a 0 ; E: not sue Hint: use dx x n e λx = n! 0 λ n + Pobem 9 Fo an eecton in ydogen wit quantum numbes n=4, =3, s=/, in a magnetic fied B=0T, te diffeence in enegy between te igest and owest states, assuming te spin-obit inteaction can be ignoed, is A: x0-3 ev; B: 4.63x0-3 ev; C:.3x0-3 ev ; D:.03x0-3 ev; E: not sue Pobem 0 Te gound state enegy fo an eecton in a cubic box is E g. Taking into account eecton spin and te Paui excusion pincipe and assuming no inteaction enegy between eectons, if tee ae 4 eectons in tis box tei tota enegy is: A: 3E g B: 36E g ; C: 4E g ; D: 8E g ; E: not sue Pobem Estimate te kinetic enegy of an eecton confined inside a nuceus of diamete 0-5 A. A: 4x0 6 ev B: x0 8 ev; C: 4x0 0 ev; D: x0 ev; E: not sue