quiz5 - PHYSICS 2D PROF. HIRSCH Formulas: QUIZ 5 SPRING...

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Unformatted text preview: PHYSICS 2D PROF. HIRSCH Formulas: QUIZ 5 SPRING QUARTER 2005 MAY 13 2005 † † † † † Lp 1 ;g= ; c = 3 ¥ 10 8 m / s 2 2 g 1 - v /c Lorentz transformation : x ' = g ( x - vt ) ; y ' = y ; z' = z ; t ' = g ( t - vx / c 2 ) ; inverse : v Æ -v uy ux - v Velocity transformation : ux ' = ; uy ' = ; inverse : v Æ -v 2 1 - ux v / c g (1 - ux v / c 2 ) Relativistic Doppler shift : f obs = f source 1 + v / c / 1 - v / c Time dilation/length contraction : Dt = g t ; L = Momentum, energy (total, kinetic, rest) : p = g m u; E = g mc 2 ; K = (g - 1)mc 2 ; E 0 = mc 2 ; E = p 2c 2 + m 2c 4 Electron : me = 0.511 MeV / c 2 Proton : mp = 938.26 MeV / c 2 Neutron : mn = 939.55 MeV / c 2 Æ Æ Electron : me = 9.109 ¥ 10-31 kg Proton : mp = 1.673 ¥ 10-27 kg Neutron : mn = 1.675 ¥ 10-27 kg Atomic mass unit : 1 u = 931.5 MeV / c 2 electron charge = -e, proton charge = e, e = 1.6 ¥ 10 -19 C ; r r r rrr Force on charge q in E and B fields : F = q( E + v ¥ B) ; centripetal acceleration = v 2 / R • Stefan' s law : R = sT 4 , R = power/unit area ; s = 5.67 ¥ 10-8 W / m 2K 4 ; R = cU / 4 , U = energy density = † Ú u(l)dl 0 † † † † † † 8p hc / l hc Planck' s law : u( l, T ) = n ( l) ¥ e ( l, T ) = 4 ¥ hc / lkB T ; Wien' s law : lm T = l e -1 4.96 k B 1 Photoelectric effect : eV0 = ( mv 2 ) max = hf - f , f ≡ work function 2 Photons : E = hf = pc ; f = c / l ; Quantum oscillator : en = nhf ; probability P (en ) µ e-e / k T n B Compton scattering : l' - l = h (1 - cos q ) mec Rutherford scattering : DN = C sin (q / 2) 4 Electrostatics : F = kq1q2 kq (force) ; U = q0 V (potential energy) ; V = (potential) 2 r r 1 1 1 = R( 2 - 2 ) l m n ; R = 1.097 ¥ 10 7 m-1 = 1 911.6 A Hydrogen spectrum : Bohr atom : E n = - ke 2 Z Z 2E ke 2 mk 2e 4 = - 2 0 ; E0 = = = 13.6eV ; E n = E kin + E pot , E kin = - E pot / 2 = - E n 2 rn n 2 a0 2h2 † † hf = E i - E f ; rn = r0 n 2 ; r0 = a0 Z ; a0 = h2 = 0.529 A ; L = mvr = nh angular momentum mke 2 † † † † † X - ray spectra : f 1 / 2 = An ( Z - b) ; K : b = 1, L : b = 7.4 Constants : h = 4.136 ¥ 10-15 eV ⋅ s ; hc = 12, 400 eV A ; k B = 1 /11, 600 eV/K ; ke 2 = 14.4 eV A hc = 1973 eV A ; e = 1.6 ¥ 10-19 C ; N A = 6.02 ¥ 10 23 Conversions : 1eV = 1.6 ¥ 10 -19 joules ; 1A = 10 -10 m = 0.1nm ; 1MeV = 10 6 eV Double slit interference : d sin q = nl (maxima) , d sinq = (n + 1 / 2)l (minima) h E 2p p2 de Broglie : l = ; f = ; w = 2pf ; k = ; E = hw ; p = hk ; E = p h l 2m i( k j x -w j t ) i( kx -w ( k )t ) wave packets : y ( x, t ) = Ú dk a( k ) e , or y ( x, t ) = Â a j e ; DkDx ~ 1 ; DwDt ~ 1 j † † † dw group and phase velocity : v g = dk ; w vp = k ; Heisenberg : DxDp ~ h ; DtDE ~ h Wave function Y( x, t ) =| Y( x, t ) | e iq ( x,t ) ; P ( x, t ) dx =| Y( x, t ) |2 dx = probability † PHYSICS 2D PROF. HIRSCH QUIZ 5 SPRING QUARTER 2005 MAY 13 2005 E -i t h2 ∂ 2Y ∂Y Schrodinger equation : + V(x, t) Y(x, t) = ih ; Y(x, t) = Y(x)e h 2 2m ∂x ∂t • h2 ∂ 2Y Time - independent Schrodinger equation : + V(x)Y(x) = EY(x) ; Ú dx Y* Y = 1 2m ∂x 2 -• † † † Square well : E n = p 2h2n 2 2 npx ; Yn ( x ) = sin( ) 2 2 mL L L ; x op = x , pop = h∂ ; < A >= i ∂x • -• Ú dxY A * op Y Eigenvalues and eigenfunctions : Aop Y = a Y ; Justify all your answers to all problems uncertainty : DA = < A 2 > - < A > 2 † Problem 1 (15 points) A particle in an infinite square well undergoes a transition from the first excited state (n=2) to the ground state (n=1) and emits a photon of wavelength l=5000A. (a) What is the wavelength of the photon emitted when this particle makes a transition from the second excited state (n=3) to the ground state? (b) What is the length of the well, in A, if the particle is (i) a proton, (ii) an electron? (mp c2 =938MeV, mec2 =0.511MeV) (c) If the square well is not infinite but has finite height V0 , will the energies of particles in this well be larger, equal or smaller? Explain using the uncertainty principle. Problem 2 (15 points) An electron is in an infinite square well of length 8A. (a) If the electron was a classical particle, what would be the probability to find it in the region x=3A to x=5A, where x is the distance from the left wall? (b) Assuming the electron is in the ground state of this well, give a rough estimate of the probability that it will be found in that interval (x=3A to x=5A), without doing an integral. State whether it is larger, equal or smaller than the result in (a) and why. (c) Calculate the probability that the electron in the ground state will be found in that interval (x=3A to x=5A), by doing an integral. Hint: use sin2 x=(1-cos(2x))/2. State whether the answer is larger, equal or smaller than the result found in (b) and why. Suggestion: answer the rest of the quiz first since this question is likely to take longest. Problem 3 ( 15 points) For an electron in an infinite square well of length L: (a) Find an expression for <p2 > for the electron in the excited state with n=10. (b) Find the uncertainty in p2 , D ( p 2 ) = < ( p 2 ) 2 > - < p 2 > 2 . (Note it is not the same as the uncertainty in p, which is Dp = < p 2 > - < p > 2 ). (c) Find <x> and <p> for an electron in the excited state with n=10. (d) Is the probability of finding the electron in the interval Dx=0.1L around x=L/4 † (measured from the left wall) largest for the state n=1, n=2 or n=3? Explain. Hint: you don't† have to do an integral for any of these questions. Justify all your answers to all problems ...
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This note was uploaded on 12/07/2009 for the course PHYS phys 2d taught by Professor Hirsch during the Spring '08 term at UCSD.

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