quiz7 - PHYSICS 2D PROF. HIRSCH Formulas: QUIZ 7 SPRING...

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Unformatted text preview: PHYSICS 2D PROF. HIRSCH Formulas: QUIZ 7 SPRING QUARTER 2005 June 3 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 7 SPRING QUARTER 2005 June 3 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 mw 2 x 2h ; x op = x , pop = h∂ ; < A >= i ∂x • -• Ú dxY A * op Y Eigenvalues and eigenfunctions : Aop Y = a Y ; Harmonic oscillator : Yn ( x ) = Cn H n ( x )e - uncertainty : ; E= DA = < A 2 > - < A > 2 p2 1 + mw 2 x 2 ; Dn = ±1 2m 2 ; 1 E n = ( n + ) hw 2 † † Step potential : R = Tunneling : ( k1 - k2 ) 2 , ( k1 + k2 ) 2 ; T = 1- R ; x2 k= -2 a ( x )dx x1 2m (E - V ) h2 ; † † † † † † † † † † 2 m[V ( x ) - E ] h2 r r r h2 ∂ 2Y ∂ 2Y ∂ 2Y Schrodinger equation in 3d : ( 2 + 2 + 2 ) + V( r ) Y(r ) = EY(r ) 2m ∂x ∂y ∂z 2 2 p 2 h 2 n12 n 2 n 3 3D square well : Y(x, y, z) = Y1 ( x ) Y2 ( y ) Y3 ( z) ; E = ( 2 + 2 + 2) 2 m L1 L2 L3 Spherically symmetric potential : Yn,l ,m ( r,q, f ) = Rnl ( r)Ylm (q, f ) ; Ylm (q, f ) = f lm (q )e imf rrr h∂ Angular momentum : L = r ¥ p ; Lz = ; L2Ylm = l(l + 1) h 2Ylm ; L z = mh i ∂f Z2 2 2 Radial probability density : P(r) = r | Rn,l ( r ) | ; Energy : E n = -13.6eV 2 n 1 Z 3 / 2 - Zr / a 0 Ground state of hydrogen and hydrogen - like ions : Y1,0,0 = 1 / 2 ( ) e p a0 Æ Æ -e eh Orbital magnetic moment : m = L ; mz = -mB ml ; mB = = 5.79 ¥ 10-5 eV / T 2 me 2 me r 1 -e r Spin 1/2 : s = , | S |= s( s + 1) h ; Sz = msh ; ms = ±1 / 2 ; ms = gS 2 2 me y ~ e -ax T ~ e -2aDx ; T~e Ú a ( x) = rrr Total angular momentum : J = L + S ; | J |= † † r r -e r rr Orbital + spin mag moment : m= ( L + gS ) ; Energy in mag. field : U = -m ⋅ B 2m Two particles : Y( x1, x 2 ) = + /- Y( x 2 , x1 ) ; symmetric/antisymmetric Screening in multielectron atoms : Z Æ Z eff , 1 < Z eff < Z Orbital ordering: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 6d ~ 5f j ( j + 1) h ; | l - s |£ j £ l + s ; - j £ m j £ j Justify all your answers to all problems PHYSICS 2D PROF. HIRSCH Problem 1 (15 points) QUIZ 7 SPRING QUARTER 2005 June 3 2005 An electron is in a box of dimensions L1, L2 = 2 L1, L3 = L1 / 2 in the x,y and z directions. (a) Find the quantum numbers and the energies of the 5 lowest energy states expressed as multiples of the ground state energy E0 . Give your answers in order of increasing energy. (b) Find two sets of quantum numbers that give rise to degenerate energy states, and the corresponding energy in terms of E0 . † (c) Find the energy (in terms of E0 ) of the lowest state for which the probability of finding the electron at height z=L3 /2 is zero for all values of x and y. Problem 2 (15 points) An electron in hydrogen is in a state with radial wave function R( r) = Cr 2e- r / 3 a 0 and angular part of the wave function Y (q, f ) = C ' sin q cosqe- if . C and C' are constants. (a) What are the quantum numbers n, l and m? (b) what are the values of the z-component of the angular momentum, Lz, and of the magnitude of the orbital angular momentum, |L|? † (c) At what value of r is the electron most likely to be found? Give your answer in terms of † a0 . Problem 3 ( 15 points) The electronic configuration of boron (B) is 1s2 2s2 2p. (a) What are the possible values of the total angular momentum quantum number j? (b) Is the energy of the states with the different values of j found in (a) the same or different? Explain. If different, state for which j the energy is lowest and why. (c) Explain why the first ionization potential of B is smaller than that of beryllium (Be), with electronic configuration 1s2 2s2 , while that of Be is larger than that of lithium (Li) with electronic configuration 1s2 2s. Justify all your answers to all problems ...
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