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Course: PHYS 2D 2D, Spring 2011
School: UCSD
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2D PHYSICS PROF. HIRSCH FINAL EXAM WINTER QUARTER 2011 MARCH 14th, 2011 ! ! ! Formulas: Time dilation; Length contraction : &quot;t = #&quot;t'\$ # &quot;t p ; L = Lp /# ; c = 3 %10 8 m /s Lorentz transformation : x'= &quot; (x # vt) ; y' = y ; z' = z ; t'= &quot; (t # vx /c 2 ) ; inverse : v \$ -v Spacetime interval : (&quot;s) 2 = (c&quot;t) 2 - [&quot;x 2 + &quot;y 2 +...

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2D PHYSICS PROF. HIRSCH FINAL EXAM WINTER QUARTER 2011 MARCH 14th, 2011 ! ! ! Formulas: Time dilation; Length contraction : "t = #"t'\$ # "t p ; L = Lp /# ; c = 3 %10 8 m /s Lorentz transformation : x'= " (x # vt) ; y' = y ; z' = z ; t'= " (t # vx /c 2 ) ; inverse : v \$ -v Spacetime interval : ("s) 2 = (c"t) 2 - ["x 2 + "y 2 + "z 2 ] uy ux " v Velocity transformation : ux '= ; uy '= ; inverse : v \$ -v 2 1" ux v /c # (1" ux v /c 2 ) Relativistic Doppler shift : f obs = f source 1+ v /c / 1" v /c (approaching) r r Momentum : p = " mu ; Energy : E = " mc 2 ; Kinetic energy : K = (" #1)mc 2 Rest energy : E 0 = mc 2 Electron : me = 0.511 MeV /c 2 ! ! ! ! ! ; E= p 2c 2 + m 2c 4 Neutron : mn = 939.55 MeV /c 2 Proton : mp = 938.26 MeV /c 2 ! ! ! ! ! ! Atomic mass unit : 1 u = 931.5 MeV /c 2 ; electron volt : 1eV = 1.6 "10 -19 J 4 Stefan's law : etot = "T , etot = power/unit area ; " = 5.67 #10\$8 W /m 2K 4 # hc etot = cU /4 , U = energy density = \$ u( ",T)d" ; Wien's law : "m T = 4.96kB 0 -E/(kB T ) Boltzmann distribution : P(E) = Ce 8\$ hc / " 8\$f 2 Planck's law : u" ( ",T) = N " ( ") # E ( ",T) = 4 # hc / "kB T ; N( f ) = 3 " e %1 c Photons : E = hf = pc ; f = c / " ; hc = 12,400 eV A ; k B = (1/11,600)eV /K Photoelectric effect : eVs = K max = hf " # , # \$ work function; Bragg equation : n% = 2d sin & Compton scattering : "'- " = h h (1# cos \$ ); = 0.0243A ; Coulomb constant : ke 2 = 14.4 eV A mec mec ! ! ! ! ! ! ! ! kq1q2 kq kq q ; Coulomb potential : V = ; Coulomb energy : U = 1 2 2 r rr r r r r Force in electric and magnetic fields (Lorentz force) : F = qE + qv " B Z2 1 Rutherford scattering : "n = C 2 ; hc = 1,973 eV A 4 K# sin (\$ /2) 1 1 1 1 Hydrogen spectrum : = R( 2 # 2 ) ; R = 1.097 \$10 7 m#1 = "mn m n 911.3A 2 2 2 2 ke Z Z ke m (ke ) m v2 ke 2 Z Bohr atom : E n = " = "E 0 2 ; E 0 = = e 2 = 13.6eV ; K = e ; U = " 2rn n 2a0 2h 2 r Coulomb force : F = hf = E i " E f ; rn = r0 n 2 ; r0 = a0 Z ; a0 = h2 = 0.529A me ke 2 ; L = me vr = nh angular momentum de Broglie : " = h E ;f = p h j ; # = 2\$f ; k = 2\$ ; " E = h# ; p = hk ; i(kx -# (k )t ) E= p2 2m Wave packets : y(x,t) = \$ a j cos(k j x " # j t), or y(x,t) = % dk a(k) e ; &k&x ~ 1 ; &#&t ~ 1 ! ! d" ; " vp = ; Heisenberg : #x#p ~ h ; #t#E ~ h dk k E -i t h2 " 2# "# Schrodinger equation : + U(x)#(x,t) = ih ; #(x,t) = \$ (x)e h 2m "x 2 "t group and phase velocity : v g = ! ! PHYSICS 2D PROF. HIRSCH FINAL EXAM WINTER QUARTER 2011 MARCH 14th, 2011 % Time " independent Schrodinger equation : - h 2 # 2\$ + U(x)\$ (x) = E\$ (x) ; 2m #x 2 2 2 2 2 -% & dx | \$ (x) |2 = 1 " square well : # n (x) = ! 2 n\$x \$ hn sin( ) ; En = L L 2mL2 # m\$ 2 x 2h ; h = 3.81eVA 2 (electron) 2me h \$ i \$x Harmonic oscillator : "n (x) = H n (x)e 1 p2 1 1 ; E n = (n + )h\$ ; E = + m\$ 2 x 2 = m\$ 2 A 2 ; %n = 1 2 2m 2 2 ! ! Expectation value of[Q] :< Q >= # " * (x)[Q]" (x) dx ; Momentum operator : p = Eigenvalues and eigenfunctions : [Q] " = q " (q is a constant) ; uncertainty : #Q = < Q2 > \$ < Q > 2 ! Step potential : reflection coef : R = Tunneling : ! (k1 " k 2 ) 2 , T = 1" R ; (k1 + k 2 ) 2 x2 k= 2m (E " U) h2 2m[U(x) - E] h2 -2 # (x )dx " (x) ~ e -# x ; T = e -2#\$x ; T =e x1 % ; # (x) = ! ! ! ! ! ! ! ! ! ! ! ! ! r r r r -i E t h2 2 \$# Schrodinger equation in 3D : " # + U( r )#( r ,t) = ih ; #( r ,t) = % ( r )e h 2m \$t 2 2 # 2 h 2 n12 n 2 n 3 3D square well : "(x,y,z) = "1 (x)"2 (y)"3 (z) ; E = ( 2+ + ) 2m L1 L2 L2 2 3 Spherically symmetric potential: "n,l,m l (r,#, \$ ) = Rnl (r)Ylm l (#, \$ ) ; Ylm l (#, \$ ) = Plm l (# )e im l\$ r r r h # Angular momentum : L = r " p ; [Lz ] = ; [L2 ]Ylm = l(l + 1)h 2Ylm ; [L z ] = mh i #\$ ke 2 Z 2 Radial probability density : P(r) = r 2 | Rnl (r) |2 ; Energy : E n = " 2a0 n 2 1 Z Ground state of hydrogen and hydrogen - like ions : "1,0,0 = 1/ 2 ( ) 3 / 2 e\$Zr / a 0 # a0 " " #e eh Orbital magnetic moment : = L ; z = #B ml ; B = = 5.79 \$10#5 eV /T 2me 2me r 1 "e r Spin 1/2 : s = , | S |= s(s + 1)h ; Sz = msh ; ms = 1/2 ; s = gS 2 2me r r "e r r r Orbital + spin mag moment : = ( L + gS ) ; Energy in mag. field : U = " # B 2m r r r r Two particles : "(r1, r2 ) = + /# "( r2 , r1 ) ; symmetric/antisymmetric Screening in multielectron atoms : Z " 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 l l Justify all your answers to all problems. Write clearly.
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