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PHYSICS 4E PROF. HIRSCH QUIZ 3 Formulas: Relativistic energy - momentum relation E = SPRING QUARTER 2009 May 8 m2 c 4 + p 2 c2 c = 3 ! 108 m / s ; Electron rest mass : me = 0.511 MeV / c2 ;Proton : mp = 938.26 MeV / c2 ;Neutron : m n = 939.55 MeV / c 2 _ 8# hc 1 ; E ( ") = 4 hc / "kB T " " e $1 Energy in a mode/oscillator : E f = nhf ; probability P(E) " e#E / kB T _ Planck's law : u( ") = n( " ) E ( ") ; n( ") = & ! ! ! ! ! ! Stefan's law : R = "T 4 ; " = 5.67 #10$8 W /m 2K 4 ; R = cU /4 , U = 0 hc Wien's displacement law : "m T = 4.96k B Photons : E = pc ; E = hf ; p = h/! ; f = c/! 1 Photoelectric effect : eV0 = ( mv 2 ) max = hf " # , # $ work function 2 h Compton scattering : ! ' -! = (1" cos # ) me c kq Q 1 "N # 4 Rutherford scattering: b = " 2 cot(# /2) ; m" v sin ($ /2) Constants : hc = 12,400 eV A ; hc = 1,973 eV A ; k B = 1/11,600 eV/K ; ke 2 = 14.4eVA kq q kq Electrostatics : F = 12 2 (force) ; U = q0 V (potential energy) ; V = (potential) r r ! ! Hydrogen spectrum : Bohr atom : E n = " ! ! ! ! ' u(%)d% 1 1 1 = R( 2 # 2 ) " m n R = 1.097 $10 7 m#1 = ; 1 911.3A ke 2 Z Z 2E ke 2 mk 2e 4 = " 2 0 ; E0 = = = 13.6eV ; E n = E kin + E pot , E kin = "E pot /2 = "E n 2rn n 2a0 2h 2 hf = E i " E f ; rn = r0 n 2 ; r0 = a0 Z ; a0 = h2 = 0.529A ; L = mvr = nh angular momentum mke 2 X - ray spectra : f 1/ 2 = An (Z " b) ; K : b = 1, L : b = 7.4 h E 2$ de Broglie : " = ; f = ; # = 2$f ; k = ; E = h# ; p = hk ; p h " group and phase velocity : v g = d" dk ; vp = " k ; Heisenberg : #x#p ~ h E= p2 2m #t#E ~ h ; "(x,t) =| "(x,t) | e i# (x,t ) ; P(x,t) dx =| "(x,t) |2 dx = probability E -i t h2 " 2# "# Schrodinger equation : + V(x)#(x,t) = ih ; #(x,t) = $ (x)e h 2m "x 2 "t % h 2 # 2$ Time " independent Schrodinger equation : + V(x) $ (x) = E $ (x) ; & dx $ *$ = 1 2m #x 2 -% Wave function ! ! ! ! ! ! " square well : # n (x) = 2 n$x $ 2h2n 2 sin( ) ; En = ; L L 2mL2 x op = x , pop = h % ; < A >= i %x " * & dx# A # op -" PHYSICS 4E PROF. HIRSCH QUIZ 3 SPRING QUARTER 2009 May 8 Eigenvalues and eigenfunctions : Aop " = a " (a is a constant) ; uncertainty : Harmonic oscillator : "n (x) = Cn H n (x)e ! Step potential : R = ! m$ 2 # x 2h (k1 " k2 ) 2 , (k1 + k2 ) 2 #A = < A 2 > $ < A > 2 2 1 p 1 1 ; E n = (n + )h$ ; E = + m$ 2 x 2 = m$ 2 A 2 ; %n = ±1 2 2m 2 2 T = 1" R ; k= 2m (E " V ) h2 b % -2 # (x )dx Tunneling : " (x) ~ e -#x ; T ~ e -2#$x ; T~e ! 3D square well : "(x,y,z) = "1 (x)"2 (y)"3 (z) ; E = a ; # (x) = # 2 h 2 n12 n 22 n 32 ( + + ) 2m L12 L22 L23 2m[V (x) - E] h2 ! Justify all your answers to all problems ! Problem 1 (10 pts) An electron is in a harmonic oscillator potential where it can absorb and emit photons of wavelength λ=3000A. Assuming the electron is in the ground state. (a) Find the classical amplitude of oscillation, in Angstrom (classical amplitude=amplitude of oscillation of a classical electron with the same total energy). (b) Calculate the uncertainty in the position of this electron, Δx, in Angstrom. Show all steps. How does it compare to the classical amplitude of oscillation? (c) Find the uncertainty in the momentum of this electron, Δp, in units eV/c. Use: m# & & 2 m# 1/ 4 % 2h x 2 $ 1 $ % (x 2 " 0 (x) = ( ) e ; ' dx e = ; dx x 2e% (x = ' $h ( 2 (3 -& -& Problem 2 (10 pts) ! 1,000 electrons V0 x x=0 A beam of 1,000 electrons of kinetic energy E=14eV is incident from the left on the potential step of unknown height V0 for x>0, shown in the figure. The wavefunction for each electron is " I (x) = e ikx + Be#ikx for x < 0 " II (x) = 1.3e ik2 x for x > 0 (except for an overall normalization constant that is unimportant). (a) Find the wavenumber of the transmitted electrons, k2, in A-1. (b) How many electrons are reflected, how many are transmitted? (c) Find the value of V0 in eV. Use h 2 /2me = 3.81 eV A 2 . ! ! PHYSICS 4E PROF. HIRSCH QUIZ 3 SPRING QUARTER 2009 May 8 Problem 3 (10 pts) A finite one-dimensional square well has height V0=10eV. It has only 2 bound states for an electron. (a) What can you say about the length of this well, L? Give an exact upper bound for L, in Angstrom. (b) If the particle in this well is a proton instead of an electron, estimate roughly the number of bound states. (c) How do the uncertainties in the position (Δx) of an electron and a proton in their lowest energy state in this well compare? Are they equal, and if not, which one is larger? Explain. Justify all your answers to all problems