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.