PHYSICS 2D
PROF. HIRSCH
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QUIZ 5
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
Proton : mp = 938.26 MeV /c 2
Neutron : mn = 939.55 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 &
h
h
(1# cos $ );
= 0.0243A ; Coulomb constant : ke 2 = 14.4 eV A
mec
mec
r
r
r r
Force in electric and magnetic fields (Lorentz force) : F = qE + qv " B ;
Drag force : D = 6#a$v
Compton scattering : "'- " =
C
;
hc = 1,973 eV A
sin (# /2)
1
1
1
1
Hydrogen spectrum :
= R( 2 # 2 )
;
R = 1.097 $10 7 m#1 =
"mn
m
n
911.3A
kq q
kq q
mv 2
Electrostatic force, energy : F = 12 2 ; U = 1 2 . Centripetal force : Fc =
r
r
r
ke 2 Z
Z 2E0
ke 2
mev 2
ke 2 Z
Bohr atom : E n = "
= " 2 ; E0 =
= 13.6eV ; K =
; U ="
2rn
n
2a0
2
r
Rutherford scattering : "n =
!
hf = E i " E f ; rn = r0 n 2 ; r0 =
!
de Broglie : " =
!
FALL QUARTER 2009
NOVEMBER 20th, 2009
h
E
;f =
p
h
4
a0
Z
; a0 =
h2
= 0.529A ; L = me vr = nh angular momentum
me ke 2
; # = 2$f ; k =
2$
;
"
Wave packets : y(x,t) = $ a j cos(k j x " # j t), or y(x,t) =
E = h# ; p = hk ;
% dk a(k) e
i(kx -# (k )t )
E=
p2
2m
; &k&x ~ 1 ; &#&t ~ 1
j
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d"
"
; vp =
; Heisenberg : #x#p ~ h ; #t#E ~ h
dk
k
E
-i t
h2 " 2#
"#
h
Schrodinger equation : +
U(x)#(x,t)
=
ih
;
#(x,t)
=
$
(x)e
2
2m "x
"t
group and phase velocity : v g =
PHYSICS 2D
PROF. HIRSCH
QUIZ 5
FALL QUARTER 2009
NOVEMBER 20th, 2009
%
h 2 # 2$
+
U(x)
$
(x)
=
E
$
(x)
;
dx $ *$ = 1
&
2
2m #x
-%
2 2 2
2
2
n$x
$ hn
h
sin(
) ; En =
;
= 3.81eVA 2 (electron)
2
L
L
2mL
2me
Time " independent Schrodinger equation : -
" square well : # n (x) =
m$ 2
x
2h
1
p2 1
1
; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
!
Harmonic oscillator : "n (x) = H n (x)e
!
Expectation value of[Q] :< Q >= # " * (x)[Q]" (x) dx ; Momentum operator : p =
!
!
#
Eigenvalues and eigenfunctions : [Q] " = q " (q is a constant) ; uncertainty :
Step potential : reflection coef : R =
!
(k1 " k 2 ) 2
, T = 1" R ;
(k1 + k 2 ) 2
#Q = < Q2 > $ < Q > 2
2m
(E " U)
h2
x2
%
-2 # (x )dx
Tunneling :
" (x) ~ e -# x
;
T = e -2#$x ;
T =e
x1
!
!
k=
h $
i $x
;
# (x) =
2m[U(x) - E]
h2
Justify all your answers to all problems
Problem 1 (10 points)
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An electron is described by the wavefunction
" (x) = 0 for x < 0 ; " (x) = Cxe#x / 2 for x $ 0
with C a constant.
(a) Find C from normalization
(b) Find <x>
(c) Find <p>, <p2> and Δp, with p the momentum operator.
$
n!
Hint: % dx x n e" #x = n +1
#
0
Problem 2 (10 points)
! Consider an electron described by the wavefunction:
" (x) = 0 for x < 0 ; " (x) = Cxe#3x for x $ 0
Consider the operator
1 #
Q= "
x #x
(a) Show that the wavefunction " is an eigenfunction of this operator, and find the
eigenvalue.
(b) Find <Q2> and the uncertainty ΔQ.
!
PHYSICS 2D
PROF. HIRSCH
QUIZ 5
FALL QUARTER 2009
NOVEMBER 20th, 2009
Problem 3 (10 points)
U(x)
1,000 electrons, E=4eV
electrons
? electrons, E=2eV
x
The potential in the figure is given by:
U(x)=0 for x<0
U(x)=3eV for 0<x<1A
U(x)=5eV for 1A<x<2A
U(x)=0 for x>2A.
Electrons are incident from the left, uniformly distributed in energy (i.e. the same number
of electrons are incident for any energy).
If 1,000 electrons are found on the right side of the barrier with energy 4eV, how many
are found on the right side with energy 2eV?
Justify all your answers to all problems