PHYSICS 2D
PROF. HIRSCH
QUIZ 5
WINTER QUARTER 2009
FEBRUARY 27, 2009
Formulas:
Time dilation; Length contraction : "t = #"t'$ # "t p ;
; c = 3 %10 8 m /s
L = Lp /#
1
!
!
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 '=
;
u
'=
; inverse : v $ -v
y
1" ux v /c 2
# (1" ux v /c 2 )
Relativistic Doppler shift : f obs = f source 1+ v /c / 1" v /c (approaching)
!
Momentum, energy (total, kinetic, rest) : p = # m u; E = # mc 2 ; K = (# $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
!
!
!
!
!
"
"
2
; electron volt : 1eV = 1.6 "10 -19 J
! Atomic mass unit : 1 u = 931.5 MeV /c
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 n / kB T
Planck : E n = nhf ; probability P(E n ) " e
; E ( $,T) = % E n P(E n ) / % P(E n )
n
!
!
!
!
!
!
!
!
!
Compton scattering : "'- " =
Electrostatics : F =
h
h
(1# cos $ );
= 0.0243A ; Coulomb constant : ke 2 = 14.4 eV A
mec
mec
kq1q2
kq q
(force) ; U = 1 2 (potential energy)
2
r
r
r
r
r
r
Force in electric and magnetic fields (Lorentz force) : F = qE + qv " B ;
Drag force : D = 6#a$v
C
Rutherford scattering : "n = 4
;
hc = 1,973 eV A
sin (# /2)
1
1
1
1
Hydrogen spectrum :
= R( 2 # 2 )
;
R = 1.097 $10 7 m#1 =
"mn
m
n
911.3A
2
2
2
2
ke Z
Z E
ke
mv
ke 2 Z
ke 2 Z
Bohr atom : E n = "
= " 2 0 ; E0 =
= 13.6eV ; K = e =
; U ="
2rn
n
2a0
2
2r
r
!
hf = E i " E f ; rn = r0 n 2 ; r0 =
!
de Broglie : " =
!
n
8$
hc / "
Planck's law : u( ",T) = N( ") # E ( ",T) = 4 # hc / "kB T
"
e
%1
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
h
E
;f =
p
h
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
!
!
group and phase velocity : v g =
d"
"
; vp =
;
dk
k
Heisenberg : #x#p ~ h ; #t#E ~ h
b
Born interpretation : P(x)dx =| " (x,t) |2 ; P(a # x # b) =
a
!
!
&
2
$ | " (x,t) |
dx ;
< f (x) >= $ | " (x,t) |2 f (x)
%&
PHYSICS 2D
PROF. HIRSCH
QUIZ 5
WINTER QUARTER 2009
FEBRUARY 27, 2009
E
-i t
h2 " 2#
"#
Schrodinger equation : + U(x)#(x,t) = ih
;
#(x,t) = $ (x)e h
2
2m "x
"t
%
h 2 # 2$
Time " independent Schrodinger equation : +
U(x)
$
(x)
=
E
$
(x)
;
& dx $ *$ = 1
2m #x 2
-%
2 2 2
2
2
n$x
$ hn
h
" square well : # n (x) =
sin(
) ; En =
;
= 3.81eVA 2 (electron)
2
L
L
2mL
2me
!
Eigenvalues and eigenfunctions : [Q] " = q " (q is a constant) ; uncertainty : #Q = < Q2 > $ < Q > 2
m$ 2
#
x
1
p2 1
1
Harmonic oscillator : "n (x) = H n (x)e 2h ; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
!
!
x2
!
%
-2 # (x )dx
" (x) ~ e -# x
Tunneling :
;
T = e -2#$x ;
T =e
x1
!
;
# (x) =
2m[U(x) - E]
h2
Justify all your answers to all (3) problems
!
Problem 1 (10 points)
An electron is in a one-dimensional potential
a
U(x) = x 4
4
with a=0.1eV/A4. Using the uncertainty principle,
(a) Estimate the uncertainty in the position of this electron when it is in the ground state
(state of lowest energy). Give your answer in A.
(b) Estimate the value of the ground state energy, in eV.
(c) Estimate the ratio of potential energy to kinetic energy in the ground state.
!
Problem 2 (10 points)
An electron is described by the wavefunction
" (x) = 0
for x < 0
" (x) = Cxe#x / 2
for x > 0
where x is in Angstrom and C is a constant.
(a) Find the value of C.
(b) Find the most probable position for this electron.
(c) Find the average position for this electron.
(d) Find the uncertainty in the position of this electron, Δx
!
#
Hint:
n "x
$xe
0
!
= n!
PHYSICS 2D
PROF. HIRSCH
QUIZ 5
WINTER QUARTER 2009
FEBRUARY 27, 2009
Problem 3 ( 10 points)
U(x)
electrons
2eV
1A
4eV
2eV
U=0
U=0
x
2A
1,000 electrons with kinetic energy E are incident from the left on the potential barrier
shown in the figure.
(a) Find the number of electrons transmitted if their energy is E=3eV.
(b) Find the number of electrons transmitted if their energy is E=1eV.
Justify all your answers to all problems