PHYSICS 2D
PROF. HIRSCH
QUIZ 4
WINTER QUARTER 2009
FEBRUARY 20, 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 4
WINTER QUARTER 2009
FEBRUARY 20, 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
n$x
$ hn
" square well : # n (x) =
sin(
) ; En =
L
L
2mL2
!
!
Justify all your answers to all problems
!
Problem 1 (10 points)
(a) An electron has de Broglie wavelength λ=1A. What is its kinetic energy, in eV?
(b) An electron has de Broglie wavelength λ=10-3A. What is its kinetic energy, in MeV?
(c) An electron is described by a wavepacket containing wavelengths in the range
9.9A " # " 10.1A . Estimate the uncertainty in the position of this electron, Δx, in A.
!
!
!
Problem 2 (10 points)
In a parallel universe, all laws of physics and physical constants are the same as in ours
except that the electrostatic interaction between an electron and a nucleus of atomic
number Z is not given by the expression U(r) = "ke 2 Z /r but rather by
ke 2 Z a0
U(r) = "
r3/2
with a0 = h 2 /(me ke 2 ) as usual. !
(a) Using the uncertainty principle, estimate the radius of an atom of atomic number Z in
this universe. Assume the radius of the atom is given by the radius of the lowest energy
orbit. Give an analytic expression in terms of Z and a0, and the numerical value in A for
Z=1.
(b) How tall do you expect the average person to be in that universe, in meters? Justify
your answer.
Problem 3 ( 10 points)
An electron is in a box of length 6A.
(a) Find the wavelength of the photon emitted when the electron makes a transition from
the n=2 level to the n=1 level. Give your answer in A.
(b) For an electron in the lowest energy state of this box, how much more likely is it to be
found at position x=3A than at position x=1A?
(c) What is the lowest energy state in this box (give its n-value) for which the electron is
equally likely to be found at x=3A as at x=1A? Justify your answer.
Justify all your answers to all problems