PHYSICS 4E
PROF. HIRSCH
QUIZ 2
SPRING QUARTER 2010
APRIL 23
Formulas and constants:
hc = 12,400 eV A ; k B = 1/11,600 eV/K ; ke 2 = 14.4eVA ; me c 2 = 0.511"10 6 eV ; m p /me = 1836
!
!
!
Relativistic energy - momentum relation E = m2 c 4 + p 2 c2
;
c = 3 ! 108 m / s
r
r
r r
Photons: E = hf ; p = E /c ; f = c / "
Lorentz force: F = qE + qv " B
1
Photoelectric effect : eV0 = ( mv 2 ) max = hf " # , # $ work function
2
%
%
2
dIn
1 '
1
x3
'4
!Integrals : In " & x n e# $x dx ;
! ; I1 =
= #In +2 ; I0 =
; & x dx =
d$
2 $
2$
15
0
0 e #1
_
_
8#
hc
1
Planck's law : u( ") = n( " ) E ( ") ; n( ") = 4
;
E ( ") =
"
" e hc / "kB T $1
Energy in a mode/oscillator : E f = nhf ; probability P(E) " e#E / kB T
&
4
!
!
!
!
!
!
!
!
' u(%)d%
0
Wien's displacement law : "m T = hc /4.96k B
h
h
Compton scattering :
"'- " =
(1# cos $ )
;
"c %
= 0.0243A
mec
mec
kq Q
1
"N # 4
Rutherford scattering: b = " 2 cot(# /2) ;
m" v
sin ($ /2)
kq q
kq
Electrostatics : F = 12 2 (force) ; U = q0 V (potential energy) ; V =
(potential)
r
r
1
1
1
1
Hydrogen spectrum
= R( 2 # 2 ) ! ;
R = 1.097 $10 7 m#1 =
! :
"
m
n
911.3A
a0
Z2
h2
ke 2
; E n = "E 0 2 ; a0 =
=
0.529A
;
E
=
= 13.6eV ; L = mvr = nh
0
Z
n
mke 2
2a0
1
ke 2 Z
ke 2 Z
v2
E k = mv 2 ; E p = "
; E = Ek + E p ; F = 2 = m
; hf = hc/# = E n " E m
2
r
r
r
mM
Reduced mass : µ =
; X - ray spectra : f 1/ 2 = An (Z " b) ; K : b = 1, L : b = 7.4
m+M
h
E
2$
p2
de Broglie : " = ; f =
; # = 2$f ; k =
; E = h# ; p = hk ; E =
; hc = 1973 eV A
p
h
"
2m
group and phase velocity : v g =
d"
dk
;
vp =
!
!
4
Stefan's law : R = "T ; " = 5.67 #10 W /m K ; R = cU /4 , U =
Bohr atom : rn = r0 n 2 ; r0 =
!
2
$8
Justify all your answers to all problems
"
k
;
Heisenberg
: #x#p ~ h
;
#t#E ~ h
PHYSICS 4E
PROF. HIRSCH
QUIZ 2
SPRING QUARTER 2010
APRIL 23
Problem 1 (10 pts)
(a) Compare the following properties of an electron in the lowest energy state in the
hydrogen atom with an electron in the first excited state of the He+ ion, ignoring the
reduced mass correction. Denoting with subindex 1 the property of the electron in the H
atom and with subindex 2 the property of the electron in the He+ ion, with property (P)
being energy, radius, etc, find the ratio P2/P1 for the following properties:
(i) total energy; (ii) speed ; (iii) angular momentum ; (iv) radius ; (v) kinetic
energy (vi) potential energy ; (vii) ionization energy=energy required to remove the
electron from the atom/ion.
Example: (i) find E2/E1 , where E1=total energy of electron in n=1 level of H, E2=total
energy of electron in n=2 level of He+. etc.
(b) Repeat (a) taking into account the reduced mass correction. Use that the ratio of the
nucleon mass (proton or neutron) to the electron mass is approximately 1836. Helium has
four nucleons.
Problem 2 (10 pts)
Electrons of de Broglie wavelength λ=0.08A hit a gold foil (Z=79) and cause X-rays to
be emitted. A plot of the intensity of emitted X-rays versus their wavelength yields a
smooth curve starting at a minimum wavelength and some sharp peaks.
(a) Find the kinetic energy of the incident electrons, in eV.
(b) Find the minimum wavelength of emitted X-rays, in A.
(c) With increasing wavelengths above this minimum wavelength, a first group of sharp
peaks is seen in the X-ray intensity versus wavelength plot. Does it correspond to the K
series or the L series? Find the value of the maximum wavelength in the group and give
the quantum numbers associated with the transition it corresponds to.
Problem 3 (10 pts)
An electron located around the origin (x=0) is described by a wavepacket with
wavenumbers in the range k=10A-1 to k=14A-1 at time t=0.
(a) Estimate the uncertainty in the position of the electron, in A.
(b) Estimate the uncertainty in the momentum of the electron, in eV/c.
(c) Find an expression for the wavefunction of this electron at t=0, assuming the
amplitude of the waves in the range 10A-1 to 14A-1 is constant. Using this expression
estimate the ratio of the magnitude of the wavefunction in the neighborhood of position
x~20A to the magnitude of the wavefunction at x=0.
Justify all your answers to all problems