PHYSICS 4E
PROF. HIRSCH
QUIZ 2
Formulas:
Relativistic energy - momentum relation E =
SPRING QUARTER 2009
APRIL 24
m2 c 4 + p 2 c2
c = 3 ! 108 m / s
;
Electron rest mass : me = 0.511 MeV / c2 ;Proton : mp = 938.26 MeV / c2 ;Neutron : m n = 939.55 MeV / c 2
_
8#
hc
1
;
E
( ") =
4
hc / "kB T
"
" e
$1
Energy in a mode/oscillator : E f = nhf ; probability P(E) " e#E / kB T
_
Planck's law :
u( ") = n( " ) E ( ")
;
n( ") =
&
!
!
!
!
!
!
Stefan's law : R = "T 4 ; " = 5.67 #10$8 W /m 2K 4 ; R = cU /4 , U =
0
hc
Wien's displacement law : "m T =
4.96k B
Photons :
E = pc ; E = hf ; p = h/!
;
f = c/!
1
Photoelectric effect : eV0 = ( mv 2 ) max = hf " # , # $ work function
2
h
Compton scattering :
! ' -! =
(1" cos # )
me c
kq Q
1
"N # 4
Rutherford scattering: b = " 2 cot(# /2) ;
m" v
sin ($ /2)
Constants : hc = 12,400 eV A ; hc = 1,973 eV A ; k B = 1/11,600 eV/K ; ke 2 = 14.4eVA
kq q
kq
Electrostatics : F = 12 2 (force) ; U = q0 V (potential energy) ; V =
(potential)
r
r
!
!
Hydrogen spectrum :
Bohr atom : E n = "
!
!
!
!
!
!
!
!
!
1
1
1
= R( 2 # 2 )
"
m
n
R = 1.097 $10 7 m#1 =
;
1
911.3A
ke 2 Z
Z 2E
ke 2 mk 2e 4
= " 2 0 ; E0 =
=
= 13.6eV ; E n = E kin + E pot , E kin = "E pot /2 = "E n
2rn
n
2a0
2h 2
hf = E i " E f ; rn = r0 n 2 ; r0 =
a0
Z
; a0 =
h2
= 0.529A ; L = mvr = nh angular momentum
mke 2
X - ray spectra : f 1/ 2 = An (Z " b) ; K : b = 1, L : b = 7.4
h
E
2$
de Broglie : " = ; f =
; # = 2$f ; k =
; E = h# ; p = hk ;
p
h
"
group and phase velocity : v g =
d"
dk
;
vp =
"
k
;
Heisenberg
: #x#p ~ h
E=
;
p2
2m
#t#E ~ h
"(x,t) =| "(x,t) | e i# (x,t ) ;
P(x,t) dx =| "(x,t) |2 dx = probability
E
-i t
h2 " 2#
"#
Schrodinger equation : + V(x)#(x,t) = ih
;
#(x,t) = $ (x)e h
2m "x 2
"t
%
h 2 # 2$
Time " independent Schrodinger equation : +
V(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
Wave function
!
' u(%)d%
PHYSICS 4E
PROF. HIRSCH
QUIZ 2
SPRING QUARTER 2009
APRIL 24
Justify all your answers to all problems
!
!
Problem 1 (10 pts)
An electron is described by the wavepacket
" (x,t) = % dke i(kx#$ (k )t )a(k)
with a(k)=A for k in the interval (k1,k2) and a(k)=0 outside that interval. k1=99A-1,
k2=101A-1 and A is a constant.
(a) Find an expression for " (x,t = 0) . What is the most probable position for this electron
at t=0?
(b) Estimate the uncertainty in the position of this electron at t=0, in A.
(c) Estimate the most probable position for this electron at t=1s.
(d) Estimate the !
group velocity of this electron, expressed as v/c.
Problem 2 (10 pts)
An electron is confined to move in a one-dimensional potential of the form
x2
V (x) = V0 2
x0
where V0=5eV and x0=0.01 A and x is the position coordinate.
(a) Assume this electron is known for certain to be within a distance of 0.01A from the
origin (x=0). Give a rough estimate of (i) its potential energy and (ii) its kinetic energy,
both in eV.
(b) Assume now the electron is in a state that minimizes its total energy (kinetic plus
potential). Estimate the uncertainty in its position and its potential and kinetic energies, in
eV.
Hint: Use Heisenberg's uncertainty principle. Use h 2 /me = 7.62 eV A 2
!
Problem 3 (10 pts)
(a) An electron is in the ground state of an infinite square well of length L. How much
more likely is it to be found at the center of the well (x=L/2) than at x=L/4?
(b) An electron in an infinite square well is in a state of energy 1000eV. Give a lower
bound for the length of this well, in A. Can you also give an upper bound?
Justify all your answers to all problems