PHYSICS 4E
PROF. HIRSCH
QUIZ 3
Formulas:
Relativistic energy - momentum relation E =
SPRING QUARTER 2009
May 8
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 = "
!
!
!
!
' u(%)d%
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
-%
Wave function
!
!
!
!
!
!
" square well : # n (x) =
2
n$x
$ 2h2n 2
sin(
) ; En =
;
L
L
2mL2
x op = x , pop =
h %
; < A >=
i %x
"
*
& dx# A
#
op
-"
PHYSICS 4E
PROF. HIRSCH
QUIZ 3
SPRING QUARTER 2009
May 8
Eigenvalues and eigenfunctions : Aop " = a " (a is a constant) ; uncertainty :
Harmonic oscillator : "n (x) = Cn H n (x)e
!
Step potential : R =
!
m$ 2
#
x
2h
(k1 " k2 ) 2
,
(k1 + k2 ) 2
#A = < A 2 > $ < A > 2
2
1
p
1
1
; E n = (n + )h$ ; E =
+ m$ 2 x 2 = m$ 2 A 2 ; %n = ±1
2
2m 2
2
T = 1" R
;
k=
2m
(E " V )
h2
b
%
-2 # (x )dx
Tunneling :
" (x) ~ e -#x
;
T ~ e -2#$x ;
T~e
!
3D square well : "(x,y,z) = "1 (x)"2 (y)"3 (z) ; E =
a
;
# (x) =
# 2 h 2 n12 n 22 n 32
( +
+ )
2m L12 L22 L23
2m[V (x) - E]
h2
!
Justify all your answers to all problems
!
Problem 1 (10 pts)
An electron is in a harmonic oscillator potential where it can absorb and emit photons of
wavelength λ=3000A. Assuming the electron is in the ground state.
(a) Find the classical amplitude of oscillation, in Angstrom (classical
amplitude=amplitude of oscillation of a classical electron with the same total energy).
(b) Calculate the uncertainty in the position of this electron, Δx, in Angstrom. Show all
steps. How does it compare to the classical amplitude of oscillation?
(c) Find the uncertainty in the momentum of this electron, Δp, in units eV/c.
Use:
m#
&
&
2
m# 1/ 4 % 2h x 2
$
1 $
% (x 2
" 0 (x) = (
) e
; ' dx e
=
;
dx x 2e% (x =
'
$h
(
2 (3
-&
-&
Problem 2 (10 pts)
!
1,000 electrons
V0
x
x=0
A beam of 1,000 electrons of kinetic energy E=14eV is incident from the left on the
potential step of unknown height V0 for x>0, shown in the figure. The wavefunction for
each electron is
" I (x) = e ikx + Be#ikx
for x < 0
" II (x) = 1.3e ik2 x
for x > 0
(except for an overall normalization constant that is unimportant).
(a) Find the wavenumber of the transmitted electrons, k2, in A-1.
(b) How many electrons are reflected, how many are transmitted?
(c) Find the value of V0 in eV.
Use h 2 /2me = 3.81 eV A 2 .
!
!
PHYSICS 4E
PROF. HIRSCH
QUIZ 3
SPRING QUARTER 2009
May 8
Problem 3 (10 pts)
A finite one-dimensional square well has height V0=10eV. It has only 2 bound states for
an electron.
(a) What can you say about the length of this well, L? Give an exact upper bound for L,
in Angstrom.
(b) If the particle in this well is a proton instead of an electron, estimate roughly the
number of bound states.
(c) How do the uncertainties in the position (Δx) of an electron and a proton in their
lowest energy state in this well compare? Are they equal, and if not, which one is larger?
Explain.
Justify all your answers to all problems