Download Chapter 3 de Broglie`s postulate: wavelike properties of particles

Chapter 3 de Broglie’s postulate: wavelike properties of particles
3.1 Matter wave: de Broglie

the total energy of matter related to the frequency ν of the wave is
E=hν

the momentum of matter related to the wavelength λ of the wave is
p=h/λ
Ex: (a) the de Broglie wavelength of a baseball moving at a
speed v=10m/s, and m=1kg. (b) for a electron K=100 eV.
(a)   h / p  6.6  10
 34
/(1  10)  6.6  10
 35
m  6.6  10
 25
(b)   h / p  h / 2mK
 34
 6.6  10 / 2  9.1  10
  (electron)   (baseball )
 31
 100  1.6  10
19
o
 1.2 A
o
A
Chapter 3 de Broglie’s postulate: wavelike properties of particles
The experiment of Davisson and Germer
(1) A strong scattered electron beam is detected at θ=50o for V=54 V.
(2) The result can be explained as a constructive interference of waves
scattered by the periodic arrangement of the atoms into planes of the
crystal.
(3) The phenomenon is analogous to the Bragg-reflections (Laue pattern).
 1927, G. P. Thomson showed the diffraction of electron beams passing through
thin films confirmed the de Broglie relation λ=h/p. (Debye-Scherrer method)
Chapter 3 de Broglie’s postulate: wavelike properties of particles
Bragg reflection:
constructive interference:
2d sin   n
o
d  0.91 A ,   50 o and   90 o   / 2  65 o
for n  1
o
   2d sin   2  0.91  sin 65  1.65 A
o
(X - ray wavelength )
for electron K  54 eV
consistent
   h / 2mK
 34
 6.6  10 / 2  9.1  10
(electron wavelength )
 31
 54  1.6  10
 19
o
 1.65 A
Chapter 3 de Broglie’s postulate: wavelike properties of particles
Debye-Scherrer diffraction
X-ray diffraction:
zirconium oxide crystal
electron diffraction :
gold crystal
Laue pattern of X-ray (top) and
neutron (bottom) diffraction for
sodium choride crystal
Chapter 3 de Broglie’s postulate: wavelike properties of particles
3.2 The wave-particle duality
Bohr’s principle of complementarity: The wave and particle models are
complementary; if a measurement proves the wave character of matter, then it
is impossible to prove the particle character in the same measurement, and
conversely
 Einstein’s interpretation: for radiation (photon) intensity
I  (1 / 0 c ) 2  hN   2  N
 2 is a probability measure of photon density
 Max Born: wave function of matter is  ( x , t ) just as  satisfies wave equation
 2 is a measure of the probability of finding a particle in unit volume at a
given place and time. Two superposed matter waves obey a principle of
superposition of radiation.
Chapter 3 de Broglie’s postulate: wavelike properties of particles
3.3 The uncertainty principle
Heisenberg uncertainty principle: Experiment
cannot simultaneously determine the exact value of
momentum and its corresponding coordinate.
p x x   / 2
Et   / 2
Bohr’s thought experiment:
(1) p x  2 p sin  '  ( 2h /  ) sin  '
x   / sin  ' ( a diffraction apparatus a   /  )
p x x  ( 2h /  ) sin  '   / sin  '  2h   / 2
(2) E  p x2 / 2m  E  2 p x p x / 2m  v x p x
x  v x t  v x  x / t  E  ( x / t )p x
 E t   p x  x  2 h   / 2
Bohr’s thought experiment
Chapter 3 de Broglie’s postulate: wavelike properties of particles
3.4 Properties of matter wave
 wave propagation velocity:
h
E
E mv 2 / 2 v
w    ( )  ( ) 


p
h
p
mv
2
w  v Why?
 a de Broglie wave of a particle
 ( x , t )  sin 2 ( x /   t ) set   1/
 ( x , t )  sin 2 (x  t )
(1) x fixed, at any time t the amplitude is one, frequency is ν.
(2) t fixed, Ψ(x,t) is a sine function of x.
n  0,1,2,.......
(3) zeros of the function are at 2 (xn  t )  n
xn  n / 2  t  xn  n / 2  ( /  )t
these nodes move along x axis with a velocity w  dxn / dt   /   
it is the node propagation velocity (the oscillation velocity)
Chapter 3 de Broglie’s postulate: wavelike properties of particles
modulate the amplitude of the waves
 ( x , t )  1 ( x , t )  2 ( x , t )
1 ( x , t )  sin 2 (x  t ),
2 ( x , t )  sin 2 [(  d ) x  (  d )t ]
d
d
( 2  d )
( 2  d )
x
t ]  sin 2 [
x
t]
2
2
2
2
d
d
for d  2 and d  2   ( x , t )  2 cos[
x
t ]  sin 2 (x  t )
2
2
(1) the velocity of the individual wave is w   / 
  ( x , t )  2 cos[
(2) the group velocity of the wave is g 
dν / 2
dν

dκ / 2
d
Chapter 3 de Broglie’s postulate: wavelike properties of particles
 group velocity of waves equal to moving velocity of particles
E
dE
1
p
dp
 d 
    d 
h
h
 h
h
d dE / h dE
g


d
dp / h
dp
1
dE
E  mv 2
p  mv  dE  mvdv 
v
2
dp
 gv
 
 The Fourier integral can prove the following universal properties
of all wave. x  1 / 4 for   1/ , and t  1 / 4
for matter wave : p  h /   1 /     p / h
x  x ( p / h)  (1 / h)xp  1 / 4
 px   / 2
uncertainty principle
E  h    E / h  t ( E / h)  (1 / h)tE
uncertainty principle
 Et   / 2
the consequence
of duality
Chapter 3 de Broglie’s postulate: wavelike properties of particles
Ex: An atom can radiate at any time after it is excited. It is found that in a
typical case the average excited atom has a life-time of about 10-8 sec. That
is, during this period it emit a photon and is deexcited. (a) What is the
minimum uncertainty  in the frequency of the photon?o (b) Most photons from
sodium atoms are in two spectral lines at about   5890 A . What is the fractional
width of either line,  /  ? (c) Calculate the uncertainty  E in the energy of the
excited state of the atom. (d) From the previous results determine, to within an
accuracy  E , the energy E of the excited state of a sodium atom, relative to its o
lowest energy state, that emits a photon whose wavelength is centered at 5890 A
(a) t  1 / 4    1 / 4t  8  10 6 sec-1
(b)   c /   3  1010 / 5890  10  8  5.1  1014 sec-1
 /  8  10 6 / 5.1  1014  1.6  10  8 natural width of the spectral line
h / 4
h
6.63  10  34
8
(c) E 



3
.
3

10
eV the width of the state
8
t
4t
4  10
(d) /  h/h   E / E  E  E /(  /  )  2.1 eV
Chapter 3 de Broglie’s postulate: wavelike properties of particles
uncertainty principle in a single-slit diffraction
for a electron beam:
sin  

y
,
py
p
 sin 
p y  p y  p sin  
 p y y 
p
y
h 

y  h 
 y
2
Chapter 3 de Broglie’s postulate: wavelike properties of particles
Ex: Consider a microscopic particle moving freely along the x axis. Assume
that at the instant t=0 the position of the particle is measured and is uncertain
by the amount x0 . Calculate the uncertainty in the measured position of the
particle at some later time t.
At t  0  p x   / 2x0
  v x  p x / m   / 2 m  x 0
At time t  x  tv x  t / 2m x0
x0   x  or t   x 
Chapter 3 de Broglie’s postulate: wavelike properties of particles

Some consequences of the uncertainty principle:
(1) Wave and particle is made to display either face at will but not both
simultaneously.
 Dirac’s
relativistic
of electron:
E   ofc 2radiation;
p 2  m02c 4
(2) We can
observequantum
either themechanics
wave or the
particle behavior

but assumption:
the uncertainty
principle
prevents
from
observing
Dirac’s
a vacuum
consists
of aus
sea
of electrons
inboth together.
(3) Uncertainty
principle
makes
predictions
onlyatofallprobable
negative
energy levels
which
are normally
filled
points inbehavior
space. of
the particles.
 The philosophy of quantum theory:
(1) Neil Bohr: Copenhagen interpretation of quantum mechanics.
(2) Heisenberg: Principally, we cannot know the present in all details.
(3) Albert Einstein: “God does not play dice with the universe”
The belief in an external world independent of the perceiving subject is
the basis of all natural science.