Transcript Document

Chapter 3: Wave Properties of Particles
De Broglie Waves
photons
E  pc
h
p
  c


E  h
E  h
For a massive particle :
h

mv
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Example 3.1 Find the de Broglie wavelengths of a 46 g golf ball with a velocity of 30
m/s and an electron with a velocity of 107 m/s.
Example 3.2 Find the kinetic energy pf a proton whose de Broglie wavelength is 1.00 fm
(10-15 m), which is roughly the proton diameter.
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Waves of what?
“normal” waves
are a disturbance in space
carry energy from one place to another
often (but not always) will (approximately) obey the
classical wave equation
matter waves
disturbance is the wave function Y(x, y, z, t )
probability amplitude Y
probability density p(x, y, z, t ) =|Y|2
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wave properties:
phase velocity
v p  
for a massive particle
h mc2
c
vp 
c c
mv h
v
for a massless particle
h E 1 pc
vp 

c
ph p 1
phase velocity does not describe particle motion
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Generic wave properties
oscillations at a particular point
Y  Acos2t
travelling waves (1 - d)
 x 
Y  Acos2 t  
 v p 
in more standard forms

x 
Y  Acos2 t  

 
or
in 3  d : Y  Acost  k  r 
Y  Acost  kx
  2 (angular frequency)
2
k
(wave number)

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phase and group velocities
simple plane wave inadequate to describe particle motion
problems with phase velocity and infinite wave train
represent particle with wave packet (wave group)
simplified version: superposition of two waves of slightly
different wavelength
-if wave velocity is independent of wavlength, each wave (and
thus the packet) travel at the same speed
-if wave velocity is depends upon wavlength, each wave travels at
a different speed, in turn different from the wave packet speed.
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Y1  Acost  kx

Y2  Acos    t  k  kx
Y  Y1  Y2


 
1
1
Y  2Acos 2   t  2k  k x cos  t  k x
2
2
with    ,  k


k 
Y  2Acos
t
x cos t  kx

2 
 2



phase velocity = wave velocity of carrier : v p 

k

group velocity = wave velocity of envelope : v g 
k
d
for more than two wave contiributions : v g 
dk
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de Broglie waves for massive particles
m0 c 2
E
  2  2  2
h
h 1  (v c ) 2
m0 v
2 2
2
k

mv

h
h 1  (v c ) 2
d d dv
vg 

dk dk dv
d dv  2
dk dv  2

m0 c 2 v
h 1  (v c )
m0 c 2


2 32
h 1  (v c ) 2

32
 vg  v
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Example 3.3 An electron has a de Broglie wavelength of 2.00 pm Find its kinetic
energy, as well as the phase and group velocity of the waves.
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electron gun
Particle Diffraction:
The Davisson-Germer experiment
scattering of electrons from annealed
surface (single crystal)
classically, diffuse scattering
waves produce constructive/destructive
interference ala x-ray diffraction
n  2d sin 
h
1

KE  qV  mv 2
mv
2
electron
detector
smaller wavelength => finer resolution as in electron microscope
Example: 54 eV electrons are scattered off of a surface with a strong maximum at an angle
of 50o with respect the incoming beam of electrons. If the spacing between the atomic
planes is .091 nm, what is the wavelength of the electrons from diffraction theory? What
is the de Broglie wavelength of the electrons?
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Particle in a box
Wave Function
Y(x)  Asin( kx)sin t
2
k
  2f
L

Boundaries  Y  0
n  2L n  1,2,
h nh
pn 

 n 2L
n2 h2
En 
8mL2
n  1,2,
Examples: electron in 0.10 nm box, neutron in
1.00 fm box, Gallis in room
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The Uncertainty Principle: limits on probabilities with wave packets
probability density |Y|2
maximum near center of wave packet (or near “average”)
non-zero near maximum=> uncertainty in position x
combination of several wavelengths => uncertainty in wave
number => uncertainty in momentum p
uncertainty principle: decreasing x (p)will eventually
drive up p (x).
It is impossible to know both the exact position and
exact momentum of an object at the same time.
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Wave function as a superpopsition of cosine waves:
at a particular instant in time

Y x   g(k)coskx dk
more generally : Yx 

ikx
g(k)e
dk


0
- Fourier Transform
xk  12
- analogous to minimum bandwidth/minimum pulsewidth
p
h

h

2
, k
2

2
2
1
k
p  k 
p  p
h
h
("h  bar")
1
xk = x p  12
xp 
2
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Example 3.6: A measurement establishes the position of a proton with an accuracy of
+/-.001nm. Find the uncertainty in the proton’s position 1.00 s later. assume v<<c.
Uncertainty principle II: measurement as interaction
observe a particle by bouncing photons off of the particle
p ~ h  photon
x ~  photon
 xp ~ h
BUT: this uncertainty is an intrinsic limit, not an artifact of
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measurement!!!
Applications of the uncertainty principle
Example 3.7: A typical atomic nucleus is about 5 fm in radius. Use the uncertainty
principle to estimate a lower limit for the energy of an electron confined to the nucleus.
Example 3.8: A a hydrogen atom is about .053 nm in radius. Use the uncertainty
principle to estimate a lower limit for the energy of an electron confined to the atom
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Energy-Time uncertainty
1
 
 Et  h
t
E  h
More precisely
Et 
2
Example 3.9: An “excited” atom gives up its excess energy by emitting a photon of a
characteristic frequency. The average time between the excitation of the atom and the
emission of the photon is 10.0 ns. What is the inherent uncertainty in the frequency of the
photon?
Chapter 3 problems: 2,3,4,5,7,9,16,17,22,24,27,28,35,37,38
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