Download waves

Light is . . .
•Initially thought to be waves
•They do things waves do, like diffraction and interference
•Wavelength – frequency relationship
cf
•Planck, Einstein, Compton showed us they behave like particles (photons)
•Energy comes in chunks
•Wave-particle duality: somehow, they behave like both
E  hf
•Photons also carry momentum
•Momentum comes in chunks
p  E c  hf c  h 
Electrons are . . .
•They act like particles
•Energy, momentum, etc., come in chunks
•They also behave quantum mechanically
•Is it possible they have wave properties as well?
p  h
The de Broglie Hypothesis
•Two equations that relate the particle-like and
wave-like properties of light
E  hf
p h
1924 – Louis de Broglie postulated that these
relationships apply to electrons as well
•Implied that it applies to other particles as well
•de Broglie could simply explain the Bohr quantization condition
•Compare the wavelength of an electron in hydrogen to the circumference of
its path
L  n  mevr  pr 
hr


2 r

cancel 
n  2 r  C
Integer number of wavelengths fit around the orbit
Measuring wave properties of electrons
•What energy electrons do we want?
p h
4.136 10 eV  s   3.00 10 m/s 

p
hc

E  mv 
2 2 
2m 2mc 
2  0.511106 eV   2
2
2
15
2 2
8
2
2
1
2

1.504 10
18
2
eV  m
2
 nm 
 1.504 eV 

  
2
For atomic separations, want distances around 0.3 nm  energies of 10 or so eV
How can we measure these wave properties?
•Scatter off crystals, just like we did for X-rays!
•Complication: electrons change speed inside crystal
•Work function  increases kinetic energy in the crystal
•Momentum increases in the crystal
•Wavelength changes
The Davisson-Germer Experiment
Same experiment as scattering X-rays, except
•Reflection probability from each layer greater
•Interference effects are weaker
•Momentum/wavelength is shifted inside the
material
•Equation for good scattering identical
e-
2d cos  m
 
d
Quantum effects are weird
•Electron must scatter off of all layers
The Results:
•1928: Electrons have both wave and particle properties
•1900: Photons have both wave and particle properties
•1930: Atoms have both wave and particle properties
•1930: Molecules have both wave and particle properties
•Neutrons have both wave and particle properties
•Protons have both wave and particle properties
•Everything has both wave and particle properties
Dr. Carlson has a mass of 82 kg and leaves this room
at a velocity of about 1.3 m/s. What is his wavelength?
h
h
6.626 1034 J  s
 

 6.22 1036 m
p mv  82 kg 1.3 m/s 
Waves: How come we don’t notice?
•Whenever waves encounter a barrier, they get diffracted,
their direction changes
•If the barrier is much larger then the waves, the waves
change direction very little
•If the barrier is much smaller then the waves, then the effect
is enormous, and the wave diffracts a lot

Light waves through a big hole
When wavelengths are
short, wave
effects are
hard to notice
l

Sound waves through a small hole

Simple Waves
•cos and sin have periodicity
2
•If you increase kx by 2, wave
will look the same
•If you increase t by 2,
wave will look the same
•Simple waves look like cosines or sines:
•k is called the wave number
  x, t   A cos  kx  t 
•Units of inverse meters
• is called the angular frequency
  x, t   A sin  kx  t 
•Units of inverse seconds
•Wavelength  is how far you have to go in space before it repeats
  2 k
•Related to wave number k
•Period T is how long you have to wait in time before it repeats
•Related to angular frequency 
  2 T  2 f
•Frequency f is how many times per second it repeats
•The reciprocal of period
Math Interlude: Partial Derivatives
•Ordinary derivatives are the local “slope” of a
function of one variable f(x)
•Partial derivatives are the local “slope”
f
of a function of two or more variables
f(x,y) in one particular direction
d f  x
f  x  h  f  x
 lim
h 0
dx
h
 x, y   lim f  x  h, y   f  x, y 
h 0
x
h
•Partial derivatives are calculated the same way as ordinary derivatives, except other
variables are treated as constant
d  Ax2  B
 Ax 2  B d
2
 Ax2  B
e
e
 Ax  B   2 Axe

dx
dx
  Ax2  Ay 2  e  Ax2  Ay 2   Ax 2  Ay 2
 Ax 2  Ay 2
 2 Axe


e
x
x
Calculate the partial derivative below:


cos  kx  t    sin  kx  t   kx  t   k sin  kx  t 
x
x
Dispersion Relations
 E
2
 E
2
2
z
z
•Waves come about from the solution of differential equations

c
2
2
t
x
•For example, for light
•These equations lead to relationships between the angular frequency 
and the wave number k
   k 
•Called a dispersion relation
What is the dispersion relationship for light in vacuum?
Need to find a solution to wave equation, let’s try:
E  A cos kx  t
Ez
 kA sin  kx  t 
x
2
 Ez
2


k
A cos  kx  t 
2
 x
z


Ez
  A sin  kx  t 
t
 2 Ez
2
  A cos  kx  t 
2
t
 2  c2k 2
 2 A cos  kx  t   c2k 2 A cos  kx  t   0
  ck
0
Phase velocity

1 
f  
T 2
k  2

vp   f 
k
•The wave moves a distance of one wavelength  in one period T
•From this, we can calculate the phase velocity denoted vp
•It is how fast the peaks and valleys move
   f 2 

vp 


T
k 2
k
What is the phase
velocity for light
in vacuum?

ck  c
vp  
k
k
Not constant for
most waves!
Adding two waves
•Real waves are almost always combinations of multiple wavelengths
•Average these two expressions to get a new wave:
 1  cos  k1 x  1t 
  x, t   12 cos  k1x  1t   12 cos  k2 x  2t 
•This wave has two kinds of oscillations:
•The oscillations at small scales
•The “lumps” at large scales
 2  cos  k2 x  2t 
Analyzing the sum of two waves:
  x, t   12 cos  k1x  1t   12 cos  k2 x  2t 
cos      cos  cos   sin  sin 
Need to derive some
obscure trig identities:
•Average these:
•Substitute:
cos      cos  cos   sin  sin 
1
2
  12  A  B 

1
2
 A  B
1
2
cos      12 cos      cos  cos 
cos A  12 cos B  cos  12  A  B   cos  12  A  B  
Rewrite wave function:
  x, t   cos  kx  t  cos  k  x    t 
 k1  k2 
  12 1  2 
k 
1
2
Small scale
oscillations
Large scale
oscillations
 k1  k2 
  12 1  2 
k 
1
2
The “uncertainty” of
two waves
Our wave is made of two values of k:
•k is the average value of these two
•k is the amount by which the two values of
k actually vary from k
•The value of k is uncertain by an amount
k
k
k
k2
k
k
•Each “lump” is spread out in space also
•Define x as the distance from the
center of a lump to the edge
•The distance is where the cosine
vanishes
cos  k x   0
Plotted at
t=0
First hint of
uncertainty principle
k1
k x  12 
x
k x 1
Group Velocity
  x, t   cos  kx  t  cos  k  x    t 
Small scale
Large scale
oscillations
oscillations
The velocity of little oscillations governed by the first factor
•Leads to the same formula as before for phase velocity:
The velocity of big oscillations governed by the second factor
•Leads to a formula for group velocity:
These need not be the same!
vp 

k

vg 
k
More Waves
One wave
•Two waves allow you to create localized
“lumps”
Two waves
•Three waves allow you to start separating
these lumps
•More waves lets you get them farther and
farther apart
Three waves
•Infinity waves allows you to make the other
lumps disappear to infinity – you have one
lump, or a wave packet
•A single lump – a wave packet – looks and
Five waves
acts a lot like a particle
Infinity waves
Wave Packets
•We can combine many waves to separate a “lump” from its neighbors
•With an infinite number of waves, we can make a wave packet
•Contains continuum of wave numbers k
•Resulting wave travels and mostly stays together,
like a particle
•Note both k-values and x-values have a spread
k and x.
Phase and Group velocity
Compare to two wave formulas:
•Phase velocity formula is exactly the same, except
we simply rename the average values of k and  as
simply k and 
•Group velocity now involves very closely spaced
values of k (and ), and therefore we rewrite the
differences as . . .
What is the phase
and group velocity
for this wave?
vp 

k

vg 
k
vp 

k
d
vg 
dk
Sample Problem
What is the phase and group velocity for this wave?
Moved 30 m
30 m
vp 
 1.0 m/s
30 s
Finish, t = 30 s
Start, t = 0 s
Moved 60 m
60 m
vg 
 2.0 m/s
30 s
Phase and Group velocity
vp 
How to calculate them:
 ck
•You need the dispersion relation: the relationship
between  and k, with only constants in the formula v p  k  k  c
•Example: light in vacuum has   ck
What’s wrong with the
following proof?
Theorem: Group velocity doesn’t
always equal phase velocity

k
d
vg 
dk
d d

vg 
 ck   c
dk dk
If the dispersion relation is  = Ak2,
with A a constant, what are the
phase and group velocity?
  kv p
d
d
dv p
vg 

vpk   v p 

k  vp
dk
dk
dk

Ak 2
vp  
 Ak
k
k
d d
2
vg 

Ak
 2 Ak


dk dk
The Classical Uncertainty Principle
•The wave number of a wave packet is not exactly one
value
•It can be approximated by giving the central
value
•And the uncertainty, the “standard deviation”
from that value
•The position of a wave packet is not exactly one value
•It can be approximated by giving the central value
•And the uncertainty, the “standard deviation” from
that value
k k
k
x x
These quantities are related:
•Typically, x k ~ 1
x
Precise Relation:
(proof hard)
xk  12
Uncertainty in the Time Domain
Stand and watch a wave go by at one place
•You will see the wave over a period of time t
•You will see the wave with a combination of angular frequencies 
•The same uncertainty relationship applies in this domain
t   12
Estimating Uncertainty: Carlson’s Rule
A particle/wave is trapped in a
box of size L
•What is the uncertainty in its
position x?
L/4
L
?
L/2
Guess of
position
L/2
•Best guess: The particle is in the center, x = L/2
•But there is an error x on this amount
Exact numbers for x:
•It is no greater than L/2
•Particle in a box: 0.181L
•It is certainly bigger than 0
•Uniform distribution: 0.289L
•Carlson’s rule: use x = L/4
•This rule can be applied in the time domain as well
Sample Problem:
A student is supposed to measure the frequency of an object vibrating at f =
147.0 Hz, but he’s late for his next class, so he only spends 0.100 s gathering
data. How much error is he likely to have due to his hasty data sampling?
•Since the data was taken during 0.100 s, the date fits into a time box of length 0.100 s
•By Carlson’s rule, we have t = 0.0250 s
•By the uncertainty principle (time domain), we have:
t   12
1
 
 20.0 s 1
2t
•Since f = /2, this causes an estimated error of
20.0 s 1
f 
 3.17 Hz
2
•Of course, the error could be much larger than this
Wave Equations You Need:
•These equations always apply
1 
f  
T 2
k  2
•Two equations describing a
generic wave
  x, t   A cos  kx  t 
  x, t   A sin  kx  t 
•Light waves only
v p  vg  c  3 108 m/s

vp 
k
d
vg 
dk
xk 
1
2
t 
1
2
Math Interlude: Complex Numbers
•A complex number z is a number of the form z = x + iy, where x and y are real
numbers and i = (-1).
•x is called the real part of z and y is called the imaginary part of z.
•The complex conjugate of z, denoted z* is the same number except the sign of the
imaginary part is changed
What’s the imaginary part
x

Re
z
 
z  x  iy
of 4 + 7i?
y  Im  z 
Note: no i
z*  x  iy
•Adding, subtracting, and multiplying complex numbers is pretty easy:
2

6

18
i

8
i

24
i
 6 10i  24  30 10i
3

4
i
2

6
i



•To divide complex numbers, multiply numerator and denominator by the complex
conjugate of the denominator
6  8i  18i  24i 2
2  6i 2  6i 3  4i
18  26i




2
9  16i
3  4i 3  4i 3  4i
25
A Useful Identity
Taylor series expansion
f  x   f  0   xf   0   x f   0   x f   0   x f
2
1
2!
3
1
3!
1
4!
4
 4
 0 
sin     3!1  3  5!1  5  7!1  7  9!1  9  
cos   1          
Apply to sin, cos,
and ex functions
1
2!
In last expression, let x  i
e  1  i 
1
2!
 i 
2

1
3!
4
1
4!
1
6!
6
8
1
8!
e  1 x  x  x  x  x 
x
i
2
 i 
3

1
4!
 i 
4

1
5!
1
2!
2
 i 

5
1
3!
1
6!
3
 i 
1
4!
6

1
7!
4
1
5!
 i 
 1  i  2!1  2  3!1 i 3  4!1  4  5!1 i 5  6!1  6  7!1 i 7 
 1  2!1  2  4!1  4  6!1  6 
 cos  i sin 
  i   3!1  3  5!1  5  7!1  7 
ei  cos   i sin 

7
5

  x, t   A cos  kx  t 
Complex Waves
Typical waves look like:
  x, t   A sin  kx  t 
1. We’d like to think about them both at once
2. We’d like to make partial derivatives as simple as possible
A mathematical trick lets us achieve both goals simultaneously:

 ik
i  kx t 
• Real part is cosine

x, t   Ae

x
• Imaginary part is sine

This makes the derivatives easier in differential equations:

  i kx t  
i  kx t 
t
  x, t   A  e

ikAe

ik

x
,
t
 

x
x

  i kx t  
i  kx t 
  x, t   A  e


i

Ae


i

x
,
t



t
t
 i
What is the dispersion relationship for light in vacuum?
2
 2 Ez

Ez
2
c
0
2
2
t
x
 i 
2
Ez  c  ik  Ez  0
2
2
 2  c2k 2
Magnitudes of complex numbers
The magnitude of a complex number z = x + iy denoted |z|, is given by:
•This formula is rarely used
2
2
z  x y
•The square of the magnitude can be written
z  x  y  x  i y   x  iy  x  iy   zz*  z
2
2
2
2
2
This is the easiest way to calculate it
2
2
Sample Problem
What’s the magnitude squared
of the following expression?
2


2a  ix 

a

exp 
 a
a  it
4a


2
2




2a  ix 
2a  ix 


a
a
2
   * 
exp 
 a  exp 
 a
a  it a  it
4a
4a




2
2
2

2a  ix    2a  ix   8a

a

exp 

2
2 2
a i t
4a


2
2
 2i 2 x 2 

a
x 

exp 

exp




2
2
2
2
a t
 4a 
a t
 2a 
a