Transcript deBroglie, probability

PH 301
Dr. Cecilia Vogel
Lecture
Review
Photon
photoelectric effect
Compton scattering
Outline
Wave-particle duality
wavefunction
probability
When do We See Which?
Wave-particle duality:
Light can show wave or particle
properties, depending on the experiment.
While propagating, light acts as a wave
while interacting, light acts as a particle.
When do We See Which?
Two-slit experiment
Light will propagate through both slits
and waves through slits interfere with
each other, but
 when it strikes the screen,
it interacts with the screen one photon at a
time.
When do We See Which?
Interference
seen if waves are coherent
Diffraction
seen if obstacle/opening about size of
wavelength
Why is the sky blue?
The sky is blue, because more blue
light is scattered by the air to our eye
(than red, yellow, etc).
Blue light is more likely to scatter than
red, because red is more likely to
diffract instead.
Less diffraction occurs for shorter
wavelengths.
Blue light has shorter wavelength, so it
diffracts less and scatters more.
Why are the clouds white?
The water droplets are much larger
than the wavelength of all visible light
(not just blue/violet)
almost no visible light is diffracted by
clouds
every color of visible light is scattered by
clouds
all colors scattered, so scattered light is
white
Matter
Matter particles, like electrons, have
particle properties (of course)
individual, indivisible particles
energy & momentum
(paintball)
Duality of Matter
Matter particles also have wave properties!
They diffract!
They interfere!
Diffract from a
crystal, interference
pattern depends on
crystal structure
...from a powder,
pattern depends on
molecular structure
Duality equations
Light/photons
E  hf
p  h/
hchc
E E 

EE
p p 
cc





Matter, e.g. electrons
Same
eqns
f  E/h
  h/ p
E

mc

E

mc
Only for matter

Cue: ‘m’
p

mv
p

mv

22
Only for light
Cue: ‘c’
Example
What is the wavelength of an electron
which has 95 eV of kinetic energy?
Note: K<<moc2, so we can use classical
equations.
Note: DO NOT USE E=hc/.
2(95eV )
2
1
K  2 mv so v  2 K / m 
0.511X 106 eV / c 2
 0.019c  5784790m/s
then p  mv  5.27X10 -24 kgm/s
then   h/p  1.26X -10m
Wave Function
For light, the wavefunction is E(x,t)
electric field (and B(x,t) = magnetic field).
For matter the wave function is Y(x,t)
like nothing we’ve encountered before.
Not an EM wave.
The matter itself is not oscillating.
Wavefunction Interpreted
For light beam, where the wave function
(E-field) is large,
the light is bright
there are lots of photons
For beam of matter particles, where the
wave function is large
there are lots of particles.
The bright spots in interference pattern
are where lots of photons or matter particles
strike.
Probability Interpretation
If you have one particle, rather than a
beam,
the wavefunction only gives probability
P(x,t) = |Y(x,t)|2.
there is no way to predict precisely where it
will be.
Where the wave function is large
the particle is likely to be.
The bright spots in interference pattern
are where a photon or matter particle is
likely to strike.
Probability Interpretation
P(x,t) = |Y(x,t)|2.
If we repeat an experiment many, many
times, the probability tells in what fraction
of the experiments, we will find the particle
at position x at time t.
Do we have to do the experiment many, many
times for the probability to have meaning?
NO!
With one particle, you can still determine
probabilities
Averages and Uncertainty
P(x,t) = |Y(x,t)|2.
If you have many possibilities with known
probabilities
Average <x> = xave=x= probability weighted
sum of possibilities

2
x
|
Y
|
dx

<x> = 
Uncertainty Dx=rms dev = root mean square
deviation
Dx =  ( x   x ) 2 
Also
Dx =  x 2    x 2
Imaginary Exponentials
What is the meaning of
e
 iy
You can do algebra and calculus on it just
like real exponentials;
just remember i2 = -1.
It is a complex number,
with real and imaginary parts.
Can be rewritten as: e  iy  cos y  i sin y
For example ei  cos   i sin 
e i  1 but ei / 2  i
Complex Algebra
z  a  ib
a and b real
To add or subtract complex numbers,
add or subtract real parts (a),
add or subtract imaginary parts (b).
To multiply, use distributive law.
To get the absolute square |z|2,
multiply z by its complex conjugate, z*.
To get the complex conjugate of z,
change the sign of all the i’s.
z
2
 a b
2
2
Complex Algebra
In general, with
z  ce
 id
z  c2
2
c and d real
Complex Example
i ( kx t )
Y  Ae
Find the absolute square, |Y|2,
which is the probability density.
Need the complex conjugate, Y*.
Y  A
2
2
The probability density is constant,
it is the same everywhere, all the time.
this particle is as likely to be a million light
years away, as here. Not localized.
Complex Example
Given |A|2 = ¼
show that
3 1
A
 i
4
4
works as well as ½.
2
2


3
3
1
1
2
1
A  


    
16 16 4
 4  4
PAL Probability
i3x
 Given the wavefunction  ( x ) 
1
e
sin( x)
1.5nm
where x is in nm
and ranges from 0 to 3 nm.
1) Find the probability density as a function of x.
2) Find <x> = the average value of x.
3) Find < x2 > = the average value of x2.
4) Find Dx = the uncertainty in x.