Download Matter Models (continued…) Examples If particles behave like

Matter Models (continued…)
• Two puzzles remain at this point:
– The wave-particle duality of light (shows
both somewhat contradicting properties).
– The physical basis for the Bohr model-why
the fixed orbits.
Actually, we need not speak of particles at all. For many experiments it is more convenient to
speak of matter waves . . . The two pictures are of course mutually exclusive, because a
certain thing cannot at the same time be a particle . . . and a wave . . . but the two complement
each other. By playing with both pictures, by going from the one picture to the other and back
again, we finally get the right impression of the strange kind of reality behind our quantum
experiments.”
~ Werner Heisenberg
PS 110 A Hatch Ch 15 -4
The de Broglie Hypothesis
• In 1923 a struggling graduate
student named Louis de
Broglie proposed that
moving matter has a wave
property too, defined by
PS 110 A Hatch Ch 15 -5
But it turned out to describe what
we observe.
• De Broglie’s idea explained the
Bohr orbitals
• The quantized orbits of the Bohr
model are predicted perfectly by
requiring electrons to exactly
wrap 1, 2, 3, etc waves around
the nucleus.
wavelength = h / (mass x speed)
where h = Plank’s constant = 6 x 10-34
Nobel Prize, 1929
originally hailed as the “French
Comedy”…
PS 110 A Hatch Ch 15 -6
PS 110 A Hatch Ch 15 -7
wavelength = h / (mass×speed)
Examples
where h = Plank’s constant = 6 x 10-34
Wavelength =
(nonsense?)
10-38 m
60 mph
100 mph
2,000 mph
Wavelength = 10-34 m
(again nonsense?)
Wavelength = 10-10 m
Diameter of an atom…
PS 110 A Hatch Ch 15 -8
If particles behave like waves ...
They ought to be able to do
some of the unique things
waves do - like …
Diffract
Interfere
Must get the right slit width.
PS 110 A Hatch Ch 15 -9
Why don’t we observe the
wave nature of matter?
• Remember, to observe wave effects, your “slits”
need to be similar to the wavelength
wavelength = h / (mass×speed)
h = 6 x 10-34
Why don’t we observe the wave
nature of matter?
• For all material objects except the very least
massive (such as electrons and protons), the
wavelength is so immeasurably small that it can
be completely ignored.
PS 110 A Hatch Ch 15 -10
Diffracting a pig
through a doorway.
PS 110 A Hatch Ch 15 -11
But what about electrons?
- to diffract, need wavelength to
be about same size as doorway
Wavelength = Planck’s constant
(mass of object)(its speed)
Pig would have to move so slowly it
would take a billion years for it to
move through the doorway.
PS 110 A Hatch Ch 15 -12
Diffraction limits how small you can
see with an optical microscope
• Let’s see if they diffract. We need slits about the
size of the electron wavelength (10-10 m) to
witness it.
• How do you make a slit that
small? You don’t. You use
the space between atoms in a
crystal.
• Fire an electron beam at a
crystal and we DO get
diffraction rings! Electrons
Video: EP #24
ARE waves!
PS 110 A Hatch Ch 15 -13
Electron Microscopes
• When objects are about the size of the
wavelength of light,~10-6 meters, light diffracts
around the object so you can’t get a clear image.
• Electron wavelengths
are 1/1000th the size
of optical
wavelengths. So
using electron beams
we can see 1000 times
smaller with the same
clarity.
PS 110 A Hatch Ch 15 -14
PS 110 A Hatch Ch 15 -15
Electrons as waves boggles my
mind!
Two-slit experiment with a
shaky machine gun:
1. With bullets
Let’s do examples with
bullets (real particles)
water (real waves)
electrons (particles or waves?)
count the hits in each location
Probability of hit for each slit
Total probability curve
P2: What if I block
the right slit?
PS 110 A Hatch Ch 15 -16
The concept of a probability
distribution
wavelength of bullet is very small
PS 110 A Hatch Ch 15 -17
For waves, we can use the amplitude as
a measure of where the wave “is”
Probability doesn’t make sense here; the wave hits a
wide area.
P3: What if you block 1 hole?
left slit
right slit
both slits
PS 110 A Hatch Ch 15 -18
Experimental double slit
experiment using electrons
• Electrons are detected like particles, a local
bright spot, but the places where they are
detected show interference patterns
These slits must be about as close as the size of
an atom
PS 110 A Hatch Ch 15 -20
PS 110 A Hatch Ch 15 -19
But what if you send the e’s through
1 at a time and detect them?
Each electron chooses a different spot to hit.
The cumulative effect of many electrons is the interference pattern
(Davisson-Germer 1927).
P4: In this experiment, which slit does a
single electron go through?
PS 110 A Hatch Ch 15 -21
Trying to fool mother nature
What do I see if I close one hole?
P5: What do I see if I detect the electrons
coming through one slit or the other?
Electron Detector
Probability Curve
Questions &
Answers (?)
Observation Point,
behind one slit
Look at a null point of the electron pattern.
Q. What is seen at the observation point with both slits open?
A. No electrons!
P6: If I close the slit on the other side (right side in picture
above) , what do I now see at this null point?
Q. Now open the second slit again, what do I see?
A. There are no electrons?
How can it be like that??
PS 110 A Hatch Ch 15 -22
Trying to Fool Mother Nature
An electron approaches two slits. It has a 25%
probability of passing through the left slit, a
25% probability of passing through the right
slit, and a 50% probability of hitting the barrier
and thus not passing through either slit.
P7: If I have a detector which detects this electron
passing through the left slit, has the probability
wave for this electron changed? What are the
probabilities associated with it now?
PS 110 A Hatch Ch 15 -23
How do we picture it?
“Electrons or photons arrive in lumps,
like particles, but the probability of
arrival of these lumps is determined as
the intensity of waves would be. It is in
this sense that the electron behaves
sometimes like a particle and sometimes
like a wave.”
Richard Feynman
Nobel Prize, 1965
PS 110 A Hatch Ch 15 -24
When we look at the electron ...
it acts like a particle -- otherwise like a wave.
How do we “look”? Must use a detector or a
probe which interacts with the electron.
e.g. light
… but a photon is like a bomb to an electron!
-- like trying to see the trajectory of a baseball
by hitting it with grenades
Making the measurement affects the
PS 110 A Hatch Ch 15 -26
result!
PS 110 A Hatch Ch 15 -25
The results we get depend on
how and what we measure.
• Don’t measure which hole the electron goes
through Æ wave.
• Do measure which hole the electron goes
through Æ particle.
• How the electron behaves depends on
whether it is observed.
• We have found the truth; and the truth
makes no sense. (G. K. Chesterton)
PS 110 A Hatch Ch 15 -27
This is a little strange
I remember discussions with
Bohr which went through
many hours till very late at
night and ended almost in
despair; and when at the end of
the discussion I went alone for
a walk in the neighboring park
I repeated to myself again and
again the question: Can nature
possibly be as absurd as it
seemed to us in these atomic
experiments?
~Werner Heisenberg
Probability wave
• The “wave” is interpreted as being the
probability of locating the particle.
• It acts like a pure wave with diffraction,
interference, refraction, etc.
• Somehow “particle” electrons “know” about the
existence of both slits even when we cannot
prove that they ever go through more than one
slit at a time.
• Clearly we need another model.
PS 110 A Hatch Ch 15 -28
PS 110 A Hatch Ch 15 -29
Probability Curve
Interference Curve
a graph showing the probability of finding a
particle at various locations in space
a probability curve from two interfering waves
0.6
0.5
0.4
0.3
0.2
0.1
y
0
PS 110 A Hatch Ch 15 -30
PS 110 A Hatch Ch 15 -31
The electron position is described
with a probability wave
Moving
probability
wave
• When we measure the position, we
find it at a certain position. We
now know more about the electron.
We refer to this as the collapse of
the probability wave.
PS 110 A Hatch Ch 15 -32
Schrodinger’s Equation
ih
∂Ψ ( x, t ) h 2 2
=
∇ Ψ ( x, t ) + V ( x, t ) Ψ ( x, t )
2m
∂t
• I don't like it and I'm
sorry I ever had
anything to do with it.
Schrodinger,
1926
PS 110 A Hatch
Ch 15 -33
The Uncertainty Principle and
waves
• Back to the Bohr model. To refine this model we need to
know how electrons move around atoms.
• To find their trajectories we must know their position and
velocity at the same time.
• But how do you locate the position of a wave/particle
electron?
Pure sine wave Æ no position
but clear frequency.
Sharp pulse Æ clear position
but unclear frequency.
The Uncertainty Principle
There is a limit to what can be known about a
single electron (or other small particle). The more
you know about where it is, the less you know
about how fast it is moving (wavelength).
The Newtonian view will not work on the small
scale. It assumes you can know both position and
motion……..
EXACTLY!
PS 110 A Hatch Ch 15 -34
Heisenberg Uncertainty Principle
• Electrons: fuzzy position and fuzzy wave
properties. How fuzzy?..
PS 110 A Hatch Ch 15 -35
An Undetected Electron
Where is it?
We cannot say exactly.
What is it doing?
We cannot say exactly.
We represent the electron mathematically as a wave.
• ..The uncertainty in position times the
uncertainty in momentum (mass x velocity)
is greater than Planck’s constant. Or
∆x ∆(mv) > h
Wave equation yields predictions about likelihood of
finding an electron at a given position.
High likelihood where the amplitude of the wave is high
PS 110 A Hatch Ch 15 -36
PS 110 A Hatch Ch 15 -37
Consequence of WaveWaveParticle Duality
Consequences
• If we try to find out where an electron is, we
know less about where it was going.
– Measuring position more accurately makes
uncertainty in momentum larger. More waves
with different momemtums. This is an
alternative explanation for electron diffraction.
• Because of the wave nature of electrons, we
CANNOT know how they move around
atoms. [This is not a measurement problem, but a
statement of fundamental limitations in nature.]
PS 110 A Hatch Ch 15 -38
Cannot tell precisely
what a single, unwatched
electron will do.
Best we can do is talk about probabilities –
then true for many electrons.
(Analogous to coin flips--we cannot predict
accurately the outcome of one coin flip, but we can
predict the outcome of 1,000,000 flips)
PS 110 A Hatch Ch 15 -39
In other words…
Conclusion
• We can predict this interaction perfectly using
Newton’s Laws of motion.
8
• We cannot predict the results of this interaction
perfectly. We can give probabilities that certain
outcomes will happen.
?
?
?
PS 110 A Hatch
Ch 15 -40
The conclusion to be drawn from the
electron experiment is that electrons, like
photons, arrive in lumps, like particles, but
the probability of arrival of these lumps is
determined as the intensity of waves would
be. Electrons have both a wave and a
particle aspect to their nature.
PS 110 A Hatch Ch 15 -41
?
So does the electron “know”
where it is and where it is going?
• Quantum mechanics
experiments demonstrate
that there is fundamental
uncertainty in nature. It is
not a matter of the
experimentalist not being
clever enough to measure
both position and
momentum at the same
time. A particle simply
cannot have an exact
position and an exact
momentum at the same
time.
Duality of Matter: Summary
1. Matter has both wave and particle properties.
2. Position and speed, as a pair, are uncertain.
3. Without making an observation there is only a
probability of where the particle is.
4. Observation affects the results of the
experiment. The wave function “collapses”.
5. The probability of finding a particle is only seen
when working with a large number of particles.
PS 110 A Hatch Ch 15 -42
Duality of Matter: Summary (cont)
6. Reality is caused by observation.
7. The universe seems to be a universe of chance,
and one must work with large numbers.
8. The chance is not a result of some underlying
structure (hidden variables).
9. These considerations apply only at atomic level.
10. Quantum mechanics does give Newton’s laws at
the larger scale.
PS 110 A Hatch Ch 15 -44
PS 110 A Hatch Ch 15 -43