Comments on Ch. 2 HW

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Comments on Ch. 2 HW Chapter 3 Wave Properties
of Particles de Broglie waves waves of
what? quiz 2
Two seemingly incompatible conceptions can each
represent an aspect of the truth.Louis de
Broglie
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Homework Problem 2-14 A silver ball is suspended
by a string in a vacuum chamber and ultraviolet
light of wavelength 200 nm is directed at it.
What electrical potential will the ball acquire
as a result.
Electrons will be ejected from the silver ball.
If the ball is in a vacuum, suspended by a
nonconducting string. As electrons are ejected,
the ball acquires a positive charge and positive
potential.
The ball will build up a positive charge until no
electrons have enough KE to escape, at which
point a steady-state situation will exist and the
potential will not change.
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Electrons stop escaping when
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symbol
units
My mathcad solution is rather sloppy.
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Suppose that a 60 W light bulb radiates primarily
at a wavelength of 1000 nm, a number just above
the optical range. Find the number of photons
emitted per second.
power total energy/time
power (energy/photon) (N of photons/time)
(N of photons/time) power / (energy/photon)
n power / (energy of a photon)
n P / hf P / (hc/?)
n 60 / (6.63x10-34)(3x108)/(1000x10-9)
If you stick with SI units, n will be in units of
s-1 i.e. photons/s.
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The Big Winners, Chapter 2
Planck, Nobel Prize, energy quanta, 1918.
Einstein, Nobel Prize, law of the photoelectric
effect, 1921.
Millikan, Nobel Prize, charge of electron and
photoelectric effect, 1923.
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Chapter 2 OSEs
All our other equations are derived from these,
the equations of relativity, and the equations of
classical physics.
I will give you more than these few equations on
your quiz.
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What kind of a clown would post something like
this for the world to see?
Potential! NOT energy!
Energy!
Energy!
Heres the trouble with physicists
We all think alike.
We dont think like ordinary humans.
We all know what each other means, so we can
afford to be sloppy like this.
Right?
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Chapter 3 Wave Properties of Particles
Overview
Einstein introduced us to the particle properties
of waves in 1905 (photoelectric effect). Compton
scattering of x-rays by electrons (which we
skipped in Chapter 2) confirmed Einstein's
theories.
One might ask "Is there a converse?" Do
particles have wave properties?
Asking such a question these days is likely to
get you ignored, but in the 1920s, so many
incredible things were happening in physics that
somebody might have listened to you.
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In fact, Louis De Broglie postulated wave
properties of particles in his thesis in 1924,
based partly on the idea that if waves can behave
like particles, then particles should be able to
behave like waves.
de Broglie
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In 1927, Davisson and Germer confirmed the wave
properties of particles by diffracting electrons
from a nickel single crystal.
http//hyperphysics.phy-astr.gsu.edu/hbase/davger.
html
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3.1 de Broglie Waves
Recall that a photon has energy Ehf, momentum
ph/?, and a wavelength ?h/p.
De Broglie postulated that these equations also
apply to particles. In particular, a particle of
mass m and momentum p has a de Broglie wavelength
whose magnitude is
Didnt you just write that same equation a
couple of lines above?
No, the equation in the first sentence was for
waves. The one in the box represents a new idea.
It is for particles.
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If the particle is moving fast enough that a
relativistic calculation is needed, use the
relativistic momentum
What made de Broglie (who was a real-life prince
and a brilliant theorist but a real klutz in the
lab) propose, seemingly just for kicks, that
particles have a wavelength?
As in my conversations with my brother we always
arrived at the conclusion that in the case of
X-rays one had both waves and corpuscles, thus
suddenly - ... it was certain in the course of
summer 1923 - I got the idea that one had to
extend this duality to material particles,
especially to electrons. And I realised that, on
the one hand, the Hamilton-Jacobi theory pointed
somewhat in that direction, for it can be applied
to particles and, in addition, it represents a
geometrical optics on the other hand, in quantum
phenomena one obtains quantum numbers, which are
rarely found in mechanics but occur very
frequently in wave phenomena and in all problems
dealing with wave motion.de Broglie, 1963
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Or, as Einstein put it
Mathematics are well and good
but nature keeps dragging us around by the nose.
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So the proposed wave nature of particles did not
come out of nowhere, but it was certainly a
daring hypothesis for a young Ph.D. student of
physics.
Now we have this equation that says particles
have a wavelength. What are we going to do with
it?
Experiment! Find experimental verification!
In order for us to observe a particle's wave
properties, the de Broglie wavelength must be
comparable to something the particle interacts
with e.g. the spacing between two slits, or the
spacing between periodic arrays of atoms in
crystals.
Also got him the 1929 Nobel Prize.
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Example find the wavelength of a 46 g golf ball
moving with a speed of 30 m/s.
Tell me something that has a physical dimension
on the order of 10-34 m, which the golf ball wave
could interact with?
Can we do an experiment which would detect the
golf ball wave?
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Example find the wavelength of an electron
moving with a speed of 107 m/s.
The wavelength is small, but roughly comparable
to atomic dimensions, so we need to consider the
wave nature of electrons when they are moving
rapidly through solids.
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Some things to think about
Collisions seem to be instantaneous, so particles
are really there and the wave associated with a
particle isnt the particle spread out.
Later we will see how a particle's wave has a
phase velocity greater than the speed of light,
c. Thus, the phase velocity cannot have a
physical interpretation.
We arent going to talk about the experimental
verification of the wave nature of particles yet,
but well get there eventually.
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If matter has a wavelength, there must be some
functiona wave functionwhich describes the
wave nature of the matter in question.
Do you think that if we could somehow find out
what this wave function is, and what mathematical
laws it obeys, then maybe we could learn
something about the matter that it describes?
Do you think Id be asking that question of the
answer were no?
That means we are going to have to spend more
time thinking about the mathematics of waves and
the functions that describe them.
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3.2 Waves of What?
This section foreshadows chapter 5.
In other words, what physical thing is it whose
variation makes up matter waves.
?
The "thing" whose variations makes up matter
waves is the wave function, ? ("psi", usually
pronounced "si").
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Huh?
The wave function of a matter wave is not
something we can see or sense. It has no direct
physical significance.
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? is, in general, complex. It cannot be directly
measured. The time and/or space average of ? is
zero. (That shouldn't bother you--the time/space
average of a sine wave is zero but you measured
sine waves in your Physics 24 labs.)
However, ? can tell us something about the matter
it represents.
In general, ? is a function of position (x,y,z)
and time.
The probability of finding the object described
by ? at the position (xyz) at time t is
proportional to the value of ?? there.
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In general, the value of ?? is between zero and
one. A small value at some position in space and
time means a small probability of finding the
object there a large value means a large
probability of finding the object there.
If ??0 at some position in space and time, then
the object is not there. If ??1 at some
position in space and time, the object is there.
Later we will find there are fundamental limits
on how precisely you can locate an object.
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Note the difference between the probability of an
event and the event itself.
If we detect an electron, it was "there." It was
not 50 "there."
If the probability of finding an electron at
(xyzt) is 50, it doesn't mean that the electron
is 50 "there." It means that half of our
measurements would find the whole electron
"there," and the other half would find no
electron at all.
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Looking ahead a bit more
For a particle or system of particles described
by the wavefunction ?, ??dV is the probability
of finding a particle (or the system) in an
infinitesimal volume element dV.
To find the probability of finding the particle
somewhere in space, we integrate the probability
over all space.
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Remember, the wavefunction tells us the
probability of finding the particle at a
particular point in space and time, but the
particle is not "spread out" in some wave.
Actually determining ? is generally a difficult
problem. We will often assume an appropriate
wavefunction without going into the details of
where it came from.
This concludes our brief diversion into the world
of quantum mechanics. We will return in chapter 5.
If we are to claim that particles are waves
(actually, have wave properties) then we had
better understand waves in detail