Transcript document

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
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.
K max = hf - hf0 = hf - φ
Electrons stop escaping when
K max = eV = hf - φ
hf - φ
V=
e
c
h -φ
V= λ
e
V=

4.14×10-15
 3×108 m/s 
eV×s 
 - 4.7eV
-9
 200×10 m 
e

V=

4.14×10-15
 3×108 m/s 
eV×s 
 - 4.7eV
-9
 200×10 m 
e

V = 1.51 V
symbol
My mathcad solution is rather sloppy.
units
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.
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.
Chapter 2 OSE’s
Ephoton = hf
hf = K max + hf0
pphoton = hf / c = h / λ
c=f λ
2GM
Rs = 2
c
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.
What kind of a clown would
post something like this for the
world to see?
Potential!
NOT energy!
Energy!
Energy!
Here’s the trouble with physicists:
We all think alike.
We don’t think like ordinary humans.
We all know what each other means, so we can afford
to be sloppy like this.
Right?
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 1920’s, so many incredible things were happening in
physics that somebody might have listened to you.
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
Werner Heisenberg and a little later Erwin Schrödinger
developed theories based on the wave properties of particles.
Heisenberg
Schrödinger
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
3.1 de Broglie Waves
Recall that a photon has energy E=hf, momentum p=h/, 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
h
λ= .
p
“Didn’t 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.
If the particle is moving fast enough that a relativistic
calculation is needed, use the relativistic momentum:
h
λ=
.
mv
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
Or, as Einstein put it:
“Mathematics are well and good…
…but nature keeps dragging us around by the nose.”
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.*
h
h
λ= =
p mv
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.
Example: find the wavelength of a 46 g golf ball moving with
a speed of 30 m/s.
A nonrelativistic calculation will do just fine here.
h
h
λ=
=
mv mv
non-relativistic: =1
6.63×10-34 J  s
λ=
(46×10-3 kg)×  30 m/s 
λ = 4.8×10-34 m
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?
Example: find the wavelength of an electron moving with a
speed of 107 m/s.
This is a fast electron, but it is moving at about 1/30 the speed
of light. A nonrelativistic calculation will suffice.
h
λ=
mv
6.63×10-34 J  s
λ=
(9.11×10-31 kg)× 107 m/s 
λ = 7.3×10-11 m
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.
Some things to think about:
Collisions seem to be instantaneous, so particles are really
“there” and the wave associated with a particle isn’t 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.
Useful equations; some from this class, some from earlier:
E = hf
p = h/
 = 2f
ħ = h/2
E = ħ
p = ħk
k = 2/
We aren’t going to talk about the experimental verification of
the wave nature of particles yet, but we’ll get there eventually.
If matter has a wavelength, there must be some function–a
“wave function”—which 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 I’d 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.
3.2 Waves of What?
This section foreshadows chapter 5.
Sound waves consist of pressure
differences in a medium. Water waves
consist of different heights of water. E&M
waves consist of measurable oscillations of
electric and magnetic fields. What about
matter waves?
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").
Huh?
The wave function of a matter wave is not something we can
see or sense. It has no “direct ” physical significance.
 is the solution to Schrödinger's
equation. Remember I mentioned
Schrödinger above, and said he
developed a theory for the wave
properties of particles. We'll learn about
“his” equation before long.
 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.
* tells us the probability of finding the body represented by
.
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.
Math review: if  is complex, then * = 2 is real (and positive).
Putting around * would be redundant.
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.
Does this way of doing things bother anybody? It
has bothered a lot of physicists. It's a radically
different way of thinking.
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.
If we have a collection of identical particles then * is
proportional to the actual density of particles. We often call
* a “probability density” even when we are talking about
one particle.*
*As we will see when we study hydrogen, the idea of a probability density
for a single electron makes a lot of sense.
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.
We assume that the probability of finding a particle
somewhere in space is 1 (otherwise, no particle, and nothing
worth doing), so that

Ψ*Ψ dV = 1 .
all space
Such a wavefunction is said to be normalized.
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…