particle in a box

1
particle in a box the uncertainty principle
"As far as the laws of mathematics refer to
reality, they are not certain, and as far as they
are certain, they do not refer to reality.A.
Einstein
Reminder Exam 1 is one week from today! Do you
want Quiz 3 on Wednesday or Friday?
2
There is a homework problem on diffraction (if I
remember correctly).
The equations you need to work the problems are
not new ones, but Ill give you them to you as
a reminder.
3
Confession
A few years ago, for dramatic emphasis, I
libelously suggested a student might have been
responsible for the mishap that oxidized Davisson
and Germers nickel sample.
In early 1925, a bottle of liquid air exploded in
their lab, shattering the glass container
containing their nickel sample and causing it to
oxidize.
Liquid air is very unstable, and doesnt need
human intervention to explode!
Hmmisnt liquid nitrogen a lot like liquid air?
Yes! Dont let someone sell you cheap liquid air
when you want liquid nitrogen!
4
3.6 Particle in a Box
Now you believe that particles have waves,
right?
Could we digress a minute. Just what does this
mean?
Is the particle the only reality, and the wave
just something physicists invented?
ruled out by experimentmatter waves shown to
interfere
Is it as Schrödinger first believedthe wave is
real, not the particle?
abandoned for many reasons, including the fact
that unlike waves, particles do not spread out
5
Is the particle real, and the wave something that
guides it (a pilot wave)?
no successful theory was found based on this idea
Is there some single reality which sometimes
manifests itself as particle and sometimes as
wave but is really both at once?
this has come to be the commonly accepted
viewbut it is possible the whole story has not
yet been told
Going much deeper into this discussion leads into
philosophy and away from physics, so back to the
physics Ohwhy study a particle in a box
6
Anyway, supposing that you believe that particles
have a wave natureand experiment certainly backs
this uplets pursue some of the consequences.
Remember your Physics 23 vibrating string
experiment. You couldn't set up arbitrary
vibrations of the string. Instead, you saw
patterns like this.
You get standing waves because the wave pulses
traveling down the string reflect (and change
phase by 180) when they reach the ends.
The standing wave consists of a series of pulses
moving up and down the string. These pulses
superpose, and when they are in phase, you see
maxima. When they are out of phase, you see
nodes.
7
You see standing waves only when the pulse speeds
are just right, and the standing wave fits on
the length of the string. Heres a static
picture (http//www.chem.uci.edu/education/underg
rad_pgm/applets/dwell/dwell.htm)
dead link January 2005but I only used it for
the picture anyway
What does this have to do with a particle in a
box?
Place a particle, represented by a wave, in a box
of length L. The particle wave moves with the
particle, reflects when it hits the wall of the
box, and then again when it hits the other end of
the box.
8
If the box is small enough (compared to the
particle wavelength), the particle wave "folds
up" over and over again every time it reflects
off a wall. Heres the best visualization I
could find http//www.zbp.univie.ac.at/schrodi
nger/ewellenmechanik/simulation.htm
The segments of the particle wave bouncing back
and forth interfere. If they interfere
constructively, our particle can fit in the box.
Otherwise -- no particle fits in the box.
Caution! Intense material! Adult content!
Shows death of wave!
9
The folding of the particle wave packet to add in
phase works when the length of the box is an
integral number of half wavelengths of the
particle wave (Ln?/2), hence equation 3.17 in
Beiser for the de Broglie wavelengths of a
trapped particle
Because KE mv2/2 and ? h / mv, the
restriction on ? also places a restriction on the
allowed particle energies
So this is a nonrelativistic calculation.
10
The permitted energies are called "energy levels"
and the number n is called a "quantum number."
Think of the box as a potential well, inside
which a particle is placed.
A free particle, outside the box, can have any
energy, and any wavelength.
When you put the particle in the box, only
certain wavelengths and energies are allowed (and
note that zero is not one of the allowed
energies). You most likely will have to add or
remove energy to put a free particle into a box.
11
Comment quantum implies something is
quantized (energy, momentum). Quantization of
properties of matter is a consequence of the wave
nature of matter. Thus, the words wave and
quantum are closely associated in my vocabulary.
All of us in this room have discrete energy
levels and quantum states.
However, as the example on page 107 shows, there
are enough energy levels to form, for all
practical purposes, the continuum that Newtonian
mechanics supposes.
12
10 gram marble in a box 10 cm wide
Minimum energy and speed are indistinguishable
from zero, and a marble of reasonable speed has a
quantum number on the order of 1030. In other
words, we can't perceive the quantum features of
the marble in the box.
13
Electron in a box 0.1 nm (10-10 m) wide (the
size of an atom)
The minimum energy is 38 eV, a significant
amount, and the energy levels are far enough
apart to be measurable.
14
Is this information useful, or is it just physics
trash talk?
Quantum well lasers (183000 pages found in Google
search, September 2002) http//nsr.mij.mrs.org/3/
1/ http//www.iis.ee.ethz.ch/research/tcad/laser_
sim.en.html http//www.chem.wisc.edu/courses/801/S
pring00/Ch1_3.html Quantum well memory (209000
pages found in Google search, September
2002) http//www.physicstoday.org/pt/vol-54/iss-5
/p46.html http//www.sciencenews.org/sn_arc99/2_27
_99/fob5.htm http//theorie5.physik.unibas.ch/qcom
p/qcomp.html
OK, so maybe we should pay attention to this
wave/quantum stuff. Is there anything else
important hidden in the wave nature of
particles?
Quantum well lasers, 388000 pages Sept. 2003
450000 pages Jan. 2005. Quantum well memory, also
388000 pages Sept. 2003 1100000 pages jan. 2005.
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3.7 Uncertainty Principle I -- derivation based
on the wave properties of particles
Consider the particle represented by this wave
group.
Where is the particle?
What is its wavelength?
The position is well-defined.
But the wavelength is poorly defined, and
therefore there is a large uncertainty in the
particles momentum (remember--wavelength and
momentum are related).
Life is uncertain. Eat dessert first. UMR
Prof. Emeritus D. M. Sparlin
16
Now consider the particle represented by this
wave group.
Where is the particle?
What is its wavelength?
The wavelength seems to be rather well-defined,
but the position is poorly defined. There is a
large uncertainty in the particles position.
To quantify the uncertainties in the wave group's
position and momentum, we need to go into much
more detail about Fourier transforms and
representation of wave groups by summations of
individual waves.
17
Beiser does this on pages 108-111. Please read
this material. You may be tested or quizzed on
major concepts (but not trivial details).
What I want you to know (backwards and forwards),
comes out of this derivation, and is called
Heisenbergs Uncertainty Principle
It is not possible to simultaneously measure,
with arbitrary precision, both the position and
momentum of a particle.
1932 Nobel Prize for creation of quantum
mechanics.
18
The quantity h/2? appears over and over again in
modern physics, so we give it a special symbol
h h /2?. The uncertainty principle can then be
written
There are fundamental limits on how precisely we
can simultaneously measure a particle's position
and momentum.
Because of the wave nature of matter, there are
fundamental limits on how precisely we can know
things.
These limits have nothing to do with our
measurement techniques they are built into
nature.
Marvelous what ideas young people have these
days. But I dont believe a word of it.A.
Einstein, referring to the uncertainty principle
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Example 3.6
A measurement establishes the position of a
proton with an accuracy of ?1.00x10-11 m. Find
the uncertainty in the protons position 1 s
later. Assume v ltlt c.
At the time of measurement, the position
uncertainty is ?x1, and
its OK to do this for v??c
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A time t later, the position uncertainty ?x2 is
About 1/10 the size of an atom, but much larger
than a nucleus.
Is v??c?
To put it bluntly, we have no clue where the
proton is 1 second later.
The proton didnt spread out, because it is
somewhere, but its wave certainly did!
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3.8 Uncertainty Principle II -- derivation based
on the particle properties of waves
I claimed above that the limits implied by the
uncertainty principle are fundamental to nature,
and are due to the wave properties of matter.
This follows cleanly and logically from the
mathematics of waves.
As humans, we are left with nagging doubts about
the uncertainty principle. How dare nature tell
us there are things we cannot know! Surely this
is just technical glitch that human cleverness
can overcome.
Heisenberg (although a theorist first, last, and
always) believed he had to specify definite
experiments for measuring an objects position
in order to validate the uncertainty principle.
Caution reading this section may be hazardous
to your grade!
22
Heisenberg proposed a thought experiment (which
can be realized in fact) lets suppose we want
to measure the position of this electron very
precisely. How do we do it?
The sphere doesnt represent the size of the
electron it represents the size of the region in
which we wish to locate the electron.
Visible light?
A real red light wave would have a much longer
wavelength than this!
The wavelength of visible light is far too large
to allow us to detect the position of the
electron. The wavelength needs to be comparable
to the position precision we seek.
You might say the electron is somewhere along the
wave, but the wavelength is so long that the
imprecision in position is enormous.
23
The last sentence contains a clue find some
kind of radiation that has a much shorter
wavelength.
Gamma rays have short wavelengths. They should
work.
But short wavelength gamma radiation carries lots
of energy and momentum. http//www.aip.org/history
/heisenberg/p08b.htm
Our gamma-ray microscope can tell us where the
electron was, but it cant tell us where it came
from or where it is heading (its momentum).
So we can forget about position (but measure
momentum), or forget about momentum (but measure
position).
24
I find this an interesting thought experiment,
but it implies that the uncertainty principle is
really only a measurement difficulty after all.
A surprising number of students tell me on tests
(through their answers to multiple choice
questions) that the uncertainty principle is just
a result of experimental difficulties.
No!-----------------------------------------------
-------------------------
-------------------------------------No!----------
-------------------------
--------------------------------------------------
--------------------- No!
The indeterminacy (of position and momentum) is
inherent in the nature of a moving body.
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Let me repeat Heisenbergs thought experiment is
bad because it implies the uncertainty is
merely some technical measurement difficulty.
3.9 Applying the Uncertainty Principle
Planck's constant is so small that we never
encounter the uncertainty principle in Newtonian
mechanics
but its consequences are manifested in materials
we constantly use in everyday life! Youll hear
about it repeatedly in this course.
To placate his critics and get his uncertainty
principle paper published, Heisenberg had to
include this thought experiment in it. The
thought experiment has appeared in (probably)
most texts, and confused thousands (millions?) of
students. All to get a paper published.
26
Frequency and time are related, and velocity and
energy are related, so we can derive an alternate
expression of the uncertainty principle
It is not possible to simultaneously measure,
with arbitrary precision, both a time for an
event and the energy associated with that event.
Prediction is very difficult. Especially about
the future.Neils Bohr and/or Mark Twain (we are
not certain who said this)
27
http//www.nearingzero.net
28
Example 3.7
A typical atomic nucleus is about 5x10-15 m in
radius. Use the uncertainty principle to place a
lower limit on the energy an electron must have
if it is to be part of a nucleus.
The problem is asking you something about the
energy of an electron confined to a region 5x1015
m in size. Obviously, the starting point is
NOT!
You are given information about the electrons
?x. In fact, you are implicitly told to use ?x
5x10-15 m. You must use
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Proceeding trustingly with the math
You could plug numbers in now. But lets keep it
symbolic and think for a bit.
Weve calculated a momentum uncertainty ?p. Does
it make sense to claim the electron has less
momentum than ?p?
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Suppose all UMR students measure the diameter of
the puck by the University Center.
Suppose all UMR students measure the diameter of
the puck by the University Center. Suppose
Physics 107 students also make the measurement.
These measurements all have uncertainties.
Suppose we calculate the difference between the
average of all UMR student results and the
average of the Physics 107 student results, and
find that the difference is -0.002 ? 0.007 m.
What can we conclude?
There is no difference between the overall
average and the Physics 107 average.
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In other words, a result of -0.002 ? 0.007 m is
consistent with zero.
What would be the minimum difference that you
would consider not consistent with zero?
A statistician would probably say 3?, or maybe
30.007 0.021.
A statistician would probably say 3?, or maybe
30.007 0.021. Ignoring the mathematicians,
because they have lots of strange ideas, most of
us could probably agree that if the result is x ?
?x, then x had better be at least as big as ?x
otherwise x is consistent with zero.
Since the mathematicians have invaded the
theory of relativity, I do not understand it
myself anymore.A. Einstein
32
This discussion was designed to get you to agree
with the statement that if an electron has a
momentum uncertainty given by
then it doesnt make sense to talk about a
momentum for the electron which is less than ?px.
33
If we agree that
then the minimum electron momentum is
Classically, KE p2 / 2m, so
Whats this business about calculating KE.
Didnt the problem ask for energy? Doesnt
that mean E? Well in this context the problem
was really asking for kinetic energy. Ask me if
you are unsure about which energy an exam
question is asking for.
34
So the minimum electron (kinetic) energy is
Units work out correctly if you keep everything
in SI. However, I encourage you to show units in
your calculations on exams, to prevent mistakes
and help me give you more points if you make a
simple math error.
35
So, KE6.11x10-11 joules. Any comments?
Hey! Isnt that kind of a big energy? I mean
for an electron?
6.11x10-11 joules x 1 eV / (1.6x10-19 joules)
3.82x108 eV.
3.82x108 eV 3.82x102x106 eV 382 MeV.
Now whats that electron rest mass again 0.511
MeV/c2.
KE p2 / 2m
36
In case you werent in class (physically or
otherwise), that big red x on that last slide
means that the previous 2 or 3 slides worth of
classical energy calculation was wrong.
37
For extremely high speeds and energies, pc gtgt mc2
so
0
Remind me again howd the sign slip in?
p
Here we go again replacing KE by E, except for
this large an energy, KEE.
38

• Four things to notice
• ?The classical calculation far overestimated the
  kinetic energy (makes senseclassical
  calculations can come up with speeds greater than
  c).
• The kinetic energy is about 20 MeV, which is much
  much greater than 0.511 Mev, so E pc was a
  reasonable approximation.
• The p in the uncertainty principle equation is
  the relativistic momentum.
• ? If you want to confine an electron to a
  nucleus, its wave nature requires that it have at
  least 20 MeV of energy.

This concludes (finally) example 3.7. Example
3.8 is similar except that a nonrelativistic
calculation is OK.
39
Example 3.9
An excited atom gives up its excess energy by
emitting a photon, as described in chapter 4.
The average period that elapses between the
excitation of the atom and the time it radiates
is 10-8 s. Find the inherent uncertainty in the
frequency of the photon.
We have time, want ?f, but E and f are related, so
Beiser uses the h version youll see why I use
this in a minute.
By not calculating numerical values right away, I
simplified the math!
40
If you measure the intensity vs. frequency of the
light emitted from this atom, the spectrum will
have at least this intrinsic linewidth.
Application it is usually desirable to have
laser lines be very sharp, i.e., the laser
emits only a single color of light. The width of
the laser line to the right depends on the design
of the laser, but not even the cleverest design
can produce a line narrower than that given by
the uncertainty principle.
intensity
frequency