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Chapter 30 The Nature of Atoms
Friday, January 14, 2011
10:03 AM
In the previous chapter we began to discuss some of the very
strange behaviour of particles/waves in the atomic and
subatomic realms. We learned about Planck's desperate
hypothesis (that elementary vibrators in a hot solid can only
emit energy in quanta that are a whole-number multiple of an
elementary quantum), and how Einstein extended this
hypothesis even more boldly by declaring that electromagnetic
radiation itself EXISTED in tiny bundles of energy, which
became known as photons. This is extremely strange, because
we're used to thinking of light as a wave, and we have
centuries of evidence to back it up.
But scientists don't just make wild speculations in a void, they
do so to solve very specific problems. So Einstein's wild
speculation did an excellent job of explaining the photoelectric
effect, and together with Planck's explanation for blackbody
radiation spectral curves and the Compton effect, provided
strong evidence for the photon hypothesis.
But no amount of evidence can ever prove a scientific
hypothesis valid. (Is your marriage red or green?) The best we
can do, and indeed what we must do, is amass as much
evidence as we can; the weight of evidence is what wins the
argument in science. And the significance of the collection of
evidence is greatly increased when there is evidence from a
greater variety of situations.
Thinking back to the photon hypothesis from the previous
chapter, we have already discussed three quite different types
of evidence in favour of photons: blackbody radiation, the
photoelectric effect, and the Compton effect. The photon
hypothesis explains all three situations handily.
However, the main tools of science are doubt and skepticism,
so inevitably, no matter how good one's explanations are,
there are clever people out there who will strongly disagree
with you, demanding even more and better evidence. This is
both a strength and a weakness of science. The harsh
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both a strength and a weakness of science. The harsh
treatment that most hypotheses receive is a strength because
it pushes their proponents to do their very best work, and it
ensures that only the best, most productive ideas are
accepted. It's a weakness because sometimes the climate
becomes so hostile that the scientific community loses good
but sensitive people, who recoil from the sometimes brutal cut
and thrust of scientific debate. Like any community, the
scientific community is full of good, kind, supportive, generous
individuals, but also contains a fair share of hostile,
argumentative egomaniacs.
Millikan in particular was very much opposed to the photon
hypothesis, and set out to design and conduct experiments to
prove Einstein wrong. However, he was surprised to discover
that his experiments actually supported the photon
hypothesis, and so he changed his stance.
And this is another important guiding principle in science: That
one honestly reports the results of ALL of one's investigations,
no matter what the results are, no matter whether they
support your favourite theory, or your preconceptions. In this
aspect, doubt and skepticism once again prove to be useful
tools, for dishonest investigators are very soon found out,
exposed, and their sham results are discarded and corrected.
"Extraordinary claims require extraordinary evidence," as they
say, and the photon hypothesis was sufficiently surprising that
it was not generally accepted for a long time. But once enough
evidence was collected from a variety of different kinds of
experiments, the weight of evidence carried the day, and the
photon hypothesis became an accepted part of the story of
physics.
This brings us to the remaining two puzzling items from last
chapter, dealing with atomic spectra and atomic structure,
which we'll discuss now.
Atomic Structure
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What is the inside of an atom like?
There were quite a number of atomic models bandied about in
the first decade of the 20th century, as physicists became
more convinced in the existence of atoms. None of them
explained the existing experimental data very well, though.
Notable is J.J. Thomson's "plum-pudding" model, which he was
developing. His discovery of the electron in 1897 gave him
hope that his model would eventually be successful in
explaining spectroscopic data.
Enter Rutherford, and the Geiger-Marsden experiment (1909).
The results of the experiment decisively scrapped Thomson's
model, and many of the others as well. But what could be put
in its place?
The following diagram is an outline of the actual apparatus.
Alpha particles fly through M towards the gold foil F; D and R is
the microscope/detector that can be rotated through various
angles about a vertical axis in the cylindrical central part of the
apparatus.
And here's a schematic diagram of the apparatus that indicates
the path of scattered alpha particles:
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the path of scattered alpha particles:
Rutherford hypothesized that most of the mass of an atom must
be contained in a very small volume, which he called the nucleus
of the atom, and which he assumed is at the centre of an atom.
He then calculated (see the diagram above) the expected
proportion of incoming alpha particles that should be scattered
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proportion of incoming alpha particles that should be scattered
at each angle. (For details of the calculation, stay tuned for
second-year mechanics, or read a good second-year mechanics
textbook over the summer.)
Rutherford then compared the distribution that he calculated
against the experimental results, and found good agreement.
This was strong support for a nuclear model of atoms, and
certainly ruled out quite a number of other atomic models,
including the very popular plum-pudding model.
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Rutherford went on to create (1911) a "planetary" atomic
model, with the nucleus playing the role of the Sun, and the
electrons playing the role of planets. This was a very good
step towards a modern atomic theory, but alas Rutherford
was unable to make quantitative predictions with his theory.
• one of the lessons/methods of science: look for the
unexpected
Also, Rutherford's model suffered from a fatal flaw: according
to classical electromagnetic theory (Maxwell's equations),
atoms should emit continuous (that is, a wide range of
wavelengths) electromagnetic radiation (they don't; rather
they emit "line spectra"), and they should be unstable
(electrons should lose energy as they emit electromagnetic
radiation, and eventually spiral into the nucleus, where they
would die a dramatic death upon joining forces with the
protons in the nucleus), which they aren't. So, although it was
clear that Rutherford was onto something, it was also very
clear that his model was wrong.
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This is another lesson of science: sometimes a theory that you
know is wrong is still worth pursuing, studying, and
understanding deeply, because it may be just the stepping
stone you need to help you discover a good theory. However,
it takes courage to publish a theory that you know is wrong; of
course, you would be honest about it and admit the fatal flaws
in the theory.
Enter Niels Bohr.
• Bohr's atomic model; see pages 921 ff in the textbook
What are line spectra?
• emission and absorption spectra:
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• emission and absorption spectra:
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&imgrefurl=http://chemicalfacts4u.blogspot.com/2011/06/atomic -orbital.html&docid=F81z3i_N2aTcM&imgurl=http://3.bp.blogspot.com/-7WaPCpa9mF4/TfgpuFxlFII/AAAAAAAAACA/x5_0HjUpn2Y/s1600/ch9orbitals1.jpg&w=576&h=388
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191&start=0&ndsp=18&ved=1t:429,r:4,s:0,i:98>
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Problem:
Collisional excitation: Suppose that two hydrogen atoms collide,
and in the process the electron in one of the atoms is knocked
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and in the process the electron in one of the atoms is knocked
into a higher orbit. Initially the electron is in the n = 1 orbit and it
ends up being knocked into the n = 3 orbit. Then the electron
"cascades" down to its ground state; that is, the electron first
makes a transition from the n = 3 orbit to the n = 2 orbit, and
then makes a transition from the n = 2 orbit to the n = 1 orbit.
(a) Determine the energy absorbed by the electron in the initial
collision.
(b) Determine the wavelengths of the emitted electromagnetic
radiation.
Solution:
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Problem: The Paschen series is analogous to the Balmer series,
but with m = 3. Calculate the wavelengths of the first three
members in the Paschen series. Which part(s) of the
electromagnetic spectrum are these in?
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Problem: The allowed energies of a simple atom are 0.0 eV,
4.0 eV, and 6.0 eV. (a) Draw the atom's energy-level diagram.
Label each level with the energy and the principal quantum
number. (b) Which wavelengths appear in the atom's
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number. (b) Which wavelengths appear in the atom's
emission spectrum? (c.) Which wavelengths appear in the
atom's absorption spectrum?
___________________________________________________
Problem: A researcher observes hydrogen emitting photons of energy
1.89 eV. What are the quantum numbers of the two states involved
in the transition that emits these photons?
Consider instead the related problem where we use
___________________________________________________
Problem: A hydrogen atom is in the n = 3 state. In the Bohr
model, how many electron wavelengths fit around this orbit?
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- the idea of electron energy levels and "shells" was later
adapted to help describe atomic nuclei as well
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___________________________________________________
Problem: Predict the ground-state electron configurations of
Mg, Sr, and Ba.
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Problem: Explain what is wrong with each electron
configuration.
(a) 1s22s22p83s23p4
(b) 1s22s32p4
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Problem: Identify the element for each electron configuration.
Then determine whether this configuration is the ground state
or an excited state.
(a) 1s22s22p63s23p64s23d9
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Then determine whether this configuration is the ground state
or an excited state.
(a) 1s22s22p63s23p64s23d9
(b) 1s22s22p63s23p64s23d104p65s24d105p66s24f145d7
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Problem: Hydrogen gas absorbs light of wavelength 103 nm.
Afterward, what wavelengths are seen in the emission
spectrum?
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BONUS MATERIAL (not covered in this running of the course)
Some concluding remarks about quantum conundrums
In classical mechanics, the state of a system of particles is
described by specifying the position and momentum of each
particle as functions of time. Typically one solves Newton's
second law of motion for the system of particles (a second order
differential equation for each particle, which results in a system
of differential equations), which then gives us the position
function of each particle. Then you differentiate the position
function to obtain the velocity function, from which you can
obtain the momentum function by multiplying by the mass.
Thus, if you specify the position and momentum of each particle
in your system, you know all about the system. For example, you
can calculate any other quantity of interest about the system
(kinetic energy, angular momentum, etc.) from the position and
momentum functions.
The situation in quantum mechanics is totally different. For
fundamental reasons (think about the Heisenberg uncertainty
principle), one can't know simultaneously and precisely the
position and momentum of a particle. Thus, it's impossible to
specify the position function and momentum function of a
particle in quantum mechanics. (In fact, one could argue that
they don't even exist, but more about that later.) How then is
one supposed to even specify the state of a system in quantum
mechanics?
After a presentation on de Broglie's ideas about matter waves by
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After a presentation on de Broglie's ideas about matter waves by
Born in 1924, Debye asked Born: if we are to take this de
Broglie's idea seriously, then what is the differential equation
that is satisfied by matter waves?
That was a good question, and undoubtedly several people
sought answers. Schrodinger was the first to bear fruit, creating
a suitable wave equation late in 1925. "Create" is the right word;
such equations are not derived, they are "guessed," although
guessing hardly does justice to this highly imaginative activity.
Here is a plausible story for how Schrodinger might have guessed
the differential equation that now bears his name. Recall from
earlier in the course that a transverse wave (a "plane" wave) can
be expressed as (see page 485 of the textbook)
or equivalently, as
As you'll learn in upper-year courses involving waves, it's
possible to subsume both expressions into a single complexvalued exponential expression as follows:
The sine and cosine descriptions can be recovered as the real
and imaginary parts of the complex function describing the wave
(look up Euler's formula if you don't know it yet).
Note that it's customary in quantum mechanics to use the Greek
letter Psi to represent this rather curious mathematical animal,
which is called a wave function. According to the standard
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which is called a wave function. According to the standard
interpretation of quantum mechanics, the wave function
"encodes" all that it is possible to know about a physical system;
therefore, the state of a physical system in a quantummechanical description is its wave function. Contrast this with
the description of the state of a system in classical mechanics in
terms of the position and momentum functions of all of particles
in the system.
Schrodinger guessed a differential equation satisfied by the wave
function. He ensured that his guess is consistent both with the
de Broglie wavelength of a particle's "companion wave" and
Einstein's relation for the energy of the wave associated with a
particle, as follows:
Now observe what happens when you differentiate the
wave function with respect to both x and t:
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In other words, differentiating the wave function with respect
to x and then multiplying by the constant iћ is the same as
just multiplying the wave function by p. A more sophisticated
way to say this is that in quantum mechanics, momentum is
an operator, and specifically its action on a wave function is
related to the action of partial differentiation with respect to
x as follows:
Sometimes a "hat" is placed above a letter to emphasize that
it's being considered as an operator, as we've done above,
but this usage is not universal.
The energy operator can be derived similarly, by observing
what happens when we differentiate the wave function with
respect to t:
In other words, differentiating the wave function with
respect to t and then multiplying by the constant iћ is the
same as just multiplying the wave function by the energy
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same as just multiplying the wave function by the energy
operator. In quantum mechanics, the energy operator is
symbolized by H (it's called the Hamiltonian operator). Thus,
Now we've done the ground work to show you Schrodinger's
equation. Start with the Newtonian expression for mechanical
energy, and substitute the quantum operators for momentum
and energy:
For a system that's free to move in three dimensions, the
Schrodinger equation is:
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All that wonderful mathematics you'll be learning in second
year (multi-variable calculus, vector calculus, differential
equations, linear algebra, complex numbers, etc.) will be
put to good use when you study quantum mechanics in
more depth!
We argued in terms of the wave function for a plane wave just as
a way of making the guess for Schrodinger's equation plausible;
this does not imply that the wave function for every single
system is a plane wave. No, you solve Schrodinger's equation for
each particular system, and the solution is the wave function for
the system. And you specify the system that you're dealing with
by identifying the potential energy function V(x).
Once you've solved the Schrodinger equation to obtain the wave
function, then the only things that you can know about the
system are things you can derive from psi by applying the right
mathematical operations to it, such as energy levels, transition
probabilities, angular momenta, etc; the results are real numbers
that you can then compare to experiments.
Everyone agrees about how to solve the Schrodinger equation,
and everyone agrees with how to extract the values of physical
quantities from the solution, and everyone agrees with how to
compare the values to experimental results. Everyone agrees
that the whole process works excellently, and agrees excellently
with experiment. Furthermore, the thought process that goes
along with quantum mechanics has been used to develop some
pretty snazzy technology (lasers, solid state devices,
microminiaturized solid state devices, semi-conductors, some
nanotechnology, etc.).
The problem is that many people don't like the philosophy
behind this program; they feel uncomfortable with its departure
from "local realism" that we are so used to in classical physics;
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from "local realism" that we are so used to in classical physics;
but maybe this is just the way it is, and we should just get used
to it, and embrace it and learn to look at the world in this way.
But then how does the classical world of our macroscopic reality
emerge from this strange microscopic reality?
Furthermore, certain physicists just don't like the quantum
philosophy, and don't accept it; they figure there is a deeper
theory out there, that would remove some of the quantum
puzzles, and return us to a more palatable interpretation.
Many other physicists ascribe to the "Shut up and calculate!"
philosophy; i.e., don't worry about interpretational issues, just
get down to business and focus on practical calculations; who
cares about "deeper meaning."
Now let's discuss what local realism is, and we'll also discuss
some of the most troublesome interpretational issues in
quantum mechanics.
The "collapse" of the wave function
So far, we've discussed an analytical (i.e., calculus-focused)
approach to the Schrodinger equation. There is another very
fruitful perspective, which we might call the algebraic approach
(and which highlights the need to get a solid grounding in linear
algebra). In the algebraic approach, the wave function is
considered to be a vector in an abstract "representation" space,
which might be finite dimensional or infinite dimensional,
depending on the system being studied.
In this approach, the state of a quantum system can be
considered to be one of the vectors in the abstract
representation space, and so vectors in this space are also often
called "state vectors." It turns out that one can choose a basis for
the abstract representation space consisting of vectors that each
have a definite energy associated with them; such basis vectors
are called "stationary states" or, equivalently, "energy
eigenstates."
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eigenstates."
Because the energy eigenstates form a basis for the
representation space, an arbitrary state of the system can be
expressed as a linear combination (i.e., superposition; quantum
mechanics is a linear theory!) of energy eigenstates.
Furthermore, the Schrodinger equation can be expressed in the
language of linear algebra as follows
where H is the Hamiltonian operator and E is a number. All you
linear algebra fans will recognize this as an eigenvalue equation,
and the solution vectors are the eigenvectors, or energy
eigenstates as we have been calling them. The eigenvalues are
the numbers E for which the eigenvalue equation has a solution.
It may be difficult at first to recognize that this eigenvalue
equation is equivalent to the Schrodinger differential equation;
remember that the Hamiltonian operator is a differential
operator, and stands for the whole left side of the analytical
Schrodinger equation. In linear algebra class we're used to
thinking of linear transformations (also called linear operators)
as matrices; we have to expand our perspective and realize that
all sorts of (initially) strange looking mathematical animals can be
considered to be linear operators acting in the right vector
space.
If you've been able to follow the previous development, we're
now ready to discuss the issues around the "collapse of the wave
function" concept, also known as the quantum measurement
problem. As a system evolves in time, its state is described by its
wave function (state vector, if you prefer), whose evolution in
time satisfies the Schrodinger equation. However, as soon as you
make a measurement of, let's say, the energy of the system, the
wave function makes a sudden change (not described by the
Schrodinger equation) to one of the energy eigenstates. That is,
what was a superposition of eigenstates has suddenly been
projected into one single eigenstate. Furthermore, the value of
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projected into one single eigenstate. Furthermore, the value of
the energy measured in the experiment is necessarily the energy
associated with the eigenstate.
The same sort of thing happens for all other observable
quantities, not just energy. In each case, the wave function goes
merrily along, evolving according to the Schrodinger equation,
until one decides to make a measurement, in which case the only
possible results of the measurement are the eigenvalues of the
corresponding operator, and the measurement suddenly
collapses the wave function into the matching eigenstate of the
operator. Furthermore, once the measurement is made, the
collapsed wave function now continues to evolve according to
the Schrodinger equation, and subsequent measurements
involve further collapses of the new wave function. Thus, the
measurement has definitely disturbed the system being
measured, and subsequent measurements reflect this
disturbance.
This was extremely puzzling, and indeed annoying, to many
physicists, including some of the founding fathers, such as
Einstein, Schrodinger, and de Broglie. The standard
interpretation described here is now called the Copenhagen
interpretation, because it was elaborated under the guidance of
Bohr (who was based in Copenhagen), and its adherents
included Heisenberg, Born, and many others.
Einstein and Schrodinger, in particular and among others, tried
their hardest to come up with logical arguments for why this
situation is unacceptable. Bohr and Heisenberg argued that the
value of a measured quantity does not even exist before the
measurement is made, and they used this principle to good
effect in winning all of their arguments with Einstein and
Schrodinger. As Einstein famously said, "Do you really believe
that the Moon is not really there until you look at it?" Indeed,
even if you believe that the microworld is beset with these
bizzare (to our macroscopically informed sensibilities)
behaviours, does the macroworld also behave in these bizarre
ways? And if not, then why not? Why does the microworld
behave bizarrely and the macroworld behaves "normally?" These
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behave bizarrely and the macroworld behaves "normally?" These
are troubling questions that have been plaguing physicists (and
philosophers of science) for nearly a century, and although
progress has been made in untangling the mess, the waters are
still rather murky.
Schrodinger devised a picturesque thought experiment with the
intention of showing how absurd the Copenhagen interpretation
of quantum mechanics is, which nowadays goes by the name of
"Schrodinger's cat." Maybe he liked cats, maybe he just wanted
to tar the reputations of the Copenhagen contingent by subtly
suggesting that they would be animal abusers; I don't know why
he chose a cat instead of some other animal.
Schrodinger suggested that an elaborate Rube-Goldberg-like
device be placed in a sealed container along with a cat. In the
device, a radioactive atom is placed with a detector, and the
detector is connected to a mechanism that smashes open a vial
of poison. Once the vial is smashed the cat will die instantly.
The radioactive atom is in a superposition of two states, one in
which the atom has not yet decayed, and one in which it has
decayed. Schrodinger argued that if you believe the Copenhagen
interpretation, you must conclude that the cat is also in a weird
superposition state of being both alive and dead. Only by
opening the container and looking (which amounts to making a
measurement) will you then collapse the wave function of the
cat and project it into one of the eigenstates, either dead or
alive. Schrodinger's argument is equivalent to Einstein's
rhetorical question, "Do you really believe the Moon is not there
until you look at it?" In other words, do you really believe the cat
is in this weird superposition alive/dead state until you look at it?
Do you really believe that the act of observing the cat kills it (or
renders it alive, from its previous ambiguous state)? Schrodinger
and Einstein certainly did not believe this.
The problem with the arguments of Schrodinger and Einstein is
that experiments with atoms and subatomic particles always
behave in the way predicted by the Copenhagen interpretation!
So they are certainly wrong at the micro level; if the Moon is
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So they are certainly wrong at the micro level; if the Moon is
really there even if I don't look at it, then they are right at the
macro level. But this is puzzling; how on Earth does the
macroworld conspire to be different from the microworld?
Many attempts have been made to sort out this mess; to learn
more, search on the terms "quantum decoherence" and
"consistent histories;" these approaches were motivated by the
earlier de Broglie-Bohm interpretation of quantum mechanics.
Wigner extended the Schrodinger cat thought experiment in an
attempt to show that consciousness is essential to collapsing
wave functions, and consciousness is essential to resolving the
measurement problem. However, this raises yet more difficult
issues; if consciousness is necessary, then how conscious does
the observer have to be? Would a dog do? How about a worm?
Would a dolphin or a monkey be sufficient? Many physicists are
not comfortable with the idea that a conscious observer is
necessary to spring reality into existence. To learn more about
this, search the key phrase "Wigner's friend."
Local Realism
The discussion so far calls into question what reality is. All of this
could be dismissed as a philosophical discussion that is a
pointless waste of time, except that every experiment with micro
objects (atoms and subatomic particles) confirms the
Copenhagen interpretation!
In 1935 Einstein gave it his last shot at punching a hole in the
Copenhagen interpretation. Together with Podolsky and Rosen
they created another thought experiment that they figured
showed, finally and emphatically, how absurd the Copenhagen
interpretation is. The world could not possibly be this way, they
argued. Their thought experiment is now known as the EPR
problem (or the EPR paradox, although it's not really a paradox).
Here's a sketch of the EPR argument: Suppose you prepare a
system containing two interacting particles which then separate
and move far apart from each other. The fact that the particles
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and move far apart from each other. The fact that the particles
interacted initially means that they have to be described by a
single wave function (i.e., they are described with a single
quantum state). Now consider the position and momentum of
each particle. From Heisenberg's uncertainty principle, we know
that it's not possible to know precisely the value of both the
position and momentum of a particle. If you measure one
quantity very precisely, then you lose all possibility of measuring
the value of the other quantity for the same particle.
When Heisenberg proposed his uncertainty principle, he
suggested that it was a consequence of inevitable disturbance in
the measurement process; for example, to measure the position
of a subatomic particle, you have to shine light on it and the
photon "kicks" the particle in an indeterminate way, thereby
disturbing the momentum of the particle. However, nowadays
Heisenberg's uncertainty principle is understood more subtly, as
a consequence of the wave aspect of particles. The point of the
EPR argument is an attempt to show that quantum mechanics is
incomplete, and they attacked Heisenberg's interpretation of his
uncertainty principle.
* * * * to be completed * * * *
photon experiments, and the analogue of Heisenberg's
uncertainty principle for other pairs of non-commuting variables
entanglement, non-locality, the EPR "paradox", Bell's inequality,
the Aspect experiments, and the demise of "local realism"
Conclusion: Quantum mechanics and its successor theories
(quantum field theories, such as QED (quantum electrodynamics)
and QCD (quantum chromodynamics)) are the most accurately
verified physical theories we have. There is clearly something
"right" about them, but they contain unresolved interpretational
issues that trouble many people. There appears to be lots of
room there for fresh ideas for both further development and
also for resolving some of the troubling issues.
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also for resolving some of the troubling issues.
Maybe some of you will decide to devote some time to thinking
about and working on some of these problems. To do this, you'll
have to learn a lot more mathematics and physics, which is fun in
itself. Others may prefer to continue with other fields of study; in
your case, I hope what you've learned in physics will be of use to
you in your favourite field of study.
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BONUS MATERIAL (not covered in this running of the course)
It turns out that quantization of energy levels is characteristic of
any bound system. The following material gives some details.
Energy Quantization For Bound Particles
Consider a string tied at both ends, such as a guitar string, or a
piano string. When the string is plucked, standing waves are
set up on the string. That is, the string vibrates in a pattern
such that the number of half-cycles in the pattern is a whole
number. That is, if L is the length of the string, then:
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Problem: Determine the length of a box in which the minimum
energy of an electron is 1.5 × 10-18 J.
Solution:
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Problem: The nucleus of a typical atom is 5.0 fm in diameter. A
very simple model of the nucleus is a one-dimensional box in
which protons are confined. Estimate the energy of a proton in
the nucleus by determining the first three allowed energies of a
proton in a box 5.0 fm long.
Solution:
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Energy Levels and Quantum Jumps
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