Nats 101 S04 #17
Reading: T&H 217-229
Quantum Mechanics
Important Points
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What does “quantum mechanics” mean?
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Measurement of a quantum object changes its state
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Heisenberg uncertainty principle
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Describe positions in terms of probabilities
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Wave-particle duality
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Two-slit experiment
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Photoelectric effect
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Bohr’s model of the atom, and wave-particle duality of electrons
The world of the very small
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“Quantum mechanics” is a phrase used to describe the study
of the motions of small bundles of matter. “Quantum” is Latin for “bundle”,
and “mechanics” is the study of motion. We have seen that electrons
and nuclei are small bundles of matter, so the study of quantum mechanics
is the study of the motions of electrons and protons, light and so forth.
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Electric charge is also quantized. It has a value of ±1. In fact,
in the atom, everything is quantized.
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This is not like our everyday, macroscopic world.
Measurement and Observation in the Quantum World
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One of the steps in the scientific method is observation. It ends up that
the quantum world requires a very different way to record observations.
One based upon probabilities.
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In general, when you measure an object, say its position, or its velocity,
then it is a fairly straightforward thing to do. You can use a tape measure,
for instance, and a stopwatch and do a quite reasonable job. Remember that
Galileo invented an accurate clock just to get the equations governing
gravity down pat.
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We use things like light to determine the position of an object. For instance,
we see it. Or even using lasers to measure the distance to the moon. In
addition, we use touch. The tape measure is put right up to the object.
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However, in the atomic world, such methods do not work. How do we record
the position of an electron, say, when the light that hits it changes its
state? The energy of the probe is too close to the energy of the thing
that we are measuring. Thus the energy of the probe changes the state of
the thing we are measuring.
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The key to these statements is that it is the consequence of small-scale
interactions that makes the difference, not that we are doing a
measurement. The measurement problems is often stated like it is the fact
that we are doing a measurement that is the problem.
The Heisenberg Uncertainty Principle
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Heisenberg (1927) (age 26) formulated the Heisenberg uncertainty
principle that is based upon the concept that the measurement alters
the object that is being measured.
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His principle says that you cannot measure the position and velocity simultaneously
to any accuracy. They are constrained to
Dx ´ Dv
> h/m,
where Dx is the error in position, Dv
is the error in velocity, h is a constant, and m is the mass of the
object being measured.
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This equation shows us that if the mass is small, then h/m is big, and the combined error in measuring position and velocity is large.
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As an example, using this equation shows us that if we locate an electron
to within the region of an atom, then we cannot know its velocity better
than within 1 million m/s.
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Conversely, if we know the position of a car, say within 5 m, then we can
obtain a precise estimate of the velocity to within 10-37 m/s.
Incredibly good!
Probabilities
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The importance of the Heisenberg uncertainty principle is that we must
alter how we describe events. For instance, in a baseball game, if someone
hits the ball, then we can tell exactly where the ball will go just by
knowing the velocity of the ball as it is struck. That is why the baseball
statisticians can tell you how far the home run was. They are using the
radar gun.
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In the quantum world, we cannot know where the ball is going if we know
the velocity well. So we describe these things by using probabilities.
For instance, the ball is hit, and we say "the odds are it may wind up
in center field, let's wait and see".
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In a similar way, we describe the location of electrons in a crystal, around
an atom in terms of electron densities. The electron density can be described
as the probability of finding an electron at any position.
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Also the thermal motion of an atom is describe in terms of probability
distribution functions. I.e. thermal ellipsoids.
Wave-particle duality
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Sometimes quantum mechanics is also called wave mechanics.
This is because these small particles display behavior that is similar
to that displayed by waves. Or, stated more precisely, the probability distribution function for the position
of a quantum object can be written in a mathematical form as a solution to the wave equation.
For this reason, we have a concept called wave-particle
duality.
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In our macroscopic world, energy can travel in one of two ways, as a particle
or as a wave. The ways to determine this are quite easy. For instance,
the two-slit experiment shows very different distributions depending upon
whether the traveling objects are particles or waves. E.g. baseballs and
water waves.
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Light shining on the two-slit apparatus produces a pattern that is consistent
with wave motion. The photoelectric effect demonstrates particle behavior.
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Any experiment done with quantum objects displays this duality. In wave
experiments they behave like waves, in particle experiments they behave
like particles.
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We should not try to constrain quantum objects to be like things in the macro world.
The quantum world is different, the problem is the way we are trying to
describe it.
The photoelectric effect
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When photons of sufficient energy strike a material, then some electrons
absorb the energy and if the energy is high enough, then the electrons
may be detached from the atom.
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If the material is in the form of a thin sheet, and if we observe light
striking one side of the sheet, then we will also observe a stream of electrons
coming off the other side. This effect is called the photoelectric
effect.
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Einstein showed that the time delay between the light striking the material
and the release of the electrons is too short for light to be a wave, so
it must be a particle. This is the origin of the word, photon.
Einstein won the Nobel Prize in 1921 for this work.
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This concept is being used a lot these days. For instance, in a camera,
a photoelectric device measures the amount of available light by measuring
the amount of electrons that come off of a thin sheet of photoelectric
material (part of a semiconductor). Then the lens aperature and shutter
speed are chosen, based upon the measurement of the number of electrons.
Wave-particle duality and the Bohr atom
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One of the quantum mechanics equations relates velocity to wavelength.
The faster an object (say an electron) is travelling, then the more energy
it has and the shorter its wavelength.
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So with this in mind, lets examine the nature of the electron in Bohr’s
model.
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If we view the electron as a particle, then the distance from the nucleus
determines its energy, and thus its speed. Just like for planets around
the sun. To stay in a given orbit, the electron must have a precise speed
that corresponds to this orbit. The further out the electron, the faster
its speed.
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If the electron is viewed as a wave, then we need to think of it in another
way. The electron has a certain energy, and therefore a certain wavelength.
It chooses the orbit radius based on making the wavelengths fit in the
circumference in an integral fashion. It forms a standing wave.
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The two models produce the exact same solutions.

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The picture above is a simplification in 2 dimensions, real electrons that are located around an atom occupy 3-dimensional
space, and their wave forms look like the images below. The equations that are used to model the shapes of the electron orbitals
are equations of standing waves, these are plots of the standing waves.
- Recently, orbitals were observed for the first time from a mineral called
cuprite.
This is the first experiment to show that the orbitals actually exist for real.
Is the quantum model correct?
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Einstein did not believe that the quantum model was correct, in spite of
the fact that he helped invent it. He did not like the fact that he could
not determine the position and velocity simultaneously to any sort of reasonable
accuracy.
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My quantum professor did not think that quantum mechanics was actually
true either. He said it was because it was too difficult to do. Nature
is always simple.
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However, you cannot argue with its success.