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# Quantum Mechanics

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```									        Quantum Mechanics
• Bohr Theory worked pretty well for one-
electron atoms (H, He+, Li++, etc.
• Failed totally for He with two electrons
• Close examination of spectra of one
electron atoms showed that some spectral
lines were actually multiple lines
• Couldn’t predict intensity of lines
• Couldn’t explain chemical bonding
Quantum Mechanics
• A new approach was needed and appeared
in two different forms
• The first was called wave mechanics and
was developed by Erwin Schrodinger
• The second was called matrix mechanics
and was developed by Werner Heisenberg
• Eventually shown to be the same
Quantum Mechanics
• Deals with the microscopic world of atoms
and light (photons)
• Blends smoothly with classical mechanics
as we approach the macroscopic world
• This is the correspondence principle
• Mathematics of quantum mechanics involve
matrices and partial differential equations
• We’ll just have to look at results
Wave Mechanics
• EM Waves have frequency, wavelength,
energy
• Relate wave to particle properties by using
E = hf
• Amplitude of an EM wave is the strength of
the electric or magnetic field and is related
to the intensity of the wave
Wave Mechanics
•   Particles have wave properties as well
•   Wavelength is h/mv
•   We say this is a matter wave
•   What is the amplitude of a matter wave?
•   Schrodinger defined a wave function 
•    is a function of position and time
Wave Mechanics
•  can be compared to the electric field
• For light the intensity is proportional to the
square of the electric field strength
• Light can be considered a stream of
particles
• Then the intensity depends on the number
of photons
Wave Mechanics
So, to say how much light we
intensity. We can either talk
about the square of the field
2
I E  N     strength or the number of
photons. The two quantities
are proportional to each other.
2
N E         We said that  is comparable
to E, so 2 must be related to
numbers of particles or
something similar.
Wave Mechanics
• The great leap is that we might consider the
probability of finding photons somewhere
in space to be proportional to E2
• Now, we make the same leap for particles
• Consider the probability of finding particles
somewhere in space to be proportional to 2
Wave Mechanics
Direct photons or particles at a
pair of slits. E2 gives the
probability of finding a photon at
the viewing screen. 2 gives the
probability of finding a particle
at the viewing screen.
Now send photons or particles
one at a time. What happens?
Place a piece of film at screen.
Wave Mechanics
We will see a dot on the film
when a photon or particle hits the
screen. If we keep sending
photons or particles and keep
watching, eventually the
interference patterns appear.
We can’t predict what any single
particle or photon will do, but we
can predict what a lot of them
will do!
Wave Mechanics
If we cover one slit for a while
and then the other for a while, no
interference pattern is seen!
So, a single photon or particle
must somehow pass through both
slits in order to interfere!
Says our macroscopic view of
waves and particles cannot be
extended to the microscopic!
Wave Mechanics
What we do know is that E2 and
2 give us the probability of
finding the photon or particle at a
point in space and time!
We can treat a wad of particles as
a wave, but we treat individual
particles by probabilities!
Uncertainty Principle
• We assume that if we measure something
we will have some small errors
• You know this from your lab work
• With better instruments and techniques you
can reduce these errors
• Heisenberg showed that there is a limit to
how small you can make the error!
Uncertainty Principle
• There are two factors involved
• One is wave-particle duality
• The other is that to measure something you have
to disturb it
• Place a ruler to measure length
• You must use pressure to line up the end of the
ruler and the end of the object
• You have to apply pressure to make the alightment
• This changes the length of the object!
Uncertainty Principle
• Example in the text about finding a ping-
pong ball in a completely darkened room
• You grope around, moving your hand
• You touch the ball during the movement
• You know where the ball was, but you don’t
know where it is after the touch
• You can’t predict the exact future position
• You gave the ball some momentum!
Uncertainty Principle
Recall the diffraction limit of light.
We can measure position to about a
wavelength of the light we use. To
get more accurate position, shorten
the wavelength which ups the
frequency. But E=hf, so the photon
has higher energy. Whacks the
electron harder and you don’t know
where it goes or how fast it is
moving. Lower the frequency and
you get more uncertainty in position.
Uncertainty Principle

x  
p  h / 
xp  h
xp  h /2
Uncertainty Principle
Particles have uncertainty in position of x=. We try to
detect with a photon that has speed c and takes a time t=
x/c= /c to pass through the uncertainty distance. So the time
of measurement is uncertain by t= /c.
Now the photon can transfer some or all of its energy to the
particle. The energy of the photon is E=hf=hc/. So, the
uncertainty in the particle’s energy after the photon hits it is
just this same amount. So the product of the two undertainties
in time and energy is Et= (hc/)(/c)=h. Heisenberg’s exact
calculation gives Et  h/2 
Uncertainty Principle
• So what does all this mean?
• When dealing with the microscopic world we
cannot simply take our macroscopic picture of
particles and waves to a smaller level
• We can’t describe in words what a photon or an
electron is
• These objects have both wave and particle
properties and we have to consider both to gain
understanding of microscopic phenomena
Uncertainty Principle
• These issues force us to deal with probabilities
rather than certainties when we discuss energies,
times, positions, momenta
• Can we violate energy of momentum
conservation?
• Sure, if we do it for a short enough time interval or
a short enough position space since we can’t
measure the failure!!!
• Is this real? YES, as we shall see!
Atoms
• So what are atoms?
• Well, Rutherford’s experiments established
the notion of a small very dense nucleus
• We know atoms contain electrons
• But the classical orbit picture is garbage!
• We only know what we can measure!!!
Atoms
• What we can measure is that atoms have definite
precise energies
• When electrons change energy levels, photons
with precise energies emerge
• The electrons have highest probability of being in
some well defined region of space with fuzzy
edges
• These probability regions are not necessarily
spherical as the classical orbits might imply
Atoms
• Atoms have other precise characteristics besides
energies
• We will take up these ideas next time when we
examine the notion of quantum numbers
• For now, we can simply say that just as a guitar
string can generate musical tones as a series of
harmonics, so the electrons in atoms can take on a
precise series of characteristics
• These properties are inherent in nature

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