quantum mechanics really comes from new ideas and experiments on the interaction
of electromagnetic radiation with matter.
But because it comes to form the
foundation of atomic and molecular
science, it's often encountered in early
stage, usually high school chemistry long
before most students learn anything in
depth about electromagnetic waves and
radiation. Because quantum mechanics is
famously weird, all this associated
craziness of quantum mechanics is just
thrown at you with no physical
intuition to build on. Now of course when
the greats like Einstein, Bohr, and
Schrodinger were creating quantum
mechanics they discovered that there
were some crazy things about the
behavior of atomic and molecular systems, but they were forced to take that step into
the unknown after trying as much as they
could- and failing- to make sense of new
experimental results with classical
ideas. And it's this contrast with
classical intuition that quantum
mechanics forces on us that is the
essence of its weirdness. But these days
students are taught, sometimes as early
as high school chemistry, about things
meant to sound mysterious and weird like
the Heisenberg uncertainty principle or
Schrödinger's wave equation, things that
because of their weirdness and coolness
have entered popular culture- and you learn
about these things without that strong
foundation in classical theory, without
that solid physical intuition. You learn
for example about emission or absorption
of a photon, and it appears, I think, in many people's minds in a cartoonish sci-fi way
a photon is this magical thing being
absorbed.  Well yes, there's some strange
things about the quantum mechanical of
understanding of emission or absorption
of a photon, but the ability of a
molecule to emit or absorb a photon
itself is not some crazy sci-fi futuristic
idea. It's a very well understood concept
which is at the core of our modern
technology and which dates to the 19th
century. The prototypical example of
emission and absorption of radiation is
to transmission or reception of radio
waves by a tower or radio. Simple, not
crazy,
perhaps amazing when it was first
discovered but not so anymore. But it
makes no sense
to learn about the quantum mechanical
description of light emission and
absorption and to hear about the weird
features of quantum mechanics without
first understanding this simple
fundamental everyday example of
radiation emission and absorption.
Usually the way the story is told is
that there were new experiments around
1900, the turn of the century, and new
equations were derived to explain these
experiments and that the equations have
little physical intuition or logic
associated with them, but that they simply
worked. And it's often said that the
equations of quantum mechanic cannot be
derived- which is basically true,
historically they were essentially just
guessed. But if you understand a little
bit about classical radiation theory
then you can appreciate what's new and
unexpected about these quantum
mechanical ideas and equations. The
fundamental principle of classical
radiation theory you must understand
before getting into quantum mechanics is
that acceleration of charges and
currents- non-uniform motion- produces
radiation- and radiation produces a force
on charges and currents, that is, it
causes them to accelerate.
Although electromagnetic radiation is
different than physical matter because
it propagates through empty space
without a medium. We can understand it byway of analogy with more physically
intuitive types of waves that propagate
a matter such as sound. When I pluck a
guitar string why does it ring out? Well
the string starts vibrating, pushing back
and forth on air molecules and inducing
a self-propelling wave in air that
propagates out. We hear it when those
vibrations of the air molecules hit our
ears and start vibrating receptor cells.
Let's take this a little further:
musicians know that when a note is
played on one instrument ,another
properly tuned instrument across the
room may ring out. How does that work? As
before playing the note, say plucking a
guitar string makes the string vibrate,
the vibrations of the string push on air
molecules, the vibrations of air
molecules travel across the room as a
sound wave. And now when they reach
another instrument, say another properly
tuned guitar, those vibrations acting on
the steel string start to push on it
inducing it to vibrate. Why only the
properly tuned
string? Because the properly tuned
string has a natural frequency that it
wants to vibrate at. If that frequency
matches the frequency of incoming air
molecule vibrations, the pushing of air
molecule vibrations is synchronized with
the pushing of the string- to put it
simply- the air molecules are always
pushing in the same direction as the
motion of the string and act so as to
speed it up. If the string is not tuned
to the same frequency of vibration as
the air molecules some of the pushes act
against the motion of the string, that is,
in the opposite direction and causes it
to slow down. Over many vibrations it
works out that only a properly tuned
string can be set in vibration by an
incoming wave with the same frequency,
and this is known as resonance.
So to recap this whole process of
emission to traveling wave to absorption:
we have a guitar string being plucked, it
vibrates. the vibrating string pushes on
air molecules, those air molecules push
on other air molecules and spread out
across the room. Then those vibrating air
molecules reach another guitar string
and push on it making it vibrate. Let's
take this even further. How does this
process result in energy transfer? In the
emission of a wave in the sound wave
analogy the string pushing on the air
results in the air pushing back on the
string, essentially an application of
Newton's third law, for every action
there is an equal and opposite reaction,
for example, like when one ball collides
with another slowing down as it sets
another in motion a force accelerating
the ball at rest comes with an opposite
force slowing down the moving ball so in
this sound wave example the string
starts shaking and as it pushes on the
air the air pushes back on it the air
molecules start to vibrate and the
vibration of the string slows down then
at the other end, with absorption, a
sound wave encounters a guitar string
and makes it ring out. We have the same
process in reverse: the vibrating air
molecules push on the string, speeding it up,
setting it in motion and the string pushes
back on air molecules slowing them down.
Energy is thus transferred from a vibrating
object to the wave and vice-versa. For
emission and absorption of light or
electromagnetic waves,
the electromagnetic field created by the
vibrating charge pushes back on the
charge, slowing it down. Say, in a radio
tower when a radio transmitter is
emitting a signal there are vibrating
electrons in the big radio antenna and
those vibrating electrons creating
electromagnetic wave that propagates
outward the wave pushes back on the
electrons and slows them down. That's why it takes energy to emit the radio signal.
And then in absorption the charge which
is set in motion by the incident wave
induces its own field which cancels out
and extinguishes the incoming field, so
when the incoming wave encounters a
radio receiver, electrons in the antenna
start to vibrate and their vibration
creates a secondary field which is canceled
out by the incoming waves.
That's why energy is absorbed from the
incoming wave when receiving the signal.
By the way the fact that the
electromagnetic wave itself carries
energy, enough energy to induce a current in the receiver antenna
and then possibly to make a small earphone vibrate from
that current to reproduce a sound- that's
why the simplest radios, crystal radios,
require no battery or external power
source. Electromagnetic wave to current
in antenna to shaking of the speaker to
hearing sound. So far this whole
discussion has been classical. We've
talked about emission and absorption and
energy transfer and the concept of
resonance and the fact that this energy
transfer between matter and wave occurs
when the vibrating object vibrates at
the same frequency as the way this is
the foundation of classical intuition as
necessary to appreciate what is newly
quantum mechanics so the foundational
rule of quantum mechanics is e equals a
chap light can only be emitted in
discrete units with energy proportional
to frequency the weirdness of relating
energy to frequency is that we would
have previously expected the frequency
of vibration to matter only to relate to
the frequency of wave emitted or
absorbed
the concept of resonance discussed
previously and as the vibration occurs
we previously expected that the energy
is continuously transferred in any
amount we don't just suddenly forget
about our classical mixture of radiation
now the new equation adds an additional
constraint on top of what we already
know we still expect that for a way to
be emitted or absorbed with frequency F
we need something vibrating at that
frequency to make this consistent with e
equals H F we can no longer have the
vibration continuously transfer energy
rather we need something to be vibrating
at frequency F in the material system as
it makes the transition corresponding to
an energy difference of 8 that's why the
whole equals hf thing is unexpected and
when the system is in a steady state of
energy we don't want it to be vibrating
at all because that would lead to
radiation being emitted the resolution
to all this is to have energy states
described by a wave function that
depends on spatial coordinates x time
oscillating function with frequency
determined by inverting a relation e
equals H F to give F equals e over H
combined with the rule that the square
modulus of the wave function represents
the probability density it all works out
very nicely in a single state of energy
the square modulus of the wave function
loses its time dependence in a
superposition state with wave function
is a linear combination of energy States
taking the square modulus of the wave
function produces an interference term
which oscillates with a frequency given
by the difference in energy of those
states divided by H so you can see how
this relation e equals H F and the
classical picture that something needs
to be vibrating a frequency F for a
photon to be emitted with that frequency
with that we can almost guess with the
correct equations of quantum mechanics
will look like we can appreciate why
they're weird
and it basically builds from there
