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Some Key Ideas in
Quantum Physics
References
• R. P. Feynman, et al., The Feynman Lectures
on Physics, v. III (Addison Wesley, 1970)
• A. Hobson, Physics: Concepts and
Connections, 4th ed. (Prentice Hall, 2006)
Nano is (typically) Quantum
Mechanical
• Four quantum phenomena that classical
models cannot explain
– The wave-particle duality of light and
matter
– Uncertainty of measurement
– Discreteness of energy
– Quantum tunneling
Quantum Mechanics
• A new theory that “replaces” Newtonian physics
• A more fundamental level of description of the natural
world
• Newtonian physics is an approximate form of QM,
very accurate when applied to large objects
– “Large” means large compared to the atomic scale
– Explains why Newton’s Laws work so well for “everyday”
phenomena
• The most precisely tested scientific theory of all time!
Essential to understanding…
• Detailed structure of atoms
– Size, chemical properties, regularities exhibited by the PT
– The light they emit
• Structure of atomic nuclei
– How protons and neutrons stick together
• Structure of protons and neutrons, and other, more
exotic particles
– Made of smaller bits still: “quarks and “gluons”
• Structural and electronic properties of materials
• Transistors, electronics
• And a host of other phenomena…
Some Key Ideas
•
•
•
•
“Wave/particle duality”
Uncertainty principle
Discrete energy levels
Tunneling
A “Thought Experiment”
• Has actually been done
many times in various
guises
• Contains the essential
quantum mystery!
• Basic setup: particles or
waves encounter a screen
with two holes (or slits)
• First, particles
One Slit Open
• Close each slit in
turn and see where
bullets hit the
backstop
• The curve shows
how many bullets hit
at a given point
• Call these N1 and N2,
respectively
N1
N2
Both Slits Open
• Bullets are localized and
follow definite paths
• Each goes through one
slit or the other
• If it goes through slit 1,
say, it doesn’t matter
whether slit 2 is open or
not
• So the combined result
is the sum of the
individual ones:
N12  N1  N2
N12
Next, Waves
• Same setup, but with
waves
• Look at a cork floating
at the backstop –
measure the energy of
its up-and-down motion
• Waves can be any
“size”, not lumpy like
particles
I1
One Slit Open
• Call the energy of the
bobbing cork “I”
– Where I is largest, the
cork bobs up and down
most vigorously
• I1 and I2 look just the
same as N1 and N2 did
I2
Both Slits Open: Interference
I12
• With both slits open, we get
an interference pattern
• Alternating regions of
bobbing and no bobbing
• A result of combining the
ripples from the two slits
• Characteristic of wave
phenomena, including light
• Note
I12  I1  I 2
Mathematics of Interference
• Call the height of the wave h (can be + or –)
• Then
h12  h1  h2
• The intensity (energy) of the wave I = h2
2
• So
I12  h12 
 h1  h2 
2
 h12  h22  2h1h2
 I1  I 2  2h1h2
Not I1 + I2!
Now try it with electrons
• Essentially the same as
with the bullets
• Electrons are “lumpy” –
we never find only part
of one
• They always arrive
whole at the backstop
• Measure how many
arrive at different
locations on the
backstop as before
One Slit Open
Both Open: Interference!?
A
• Notice that at some places (e.g. A) there are
fewer electrons arriving with both open than
there were with only one open!!!
An Implication
• Proposition: Each electron either goes through
slit 1 or slit 2 on its way to the backstop
• If so, then for those that pass through slit 1,
say, it cannot matter whether slit 2 is open or
closed (and vice versa)
• The total distribution of electrons at the
backstop is thus the sum of those passing
through slit 1 with those passing through slit 2
• Since this is not what is observed, the
proposition must be wrong!
An Implication
• Electrons (and other objects at this scale) do
not follow definite paths through space!
• They can be represented by a kind of wave,
that exhibits interference like water waves
• They also behave like particles, in the sense
that they are indivisible “lumps”
• “Wave-particle duality”: Is it a wave or a
particle? It’s both! And neither…
Surely we can check this…
• Let’s find out whether the electrons go through slit 1
or 2
• Put a detector behind the slits, e.g. a light source
– Electrons passing nearby scatter some light
– We see a flash near slit 1 or 2 – tells us which one it came
through
Light source
What do we see?
• When we can tell which slit they go through,
there is no interference!
Okay, maybe…
• …the light hitting the electrons affects them in
some way, changing their behavior?
• How can we reduce this effect?
• We can reduce the energy carried by the
light; this reduces any “kick” that the light
gives the electrons
• This requires that we increase the
wavelength of the light
A Funny Thing
• We can only “see” things that are comparable
to or larger than the wavelength of the light
• When the wavelength becomes larger than
the spacing between the slits, we can’t tell
which slit the flash is near!
– We get a diffuse flash that could have come from
either
• The interference pattern now returns!!
• When we “watch” the electrons, they behave
differently!
Another Implication
• Observing a system always has
some effect on it
• This effect cannot be eliminated
– No matter how clever we are at
designing experiments!
– With baseballs, e.g., the effect is
too small to be noticeable
• The observer is part of the
observation!
We have to remember that what
we observe is not nature in itself,
but nature exposed to our method
of questioning. – Heisenberg
Werner Heisenberg
Quantum
Mechanics
• Heisenberg, Erwin
Schrödinger and Max Born
showed how to determine
the behavior of the
quantum “waves”
• Showed that the QM
version of the “planetary
atom” was stable!
Max Born
Erwin Schrödinger
Hydrogen Atom Wave Patterns
• Characteristic
patterns and
frequencies
• Like musical
notes!
• The chemical
properties of
the elements
are related to
these patterns
“Hearing” the Tones
• Electrons can “jump” from
one waveform to another
• In this process, light is
emitted
– Frequency = difference in
waveform frequencies
• Since different elements
have different characteristic
waveforms, each produces
a different “spectrum” of
light
• The “fingerprints” of the
elements
Another Implication
• If we carefully set up the electron gun so that the electrons it
produces are identical, we still get the same interference pattern
• So the same starting conditions lead to different outcomes!
• What causes this? Nothing – the electrons are identical!
• A fundamental feature of the microscopic world: randomness
• The overall pattern is what is predictable, not behavior of
individual particles
A philosopher once said “It is necessary for
the very existence of science that the same
conditions always produce the same results.”
Well, they don’t!
– Richard Feynman
The Uncertainty Principle
• In QM, particles are described by waves
– Usually called the “wave function”
• Waves for a faster-moving particle have shorter
wavelength
• Those for a slower-moving particle have longer
wavelength
Faster
Slower
Uncertainties
• The wave is spread out in space – the particle can be
found wherever the wave is not zero
• There is an “uncertainty” in the location x of the
particle
x
(Think of this as the size of the region in space where
the particle is likely to be found.)
• A wave spread over all space would have infinite
uncertainty – not a real particle
Real Waves for Real Particles
• To make a useful wave, we can add many of these “pure”
waves together:
Real Waves, continued
• But now we don’t have a single speed (wavelength),
it’s a mixture!
• So for a real particle there is an uncertainty in the
speed as well:
s
If we measure the speed we will get a range of
possible results, with a variation of about s
• Both the speed and location are uncertain
– Remember: no definite trajectories!
The Uncertainty Principle
• For any particle
(x)(s)  h m
where h is a fundamental constant of nature
(“Planck’s constant”) and m is the mass of the
particle
– Strictly speaking, the above is h/m at a minimum; it can be
larger
• What does this mean?
The “Range of Possibilities”
• Let’s call the product (x)(s) the particle’s “range
of possibilities” (not standard terminology!)
Speed
Speed
s
Position
x
• The HUP says the area of the rectangle is fixed,
equal to h/m
Position
Localizing a Particle
• Say we make (x) smaller; then (s) must get larger:
Speed
Speed
Rectangle must
have the same
area as before
s
s
x
Position
• And vice versa, of course
x
Position
What it Means
• The HUP means that the
more precisely we localize a
particle (know where it is), the
more uncertain is its speed,
and vice versa
• Note that heavier particles
have a smaller realm of
possibility
– Shows why e.g. baseballs do
appear to have a precise
location and speed!
(s)(x)  h m
Baseball RoP (not to scale!!)
Electron RoP
Proton RoP
Area of the rectangle is
reduced if m is large!
Exercise
Arrange these objects in order, beginning
with the object having the largest “realm of
possibilities” and ending with the one having
the smallest: proton; glucose molecule
C6H12O6; helium atom; baseball; electron;
grain of dust; water molecule; automobile.
Quantum Reality
• Atomic-scale phenomena are weird 
– Particles “everywhere and nowhere” until found
– Essential randomness
– Influence of observer on observed
• Macroscopic (big) objects don’t act like this,
apparently
• Can/does quantum weirdness extend into the
macroscopic world?
• If so, why is it not apparent?
– See “Mr. Tompkins in Wonderland” by G. Gamow
Schrödinger’s Cat
• Erwin Schrödinger was an early
pioneer of QM
– Austrian; later moved to Ireland
– Nobel 1933
– Basic equation governing QM waves
called the “Schrödinger equation”
• A thought experiment – not actually
done, at least with cats 
• Designed to show the paradoxical
nature of QM in the macroscopic
world
Experimental Setup
How it Works
• Let’s assume that radioactive decay of the nucleus
happens with probability ½ in a minute
• Decay is a QM process – random!
• Until we observe the nucleus, it “goes both ways”
• After a minute the nucleus is neither “undecayed” nor
“decayed”, it is a mixture of the two
– Just as the particles go neither through slit 1 or 2, but rather
through both, in a sense
• When we observe it, the state “collapses” to one or
the other outcome, with probability ½ for each
The Poor Cat
• Since the nucleus is not in a definite state until we
observe it, neither is the cat!
• It is neither dead nor alive, until we observe it!!
– The rules say it is in a “superposition” (mixture) of the two
• Schrödinger (rightly) considered this absurd
• Special role of observation in the theory
– The “Copenhagen interpretation” – Bohr
• Is consciousness required for measurements? Is the
cat conscious? Is a bug?
Modern Interpretation
• “Measurement” occurs when the microscopic system
interacts with a macroscopic object, here the Geiger
counter
– And of course the cat too!
• Such macroscopic objects “decohere” very quickly
– The quantum superpositions get “washed out” due to the
enormous numbers of particles
• They act classically!
• The basis for modern interpretations of QM
“Many Worlds” Interpretation
• The most “exotic” interpretation of QM 
• Both states persist
– One with nucleus decayed/dead cat
– Another with nucleus intact/live cat
•
•
•
•
•
The decohere so they cannot “interact”
Both go on their (merrry?) ways
As though the universe splits into two
Every decohering process leads to further splitting
All possible outcomes are realized somewhere in
this “multi-verse”!
The Situation Today
• Rules for calculating with QM are well established,
work beautifully
• Problems of interpretation not fully resolved
• Decoherence is the key to understanding the
interaction of QM systems with the macroscopic
world – well understood
• Most physicists regard the problem as interesting and
fundamental but not critical for most research
Some physicists would prefer to come back to the idea of
an objective real world whose smallest parts exist
independently in the same sense as stones or trees exist
independently of whether we observe them. This
however is impossible… Materialism rested on the
illusion that the direct “actuality” of the world around us
can be extrapolated into the atomic range. This
extrapolation, however, is not possible – atoms are not
things. [emphasis added]
– Werner Heisenberg
Energy of Quantum Systems
• Particles associated with waves
– Wave frequency corresponds to energy, a lá
E = hf
• The waves are described by Schrödinger’s
equation
• Solutions for “bound” quantum systems
typically have discrete energy levels
• Can we understand this qualitatively?
Standing Waves
• For bound systems the quantum wave
must vanish outside some region
• Then only waves with appropriate
wavelengths will “fit”
• Like standing waves on a string
• A discrete set of energies
Quantum Particle in a 1D Box
Higher Dimensions
• Analogy: standing
waves on a
drumhead
• Discrete frequencies
(energies)
• There may be
several modes of
oscillation with the
same frequency –
“degeneracy”
A Caveat
• In realistic situations, the quantum wave
need not strictly vanish outside the
“bound” region
– It decays exponentially there
• Result is still that solutions have
discrete frequencies
• Also: “tunneling”
Tunneling
• Roller coaster:
Maximum height
(KE = 0)
Too slow!
“Classically
forbidden”
region
(KE would
be < 0)
Quantum Mechanically
• QM wave decays in
the forbidden zone,
but isn’t zero!
• “Leaks” through to
other side
• Hence some
probability to tunnel
through!
An Optical
Analogy
• Schrödinger’s equation
describes a sort of
wave, similar to light
waves
• Look in window –
some light transmitted,
some reflected
• Typical wave behavior