Some Key Ideas in Quantum Physics

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Some Key Ideas in Quantum Physics
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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)

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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

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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 Newtons Laws work so well for
  everyday phenomena
• The most precisely tested scientific theory of
  all time!

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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

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Some Key Ideas

• Wave/particle duality
• Uncertainty principle
• Discrete energy levels
• Tunneling

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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

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One Slit Open
N1

• 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

N2
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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 doesnt matter
  whether slit 2 is open or not
• So the combined result is the sum of the
  individual ones

N12
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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

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One Slit Open
I1

• 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
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Both Slits Open Interference

• 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
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Mathematics of Interference

• Call the height of the wave h (can be or )
• Then
• The intensity (energy) of the wave I h2
• So

Not I1 I2!
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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

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One Slit Open

• Just like the bullets

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Both Open Interference!?

• Notice that at some places (e.g. A) there are
  fewer electrons arriving with both open than
  there were with only one open!!!

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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!

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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? Its both! And neither

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Surely we can check this

• Lets 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
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What do we see?

• When we can tell which slit they go through,
  there is no interference!

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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

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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 cant 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!

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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!

Werner Heisenberg
We have to remember that what we observe is not
nature in itself, but nature exposed to our
method of questioning. Heisenberg
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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
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Hydrogen Atom Wave Patterns

• Characteristic patterns and frequencies
• Like musical notes!
• The chemical properties of the elements are
  related to these patterns

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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

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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 dont! Richard Feynman
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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

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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
• (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

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Real Waves for Real Particles

• To make a useful wave, we can add many of these
  pure waves together

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Real Waves, continued

• But now we dont have a single speed
  (wavelength), its a mixture!
• So for a real particle there is an uncertainty in
  the speed as well
• 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!

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The Uncertainty Principle

• For any particle
• where h is a fundamental constant of nature
  (Plancks 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?

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The Range of Possibilities

• Lets call the product (?x)(?s) the particles
  range of possibilities (not standard
  terminology!)
• The HUP says the area of the rectangle is fixed,
  equal to h/m

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Localizing a Particle

• Say we make (?x) smaller then (?s) must get
  larger
• And vice versa, of course

Rectangle must have the same area as before
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What it Means
Baseball RoP (not to scale!!)

• 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!

Electron RoP
Proton RoP
Area of the rectangle is reduced if m is large!
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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.

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Quantum Reality

• Atomic-scale phenomena are weird ?
• Particles everywhere and nowhere until found
• Essential randomness
• Influence of observer on observed
• Macroscopic (big) objects dont 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

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Schrödingers 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

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Experimental Setup
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How it Works

• Lets 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

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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?

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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

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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!

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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

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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
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Energy of Quantum Systems

• Particles associated with waves
• Wave frequency corresponds to energy, a lá
• E hf
• The waves are described by Schrödingers equation
• Solutions for bound quantum systems typically
  have discrete energy levels
• Can we understand this qualitatively?

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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

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Quantum Particle in a 1D Box
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Higher Dimensions

• Analogy standing waves on a drumhead
• Discrete frequencies (energies)
• There may be several modes of oscillation with
  the same frequency degeneracy

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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

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Tunneling

• Roller coaster

Classically forbidden region (KE would be lt 0)
Maximum height (KE 0)
Too slow!
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Quantum Mechanically

• QM wave decays in the forbidden zone, but isnt
  zero!
• Leaks through to other side
• Hence some probability to tunnel through!

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An Optical Analogy

• Schrödingers equation describes a sort of wave,
  similar to light waves
• Look in window some light transmitted, some
  reflected
• Typical wave behavior