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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)

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

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

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

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

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

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

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!

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

- Just like the bullets…

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!!!

An Implication

- Proposition: Each electron either goes through slit 1 or slit 2on 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!

Werner Heisenberg

We have to remember that what

we observe is not nature in itself,

but nature exposed to our method

of questioning. – 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

Slower

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

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

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:

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

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?

Speed

s

Position

Position

x

The “Range of Possibilities”- Let’s call the product (x)(s) the particle’s “range of possibilities” (not standard terminology!)
- The HUP says the area of the rectangle is fixed, equal to h/m

Speed

s

s

Position

Position

x

x

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

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!

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

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

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:

“Classically

forbidden”

region

(KE would

be < 0)

Maximum height

(KE = 0)

Too slow!

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

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