HUMS4100: Quantum Mechanics and Reality

I think I can safely say that nobody understands quantum mechanics. Richard Feynman.

Waves and Particles

• We have used quantum mechanics as a tool: does it just disguise something deeper?
• Or "Shut up and calculate!"
• So what is this wave thing?

Heisenberg's Uncertainty Principle

Heisenberg 1927

If an electron is a wave, how can we define its position?

• Suppose we try to measure position of electron by confining it to box

• Uncertainty in position δx = L but there is also an uncertainty in momentum δp~2p~2h/λ=h/L

• Hence

  δxδp = L h/L = h
  

• if we squeeze walls together to measure position better, momentum becomes more uncertain.

• if we could do it with one photon then the position uncertainty ~ wavelength

• δx >λ So decrease wavelength to get position better, but photon carries momentum p=h/λ and some of it gets transferred

• δx δp >λ(h/λ) >h
  

• (uncertainty in position.) x (uncertainty in mom.) > ħ

• This is a fundamental limitation on human knowledge: can always do worse but cannot do better!

• There is another form of H's uncertainty principle, which is going to become important later
• This says that to measure the energy of anything perfectly, you need to take an infinite time.

  δE δt > h
  

• e.g two signals of 400 and 403 Hz

  0.08s

  Sound files courtesy John Tann

• 0.33s

  Sound files courtesy John Tann

• 2.00s

  Sound files courtesy John Tann

• 5.00s

  Sound files courtesy John Tann

Which slit did the electron go through?

• We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of the quantum mechanices. In reality it contains the only mystery...Any other situation in QM, it turns out, can always be explained by saying, "You remember the case of the experiment with the two holes? It's the same thing." Richard Feynman, the Characeter of Physical Law

• Since the electron behaves as a wave, can do a Young's slits type experiment, and get usual 2 slit interference pattern:

• Remember: it builds up as discrete events

• But the electron is a particle, with charge. It must go through one slit or the other...

  Suppose we close off one slit:

  ---

  Suppose we close the other slit:
  Note when we add together two one slit patterns, we do not get two-slit patterns.

• Now we get sum of one slit patterns, but not a 2 slit pattern!
• What happens if we use a detector that only picks up one electron in two?
• More worrying than this: we can do a "delayed choice" experiment: don't try to observe the electron until after it has gone through one of the slits...that still destroys the pattern.
• Conclusion We cannot decide which slit the electron went through without destroying the pattern. Observing something fundamentally changes it!

• There was a young man who said "God
  Must think it exceedingly odd
  That this tree
  Continues to be
  When there's no one about in the Quad"

• Kerner: Now we come to the exciting part. We will watch the bullet to see how they make waves ...The wave pattern has disappeared
  Because we looked. Every time we don't look, we get wave pattern. Every time we look to see how we get wave pattern we get particle pattern
  Hapggod (Tom Stoppard)

• Uncertainty principle largely misunderstood by philosophers: the practical limits on measurement are much larger. However the idea that observing a system automatically changes it has had significant impact e.g.
  • Anthropologists know they cannot observe an primitive society without altering it.
  • Hawthorne effect in psychology

What Waves?

• so that if it hits a potential hill

  slow electron is always reflected (low energy) fast electron always goes over (high energy)

• Waves do not behave like this: they get partially reflected (remember what happens to light hitting glass).

• If electron is literally the wave,

• this would imply we see 1/2 electrons

Probability Interpretation

• P = |Ψ(x)|²

• e.g. for the electron in a box, in the first excited state

• Note Electron must be somewhere: i.e. probability of detecting it somewhere = 1
• Think of a die:
• probability of any given face = 1/6
• probability of any face being uppermost = 1

• Back to barrier problem

  Probs must add to 1: P₁ = prob. that electron hits detector 1: P₂ = prob. that electron hits detector 2 P₁ + P₂ = 1

• • 481 could hit 1
  • 519 ------------ 2
  (Maybe)
• 1000 will hit 1 or 2
• But we cannot say what any individual electron will do
• Classical Determinism
• Given state of solar system in (say) 100 A. D., can use Newtonian mechanics to predict earth's position now
• Quantum mechanics:
• Can only predict most likely (probable) position now.
• Morals
  1. Macroscopic (i.e. large) objects are predictable, electrons aren't!
  2. Cannot ask "what happens?": can only ask "what can we measure?"
  3. No reason to assume that rules deduced for macroscopic objects are true for very large/very light/very fast objects. "What colour is an electron?"

Tunnelling

If a ball rolls up to a barrier, it gets reflected.

• In quantum mechanics it can actually get through!
• Where is it when it where it's not allowed to be?
• You can't see it without supplying enough energy to get it over the barrier!

• E.g.: suppose we measure the position of a particle and it was at C

                       x                 
                       C
  

• Where was it just before?
• Classical Mechanic At C.
• Quantum Mechanic Somewhere: it was only measuring it that fixed its position . Where is a candle flame after it is blown out?

Measurement

• In classical mechanics, we believe that a object is the same whether we measure it or not.
• In quantum mechanics, until we have measured it, its condition is indeterminate.

• e.g a H atom in the first excited state (n = 2)will decay to the ground (n = 1) state in about 10-10 s. and emits a photon

• Can measure this in various ways: e.g
• By seeing the photon
• By measuring the radius of the atom (n = 1 is half the size).

• if we create it in the excited (n = 2) state, and wait, it will start being a mixture of the 2 states.

  Initially it is almost all the upper state

• After some time it's a 50-50 mixture

• Finally it is almost all the lower state

By measuring the atom, we can decide which of the two states it is in

e.g. consider light going through 2 sheets of polaroid at 90°.

Classical Mechanic: First sheet eliminates all vertically polarized light Second sheet eliminates all horizontally polarized light Result: darkness Quantum Mechanic: First sheet measures how much of light is polarized in horizontal direction, and produces a new wave polarized horizontally Second sheet measures how much of light is polarized in vertical direction, but there isn't any.. Result: darkness .

Now insert a third sheet at 45° between the two

• Classical Mechanic:
  1. First sheet eliminates all vertically polarized light
  2. Third sheet eliminates all light polarized at 45⁰
  3. Second sheet eliminates all horizontally polarized light
  4. Result: darkness
• Quantum Mechanic:
  1. First sheet measures how much of light is polarized in horizontal direction, and produces a new wave polarized horizontally
  2. Third sheet measures how much of light is polarized at 45⁰ direction, and produces a new wave polarized at 45⁰
  3. Second sheet measures how much of light is polarized in vertical direction, produces a new wave polarized vertical
  4. Result: light
• 
• D uuuuuuh!!!!

• They said, "You have a blue guitar,
  You do not play things as they are."
  The man replied "Things as they are
  Are changed upon the blue guitar."
  Wallace Stevens

EPR

• E.g. go back to our wave function example:

• This seemed to say that the electron gets split in half, but we interpreted it as a probability.

• But when did the electron decide which way it was going?

• The Einstein-Podolsky-Rosen paradox (EPR) is a more sophisticated version of this.

• We can have a particle with no spin which decays into 2 particles with spin

• There are two possibilities for the way the spins can arrange themselves: up-down

• There are two possibilities for the way the spins can arrange themselves: down-up

  .
• When does the electron and positron make up their minds as to which way they are going to spin?

• Classical Mechanic Obviously when they are created, their spins will be fixed.

• Quantum Mechanic It is indeterminate until you measure them

• EPR Don't be stupid. Measuring the spin of one would automatically defines the spin of the other. They could be separated by any distance, how does the second one know how to fix its spin?

• God does not play dice. Einstein

• Hence the only logical way out is "hidden variables": underneath quantum mechanics, there is a much more complicated theory where everything is deterministic.
• it only looks random on the surface.

• Stern-Gerlach experiment is the classic example: measures spins with magnetic fields: note Start with \color{red}{ \uparrow } in z-direction
• Measure in z-direction: gives 100% $ \uparrow $
• Measure in x-direction: gives 50% \color{red}{ \to } -50% \color{red}{ \leftarrow } mix.
• Subsequent measure in z-direction gives 50% \color{red}{ \uparrow } + 50% \color{red}{ \downarrow }

• suppose there is a particle which "is" a cylinder,

• but we can only see it as square

• or round

• depending on the observing angle from a detector

Create two of these, send them in opposite directions.

• Look at them with two detectors: If the detectors are set at the same angle, must always measure same shape

• Look at them with two detectors: If the detectors are set at random angle, they will measure same shape 1/2 of time

• Now suppose the information is encoded on the particles, so they "know" which way the other is pointing.
• Must always look the same if the dectors are identical: e.g. for the above example, we could write it as (ssr):
  • square as seen from angle 1 and 2
  • round from angle 3
• This gives
  • 5 settings where they measure the same shape (11,22,33,12,21)
  • 4 where they measure a different shape (13,31,23,32)
  so same shape 5/9 (> 1/2 time).
• Some other codings (rss,srs,rrs,rsr,srr) give same result, but two (rrr) and (sss) always give same shape.
• i.e. if we set up our local system to agree when the detectors are set in a similar way, the result must be differ when they are set at random!
• Experiment agrees with QM: Detectors always agree if they are oriented similarly, but agree only 50% of the time if they are differently oriented. This is not possible with any local hidden variable theory: i.e. no clockwork can reproduce the effect!

• EPR thought the states must be separate

• They are actually one "entangled" state

• and a measurement destroys it: e.g.

Schrödinger's Cat

The trivial version: you have a box, with a lid: when it is opened, cyanide gas is released.

• Take a cat.
• Put it in the box and close the lid.
• Is the cat dead?
• Why don't you look?

The sophisticated version: you have a box, with a lid and a single radioactive atom: when the atom decays, cyanide gas is released.

• Take a cat
• Put it in the box and close the lid.
• Is the cat dead or alive?
• Classical Mechanic Obviously its either dead or alive
• Quantum Mechanic It is indeterminate until you measure it . More exactly, the cat is a mixture of alive and dead cats: the measurement fixes it.
• Schrödinger Don't be stupid.

• Bell's theorem shows that there is a measurement that you can do on the polarizations of the particles which is incompatible with any conceivable hidden variable theory.
• Aspect did the experiment.
• The Schrödinger's Cat experiment has been done:
• No animals were injured in the making of this movie.
• One atom: process is totally random, so you can't decide if a one-atom cat is alive or dead without measuring it(!)
• Many atoms (10²⁹): constitutes an independent measuring system, so the cat measures it's own deadness
• Few atoms (2-20): process becomes steadily more predictable

• God not only plays dice, but throws them where they cannot be seen. Hawking

So what happened to Schrödinger?

• A Long Beach, California, man was arrested last night in a spectacular commando raid. A police SWAT team stormed the home of Erwin Schrodinger at about 9:30 PM after receiving tips from his neighbors about sinister activities on the property. A local judge issued a warrant when police presented geiger counter measurements taken from the sidewalk showing the presence of radioactive materials somewhere on the premises.
• Police have released very little information, but so far it appears the elderly Mr. Schrodinger faces felony charges of cruelty to animals, possession of fissionable materials, and possession of lethal toxins.
• Investigators were seen removing evidence from his home following the arrest. A neighbor described one of the items as a "black box with all kinda gadgetry sticking out of it". Paul Fooster, who lives next door, described Mr. Schrodinger as "eccentric" and "possibly dangerous". He said the suspect tended to ramble to himself about things collapsing and "super positions" and "eigen" states, which he thought might be a code word for terrorist-friendly countries.

• Said Mr. Fooster, "I felt chilled to the bone one time when I overheard him babbling on about sticking cats in a box with cyanide. Isn't that what them Al Qaida folks were doing with dogs? I mean, I've got children, you know?" Another neighbor, Arlene Monigan, reported similar behavior and worried that Mr. Schrodinger might have ties to terrorists. When she saw an eerie blue glow emanating from within the house, she finally notified authorities.
• Mr. Schrodinger is being held without bail pending charges. A distraught Mrs. Schrodinger, who says Mr. Schrodinger is harmless, states she has petitioned several times to see or at least talk with her husband, to no avail. "As it stands," she said, "I don't know whether he's dead or alive!"

The quantum Zeno effect

or the "watched pot effect":

• If measuring something really matters, then we can "reset the clock" by making sure that nothing has happened.
• Hence watching a cold pot on a stove makes sure it never boils:
• in more sophisticated language, if you continuously measure the state of an atom to make see if it hasn't decayed, then it can't!

Quantum Teleportation/ Computing/ Encryption

• Bit (Classically) is Binary digit = 0 or 1.
• Can be represented by (e.g)
  • on or off switch,
  • up or down mag. field
  • hole or no hole in punched card (if you were around with the dinosaurs!)

• Qubit is Quantum bit: can be encoded in (e.g.) spins \color{red}{ \psi = \alpha \left| \uparrow \right\rangle + \beta \left| \downarrow \right\rangle } Classical bit corresponds to poles, qubit to any point on sphere: hence contains much more info

• Then we can make entangled qubits A (Alice's) and B (Bob's)
  \color{red}{ \psi ^ + = \frac{1}{{\sqrt 2 }}\left( {\left| { \uparrow _A \uparrow _B } \right\rangle + \left| { \downarrow _A \downarrow _B } \right\rangle } \right)}
• Then if Alice measures the state of her qubit (say \color{red}{{ \uparrow _A }} then entanglement says Bob must get the same \color{red}{{ \uparrow _B }}
• Teleportation (like most psi) violates conservation of energy, causality and common sense. Can use entangled state to transmit qubit exactly (but not clone it!)
• Q Encryption: by sending over one part of an entangled pairs, can detect eavesdropping. Can transmit secret key by comparing successful measurements of Bell states

• Q Computing: Since data is stored as comlex numbers can in principle store huge amounts: algorithms are not similar to digital ones but can (e.g) factor 600 digit numbers.
• Can't be built yet!
  Quantum Computation - Or How to Take Advantage of Quantum Strangeness
  • Alexandre Blais
  • Departement de Physique
  • Universite de Sherbrooke
  • Friday 2007 March 23 11:30
  • room Azrieli Theatre 101.
  • Quantum computers would take advantage of the "most bizarre aspects of quantum mechanics" (superposition of states and entanglement) in order to, in principle, perform calculations exponentially faster than any classical computer could. The experimental realization of such a quantum computer is however an enormous challenge. Assuming no knowledge of quantum mechanics, I will present some of the fundamental ideas of quantum information. I will also discuss how these ideas could be realized experimentally.

Many worlds theory

• How does it know it is being measured?.
• Do we need an actual conscious observer? Is there a link between consciousness and QM? A way out is

• Many-worlds theory: Everett (1957) . Every time a measurement is made, the universe subdivides into separate universes that correspond to every possible outcome

• Avoids observation problems, but not testable (?) and not very economical!

• In all fictional works, each time a man is confronted with several alternatives, he chooses one and eliminates the others; in the fiction of Ts'ui Pên, he chooses-simultaneously-- all of them. He creates in the diverse way, diverse futures..which themselves also proliferate and fork. The Garden of Forking Paths, Borges.

• What might have been is an abstraction
  Remaining a perpetual possibility
  Only in a world of speculation.
  What might have been and what has been
  Point to one end, which is always present.
  Footfalls echo in the memory
  Down the passage which we did not take
  Towards the door we never opened
  Into the rose-garden.
  T. S. Eliot (Burnt Norton)

Conclusions:

• Either

  Quantum mechanics is correct, and there is no "simpler" system

• Or

  Reality is even uglier than we thought: e.g.

• non-local hidden variables: every bit of the universe is involved with every other bit: very Zen, but totally wipes out free will!
• ????????????
• (Ugh!) Does it bother you that 20th century technology depends fundamentally on something no-one understands?

• I can only say, there we have been: but I cannot say where.
  And I cannot say, how long, for that is to place it in time.
  T. S. Eliot (Burnt Norton)

• Obviously quantum mechanics should affect the way in which we understand the origin of the universe