Interpretations of Quantum Mechanics Scott Johnson Intel.

Interpretations ofQuantum Mechanics

Scott Johnson

Intel

Mysteries ofQuantum Mechanics

Scott Johnson

Intel

Dec 9, 2005 Johnson 3

Outline• Motivation: What the Bleep

– What are the mysteries of quantum mechanics?

• Mystery #1: Wave or particle?• Mystery #2: What is a measurement?• Mystery #3: Non-locality


“What the Bleep” Movie• A locally produced movie • Thought-provoking and

entertaining– I liked it

• Physics conclusions are speculative– Not a science documentary– Some good quotes

• Good quantum measurement scene


“What the Bleep” Quantum Measurement

• Reasonable dramatization of Copenhagen interpretation– Except big object like basketball would have really small spread

• How good of a description of quantum mechanics?

When she does not look,there is a wave function.

When she does look,it collapses to a single location.


“What the Bleep” Clip

• So, what are the mysteries of quantum mechanics?


Physics Is…

• …using math to model the world• We don’t know why math is the best thing to

use, but it works well

Eugene Wigner


Classical Physics

• Mathematical model of the way things move

tkx 2cos


Classical Physics

• Clear connection between the model and the real world


Quantum Physics

• Also a mathematical model of the way things move

• Very different form – probabilities!

xxP 2cos2


Two-Slit Interference

• Result is different from classical if we use elementary particles


Wave Mechanics

• This is the same behavior we see from classical waves

Wolfg

an

g Ch

ristian

, Dicke

nso

n C

olleg

e


Single Particle Interference

• Not a wave of particles• Single particles interfere with themselves


Quantum Mechanics

• Quantum mechanics is the mathematics of a wave function ψ– Wave function squared |ψ|2 gives the probability of finding the

particle

• Wave function has all the information we know about a particle


Quantum Mechanics

• Wave packet travels, but still a probability


Quantum Measurement

• How do we go from a probability to an actual event?

• Standard answer:– Copenhagen interpretation– Wave function collapse


Quantum Measurement

• Two-slit wave packet collapsing• Eventually builds up pattern


Wave or Particle?

• Let’s ask a few questions that might help us to decide– Which path does particle follow through the 2 slits?– Does a particle in a ground state move?


Which Path?

• A classical particle would follow some single path• Can we say a quantum particle does, too?• Can we measure it going through one slit or another?


Which Path?

• Short answer: no, we can’t tell• Anything that blocks one slit washes out the

interference pattern


Which Path?

• The wave function is that of one slit


Which Path?

• Einstein proposed a few ways to measure which slit the particle went through without blocking it

• Each time, Bohr showed how that measurement would wash out the wave function

Movable wall;measure recoil

Source

Crystal with inelastic collision

Source

No:Movement of slit washes out pattern

No:Change in wavelength washes out pattern


Which Path?

• Now possible to measure which slit a particle went through without disturbing its momentum at all– Not quite two slits, and fairly difficult to do

• And the result … interference is still washed out!• Something more fundamental than disturbing momentum

is at work here

Source


• Any which-path measurement destroys the interference pattern

• We cannot determine which slit the particle goes through

Path is measured at one or both slits:

Which Path?


Particle in Stationary State Move?

• Waves and wave functions have ground states– The wave is stationary in time


Electron In Atom Move?

• Example ground state is electron in an atom• Does the electron in the ground state move?

– Quantum formalism says yes, but do we really know?

More accurate pictureof electron wave function

Proton

Electron

Diagram of hydrogen atom



• Great test: give the particle a clock and see if it runs slow– This is from relativity – fast clocks run slow

• This test can actually be done– Make atom with muon instead of electron– Muon like a heavy electron– Muons have short lifetimes, ~2.2μsec– If their lifetimes increase, they are moving fast

ProtonElectron

Hydrogen atom

Proton Muon

Muonic hydrogen “atom”



• Muonic atoms made with heavier nucleii should be smaller and the muons should move faster

• The result ... • Muons around heavier nucleii do live longer• The particle in a ground state is really moving!

– ...at least according to Einstein’s special relativity

Muon around proton (muonic hydrogen)

Electron around proton (hydrogen)

Muon around heavier nucleii


Wave or Particle?• So, from these last two experiments...

– A particle is indeed moving, but– We can’t tell what path it follows

• Could it follow a path but we just can’t see it?– Well, maybe. Here’s what such a path might look like:

This path gives the correct position and momentum probability distribution for the ground state of the harmonic oscillator


Wave or Particle?

• So, which is it, wave or particle?– Best answer is probably “neither”– It is something else that we don’t fully understand yet

• Another way of asking that:– Is the wave function a real thing that collapses?– Or is it a statement about our knowledge of the

particle?


Philosophy

Niels Bohr Albert Einstein

Positivism

Sense perceptions are the only admissible basis of human knowledge and precise thought.

Realism

Physical objects continue to exist when not perceived.


Photomultiplier Tube

• Measurement requires interaction with other particles

100V 300V 500V 700V

200V 400V 600V 800Vphoton

electrons


What is a Measurement?

• How do we differentiate between a measurement and a quantum interaction?

• Measurement devices, including our eyes, are all quantum mechanical

• Is a consciousness required for measurement?– Is a human required? A chimp? A cockroach?– You may be a physicist if:

• …• You’re afraid that if you look at something, you’ll collapse its

wave function• …


Schrödinger's Cat Paradox

• Why don’t we see superpositions of objects like cats?

Paradox: A seemingly contradictory statement that may nonetheless be true

From John GribbonIn Search of Schrödinger's Cat

Detector 1 releases poison

Detector 2 prevents its release


Schrödinger's Cat Paradox

• Note that a superposition is quite different than a pure probability, but both are still weird


Multi-Particle Wave Function

• To investigate measurement, we need a new tool– Multi-particle wave function– Single wave function that describes multiple particles


Quantum Multi-ParticleOne 1D particle requires One 1D wave function

One 2D particle requires One 2D wave function

Two 1D particles require Two 1D wave functions?NO!


Classical Multi-Particle

• Two 1D particles can be tracked with a single point on a 2D plane



• Another example


Quantum Multi-Particle

• 2 particles in 1D requires a 2D wave function!• This was a disappointment to Schrödinger

Par

ticle

2

Particle 1

Particle 1with fixedparticle 2

Note: this is draw

n, not calculated

Erwin Schrödinger

P2

P1


Many-Particle Wave Functions

• These are not spatial dimensions!– Purely mathematical “wave function space”

dimensions

Space Wave function# particles dimensions dimensions 1 particle 1D 1D wave function2 particles 1D 2D wave function1 particle 3D 3D wave function2 particles 3D 6D wave function10 particles 3D 30D wave function1023 particles 3D 3x1023D wave function



• These 2 particles are described by one 2D wave function

• Projecting (integrating) the 2D function onto each axis gives 1D wave functions



• Sometimes the 2D function separates neatly into two 1D wave functions…



• But not in general• These two particles are correlated or entangled

– The 1D probability densities don’t have complete info



• This “classical state” is very useful because it keeps its shape as it oscillates– Only available for a harmonic oscillator



• Particles can stay separable– Don’t need 2D function (two 1D functions are good

enough), but can plot one anyway



• Particles usually don’t stay separable– They usually become entangled with other particles– They always become entangled when being

measured


Decoherence

• Schrödinger's cat is a good problem because it is specific and physical– Why don’t we see superpositions of macroscopic

objects like cats?

• The answer has recently (last 10-15 years) been appreciated as decoherence


Decoherence

• Note the difference between these two graphs• Can a superposition become a mixed state?

Superposition Mixed State


Decoherence

• Yes! Decoherence turns a superposition into a mixed state


Decoherence

• We can look at the second particle, too


Decoherence for 2-Slit

• 2-slit is a 2D system• Need a 3rd dimension for the “environment”

particle

Source

A particle in here flips states


Decoherence for 2-Slit

P1 x

P1 y

P2

2D particle going through slits shown on this face in red

Slices of 3D total function shown here in blue

Measurement particle shown along this axis


2-Slit Decoherence

P1 x

P1 y

P2 P1 x

P1 y

P2

No measurement Measurement

Measurement moves wave function in 3rd dimension – no longer overlap

Wave function stays in one region in that 3rd dimension


Effect of Measurement

• Measurement shifts the wave function so it no longer overlaps


P1 x

P1 y

P2 P1 x

P1 y

P2


Effect of Measurement

• Origin not as clear away from slits


P1 x

P1 y

P2 P1 x

P1 y

P2


2-Slit With Partial Measurement

• Partial transfer of wave function• Interference pattern is washed out but still there

P1 x

P1 y

P2


Decoherence…• …is fast

– A molecule interacting with heat photons in a lab vacuum will decohere in ~10-17 seconds

• Faster than any possible measurement we can make• Possibly the most efficient process known

• … Solves Schrödinger's Cat– Any macroscopic object will decohere long before we

can see a macroscopic superposition• People are trying to get superpositions of fairly macroscopic

objects – work in progress

• …is holding up practical quantum computers


Quantum Computing

• Much faster than a regular computer for some problems

• Use superpositions to represent all numbers at once

• Catch is, only get one random output at a time

• Shor showed how to use this to factor big numbers very quickly

1 0 0 1 0 1 1 0

Classical computer Quantum computer

All 8-bit numbers at once!(superposition)

x

P


Measurement Still Has a Mystery

• Decoherence leaves us with two (or more) outcomes as proper probabilities– Probabilities are less mysterious than superpositions

• It does not say how nature chooses among these probabilities

• I also does not say when the choice is made


Wave Function Collapse

• If A detects particle, wave function collapses instantaneously so B cannot detect it

• If collapse is instantaneous, this violates causality• Explanation is from relativity

detector A

detector B


Relativity of Simultaneity

• In one reference frame, A and B take place at the same time– No problem yet for A instantaneously stopping B from

detecting particle

x

t

A B



• In another reference frame, A happens first– Still no problem, A can stop B

x

t

A

B



• In this reference frame, though, B happens first!– How can A stop B if B happens first?– Violates causality

x

t

AB


Fate?

• Violating causality might imply fate• Not so bad – classical physics had fate

– “Determinism”– Can predict every particle’s location

location

time

particles colliding


Bell’s Theorem• OK, maybe wave functions don’t

collapse instantaneously• So, is quantum mechanics local?

• John Bell devised a way to test for non-locality (Bell’s theorem)– Compares “local hidden variables” to QM

• Some of these experiments have been carried out

• The verdict … • Quantum mechanics is non-local!

John Bell


Small Source

• Back-to-back 2-slit with correlated particles• With small source, separate interference patterns

Source

Particle 2

Par

ticle

2

Par

ticle

1

Par

ticle

1


Large Source

• Same back-to-back 2-slit w/ correlated particles• With large source, correlated interference pattern

Source

Particle 2

Par

ticle

2

Par

ticle

1

Par

ticle

1


Small Source

• Change slit width 1, only pattern 1 changes

Particle 2

Par

ticle

1

Particle 2

Par

ticle

1

Change particle 1’s slit


Large Source

• Change slit width 1, correlated pattern changes

Particle 2

Par

ticle

1

Particle 2

Par

ticle

1

Change particle 1’s slit


Correlated Pairs

• We could (in principle) change the slit width after the particles were launched!

• This is a non-local correlation

Particle 2P

artic

le 1

Particle 2

Par

ticle

1… … … … … … … … … …Say thatparticle 1lands here

Possibilitiesfor particle 2 depend on particle 1’s slit!


Another Quantum Mystery

• Quantum mechanics non-locality cannot be used for faster-than-light communication– More subtle, but still non-local

• One group has “teleported” a single particle– Again, not faster than light

• What does this non-locality mean philosophically?


What Does Non-Locality Mean?

• Non-local “hidden variables”? – Just like classical physics, each angle is fixed– Value of fixed angle is not the same for each vertex with same

input conditions– Although appealing to me, this idea is not popular

• Transactional interpretation– The present transacts with the future much like with the past– Wave function from the future + wave function from the past

location

time

particles colliding


Status of Mysteries• Mystery #1: Wave or particle?

– Unsolved; wave function gives probabilities only

• Mystery #2: What is a measurement?– Solved; interactions with decoherence give pure

probabilities

• Mystery #3: Non-locality– Unsolved; universe is non-local; what does that

mean?


Retrospect: What the Bleep

• So, how good is What the Bleep’s picture of quantum measurement?

• The good:– Striking and easy to understand– Captures the spirit of Bohr’s Copenhagen interpretation

• The bad:– Implies that consciousness is needed to collapse wave function

• Eyes closed, or even back of head, would have the same effect on the wave function

– Vastly exaggerates size of spread for a basketball-sized object• Would be too small to see, even un-collapsed


Q&A




• Mystery #1: Wave or particle?– 1-particle wave function– Which way?– Does an electron in an atom move?– Does an atom really jump from state to state?

• Mystery #2: What is a measurement?– Multi-particle wave function, entanglement– Schrödinger's cat– Decoherence

• Mystery #3: Non-locality– Wave function collapse– Relativity of simultaneity– Bell’s theorem


Shor’s Algorithm• Picture a 250-bit number; with a quantum computer, make that a

superposition of every 250-bit number, all 2250 of them at once!– Call each of these 2250 numbers by variable name a

• Now say we have some function f(a)=ka mod N with a really long repeat period, like 1040 – This long repeat period can be used to find the prime factorization of N

• Act with this function on the superposition once and you have effectively done the calculation 2250 times, a phenomenal speed-up– The catch is you can only read out one randomly chosen answer at any

one time

• Do an FFT on the number (which is also the function)– Even multiples of the period will be large; other values small

• Read out the value of all bits– This will be one possible answer

• Repeat several times to get a approximation of the function

Peter Shor, 1994

Interpretations of Quantum Mechanics Scott Johnson Intel.

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