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Interpretations of
Quantum Mechanics
Scott Johnson
Intel
Mysteries of
Quantum Mechanics
Scott Johnson
Intel
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
Dec 9, 2005
Johnson
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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
Dec 9, 2005
Johnson
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“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
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“What the Bleep” Quantum Measurement
When she does not look,
there is a wave function.
When she does look,
it collapses to a single location.
• Reasonable dramatization of Copenhagen interpretation
– Except big object like basketball would have really small spread
• How good of a description of quantum mechanics?
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“What the Bleep” Clip
• So, what are the mysteries of quantum
mechanics?
Dec 9, 2005
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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
Dec 9, 2005
Johnson
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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
Dec 9, 2005
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Classical Physics
x  cos2 k t 
• Mathematical model of the way things move
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Classical Physics
• Clear connection between the model and the
real world
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Quantum Physics
Px   cos 2 2 x 
• Also a mathematical model of the way things
move
• Very different form – probabilities!
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Two-Slit Interference
• Result is different from classical if we use
elementary particles
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Wave Mechanics
Wolfgang Christian, Dickenson College
• This is the same behavior we see from classical
waves
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Single Particle Interference
• Not a wave of particles
• Single particles interfere with themselves
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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
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Quantum Mechanics
• Wave packet travels, but still a probability
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Quantum Measurement
• How do we go from a probability to an actual
event?
• Standard answer:
– Copenhagen interpretation
– Wave function collapse
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Quantum Measurement
• Two-slit wave packet collapsing
• Eventually builds up pattern
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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?
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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?
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Which Path?
• Short answer: no, we can’t tell
• Anything that blocks one slit washes out the
interference pattern
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Which Path?
• The wave function is that of one slit
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Which Path?
Crystal with
inelastic collision
Movable wall;
measure recoil
Source
Source
No:
Change in wavelength
washes out pattern
No:
Movement of slit
washes out pattern
• 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
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Which Path?
Source
• 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
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Which Path?
Path is measured at one or both slits:
• Any which-path measurement destroys the interference
pattern
• We cannot determine which slit the particle goes through
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Particle in Stationary State Move?
• Waves and wave functions have ground states
– The wave is stationary in time
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Electron In Atom Move?
Electron
Proton
Diagram of
hydrogen atom
More accurate picture
of electron wave function
• 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?
Dec 9, 2005
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Electron In Atom Move?
Electron
Proton
Proton
Muon
Muonic hydrogen “atom”
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
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Electron In Atom Move?
Electron around
proton (hydrogen)
Muon around heavier nucleii
Muon around proton
(muonic hydrogen)
• 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
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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
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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?
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Philosophy
Positivism
Realism
Sense perceptions
are the only
admissible basis of
human knowledge
and precise thought.
Physical objects
continue to exist
when not perceived.
Niels Bohr
Dec 9, 2005
Albert Einstein
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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
Dec 9, 2005
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Photomultiplier Tube
photon
200V
400V
600V
800V
electrons
100V
300V
500V
700V
• Measurement requires interaction with other
particles
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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
• …
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Schrödinger's Cat Paradox
Paradox: A seemingly contradictory statement that may nonetheless be true
Detector 1
releases poison
Detector 2 prevents
its release
From John Gribbon
In Search of Schrödinger's Cat
• Why don’t we see superpositions of objects like
cats?
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Schrödinger's Cat Paradox
• Note that a superposition is quite different than a
pure probability, but both are still weird
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Multi-Particle Wave Function
• To investigate measurement, we need a new
tool
– Multi-particle wave function
– Single wave function that describes multiple particles
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Quantum Multi-Particle
One 1D particle
requires
One 1D wave function
One 2D particle
requires
One 2D wave function
Two 1D particles
require
Two 1D wave functions?
Dec 9, 2005
NO!
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Classical Multi-Particle
• Two 1D particles can be tracked with a single
point on a 2D plane
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Classical Multi-Particle
• Another example
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Classical Multi-Particle
• Another example
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Classical Multi-Particle
• Another example
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Classical Multi-Particle
• Another example
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Quantum Multi-Particle
P2
Particle 2
Note: this is drawn, not calculated
Particle 1
with fixed
particle 2
P1
Erwin Schrödinger
Particle 1
• 2 particles in 1D requires a 2D wave function!
• This was a disappointment to Schrödinger
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Many-Particle Wave Functions
Space
# particles
dimensions
1 particle
1D
2 particles
1D
1 particle
3D
2 particles
3D
10 particles
3D
1023 particles
3D
Wave function
dimensions
1D wave function
2D wave function
3D wave function
6D wave function
30D wave function
3x1023D wave function
• These are not spatial dimensions!
– Purely mathematical “wave function space”
dimensions
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Quantum Multi-Particle
• These 2 particles are described by one 2D wave
function
• Projecting (integrating) the 2D function onto
each axis gives 1D wave functions
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Quantum Multi-Particle
• Sometimes the 2D function separates neatly into
two 1D wave functions…
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Quantum Multi-Particle
• But not in general
• These two particles are correlated or entangled
– The 1D probability densities don’t have complete info
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Quantum Multi-Particle
• This “classical state” is very useful because it
keeps its shape as it oscillates
– Only available for a harmonic oscillator
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Quantum Multi-Particle
• Particles can stay separable
– Don’t need 2D function (two 1D functions are good
enough), but can plot one anyway
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Quantum Multi-Particle
• Particles usually don’t stay separable
– They usually become entangled with other particles
– They always become entangled when being
measured
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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
Dec 9, 2005
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Decoherence
Superposition
Mixed State
• Note the difference between these two graphs
• Can a superposition become a mixed state?
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Decoherence
• Yes! Decoherence turns a superposition into a
mixed state
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Decoherence
• We can look at the second particle, too
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Decoherence for 2-Slit
A particle in here
flips states
Source
• 2-slit is a 2D system
• Need a 3rd dimension for the “environment”
particle
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Decoherence for 2-Slit
Slices of 3D total
function shown
here in blue
P1 y
2D particle
going through
slits shown on
this face in red
P1 x
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P2
Measurement
particle shown
along this axis
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2-Slit Decoherence
Measurement
No measurement
P1 y
P1 y
P1 x
P2
P1 x
Wave function stays
in one region in that
3rd dimension
Dec 9, 2005
P2
Measurement moves wave
function in 3rd dimension –
no longer overlap
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Effect of Measurement
Measurement
No measurement
P1 y
P1 y
P1 x
P2
P1 x
P2
• Measurement shifts the wave function so it no
longer overlaps
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Effect of Measurement
Measurement
No measurement
P1 y
P1 y
P1 x
P2
P1 x
P2
• Origin not as clear away from slits
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2-Slit With Partial Measurement
P1 y
P1 x
P2
• Partial transfer of wave function
• Interference pattern is washed out but still there
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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
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Quantum Computing
Quantum computer
Classical computer
1
0
0
1 0
1
1
0
All 8-bit numbers at once!
(superposition)
• 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
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P
x
65
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
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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
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Wave Function Collapse
detector A
detector B
• If A detects particle, wave function collapses
instantaneously so B cannot detect it
• If collapse is instantaneous, this violates causality
• Explanation is from relativity
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Relativity of Simultaneity
t
A
B
x
• In one reference frame, A and B take place at
the same time
– No problem yet for A instantaneously stopping B from
detecting particle
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Relativity of Simultaneity
t
x
B
A
• In another reference frame, A happens first
– Still no problem, A can stop B
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Relativity of Simultaneity
t
A
B
x
• In this reference frame, though, B happens first!
– How can A stop B if B happens first?
– Violates causality
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Fate?
particles colliding
time
location
• Violating causality might imply fate
• Not so bad – classical physics had fate
– “Determinism”
– Can predict every particle’s location
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Bell’s Theorem
• OK, maybe wave functions don’t
collapse instantaneously
• So, is quantum mechanics local?
• John Bell devised a way to test for nonlocality (Bell’s theorem)
– Compares “local hidden variables” to QM
• Some of these experiments have been
carried out
John Bell
• The verdict …
• Quantum mechanics is non-local!
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Particle 1
Source
Particle 1
Particle 2
Small Source
Particle 2
• Back-to-back 2-slit with correlated particles
• With small source, separate interference patterns
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Particle 1
Particle 1
Particle 2
Large Source
Source
Particle 2
• Same back-to-back 2-slit w/ correlated particles
• With large source, correlated interference pattern
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Small Source
Particle 1
Particle 1
Change particle 1’s slit
Particle 2
Particle 2
• Change slit width 1, only pattern 1 changes
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Large Source
Particle 1
Particle 1
Change particle 1’s slit
Particle 2
Particle 2
• Change slit width 1, correlated pattern changes
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……………
Particle 2
Particle 1
Say that
particle 1
lands here
Particle 1
Correlated Pairs
……………
Particle 2
Possibilities
for particle 2
depend on
particle 1’s slit!
• We could (in principle) change the slit width after
the particles were launched!
• This is a non-local correlation
Dec 9, 2005
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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?
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What Does Non-Locality Mean?
particles colliding
time
location
• 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
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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?
Dec 9, 2005
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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
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Q&A
Dec 9, 2005
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Outline
• Motivation: What the Bleep
– What are the mysteries of quantum mechanics?
• 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
Dec 9, 2005
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Shor’s Algorithm
Peter Shor, 1994
• 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
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