Transcript Copenhagen Interpretation

Quantum Weirdness
The Copenhagen Interpretation
Entangled States
The Einstein-Podolsky-Rosen
experiment
EPR Results and Conclusions
Local Realism
Bell’s Theorem
Applications of Quantum
Weirdness
Albert Einstein (1879-1955)
The more success the quantum theory has, the sillier it looks.
-Albert Einstein
Prof. Rick Trebino, Georgia Tech, www.frog.gatech.edu
The Copenhagen Interpretation
Quantum mechanics is one of
the most successful theories
in history. But while its
predictions are clear, its
interpretation is not.
The Copenhagen
Interpretation is an
interpretation of quantum
mechanics. It arose out of
discussions between Bohr
and Heisenberg in 1927 and
was strongly supported by
Max Born and Wolfgang
Pauli.
Max Born (1882-1970)
The Copenhagen Interpretation
1. A system is completely described by a wave function Y, which
represents an observer's knowledge of the system. (Heisenberg)
2. The description of nature is probabilistic. The probability of an event
is the mag squared of the wave function related to it. (Max Born)
3. Heisenberg's Uncertainty Principle says it’s impossible to know the
values of all of the properties of the system at the same time;
properties not known with precision are described by probabilities.
4. Complementarity Principle: matter exhibits a wave-particle
duality. An experiment can show the particle-like properties of
matter, or wave-like properties, but not both at the same time. (Bohr)
5. Measuring devices are essentially classical devices, and they
measure classical properties such as position and momentum.
6. The correspondence principle of Bohr and Heisenberg: the quantum
mechanical description of large systems should closely approximate
the classical description.
Uncertainty in spin components
Recall that the z-component and total angular momentum of the
spin are precisely knowable. But the x- and y-components are not.
It can be shown that:
 S x , S y   S x S y  S y S x  i S z
A B 
Using:
1
2

Y*  A, B  Y
For electrons,
positrons,
protons, and
neutrons ms =
±1/2
We find:
S x S y 
1
2

Y *i S z Y 
1
2

Y *i (mS )Y 
mS
2
2

1
4
So, as long as mS ≠ 0 for a given particle, there’s an Uncertainty
relation between the x and y components of its spin.
This means that we can measure one component, calling it Sz, (and
obtaining ±ħ/2), but doing so randomizes the other two components.
2
Objections to the Copenhagen Interpretation
Many physicists objected to the
Copenhagen interpretation’s
nondeterministic nature.
There were also objections to the vague
measurement process that converts
probability functions into
nonprobabilistic measurements.
Some who rejected this interpretation
were Albert Einstein, Max Planck, Louis
de Broglie, and Erwin Schrödinger.
Einstein said to Born:
“I, at any rate, am convinced that God does not play dice (with the
universe).”
Superpositions of states
Energy
Excited
level, E2
Stationary states are stationary. But an
atom can be in a superposition of two
stationary states, and this state moves.
E = hn
Ground
level, E1
Y(r , t )  a1 1 (r ) exp(iE1t / )  a2 2 (r ) exp(iE2t / )
where |ai|2 is the probability that the atom is in state i.
A superposition means that the atom is vibrating:
Y (r , t )  a1 1 (r )  a2 2 (r ) 
2
2
2
2 Re a1 1 (r )a2* 2* (r ) exp[i( E2  E1 )t / ]
Vibrations occur at the frequency difference between the two levels.
Wave-Function Collapse
It’s our lack of knowledge of which state a system is in that puts
it into a superposition state:
Y(r , t )  a1 1 (r ) exp(iE1t / )  a2 2 (r ) exp(iE2t / )
Making a measurement of the energy of the above state will
produce either E1 or E2, with probabilities |a1|2 and |a2|2,
respectively. In the Copenhagen interpretation, the state
collapses to the measured one, and the unobserved state is
removed from further consideration.
This is called wave-function collapse.
Unmeasured possible states simply disappear from sight like
losing lottery tickets.
Schrödinger’s Cat
To reveal what he considered
its absurdity, Schrodinger
proposed (but fortunately
never implemented!) putting a
cat in a sound-proof box and
killing it with a ½ probability.
Before we open the box, is
the cat alive or dead?
Even though the cat may feel otherwise, quantum mechanics
says the cat is both! It’s in a superposition of “alive” and “dead.”
1
1
Y 
alive 
dead
2
2
Making a measurement on the system (peaking into the box)
collapses the cat’s state to either “alive” or “dead.”
The EPR
Paradox
It seems that our
consciousness
plays a role in
quantum
mechanics.
Einstein became uneasy about such implications and, in later years,
organized a rearguard action against quantum mechanics. His
question, “Do you really think the moon isn't there if you aren't
looking at it?” highlights the depths of his distaste for the role of
the consciousness.
His strongest counter-argument was a paradoxical implication of
quantum mechanics now known as the Einstein-Podolsky-Rosen
(EPR) Paradox.
The Einstein-Podolsky-Rosen Paper
Einstein believed that, while quantum mechanics could be used to
make highly accurate statistical predictions about experiments,
it’s an incomplete theory of physical reality.
In 1935, Einstein, working with physicists Boris Podolsky and
Nathan Rosen, published the paper, “Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?”
In this paper, they devised a clever thought experiment that “beat”
the Uncertainty Principle. So they concluded that there must be
more going on than quantum mechanics knew about, concluding:
The quantum-mechanical description of reality given by
the wave function is not complete, that is, there must be
Hidden Variables that we don’t know about and hence
don’t measure that cause the uncertainty.
Hidden Variables
Suppose that you’re modeling
a baseball pitch, but you don’t
know about air.
Air, combined with the ball’s
spin, causes it to curve, and
variations in air pressure
cause it to wobble away
from your theoretically
perfect parabolic path.
You’d find that the pitch arrives
in a somewhat random
position.
Your theory is incomplete,
and the air pressure vs.
position is a hidden variable.
Locations of actual
pitches with the
same initial position
and velocity
Theoretically
calculated
location for
the initial
position and
velocity
Strike zone
EPR: Entangled States
Imagine a pair of
particles whose quantum
spins are known to be
opposite. We can
actually know that the
total spin S of the twoparticle system is zero if
it’s in an S = 0 or “singlet”
state. So one is spin-up,
and the other is spindown, but we don’t know
which is which.
Initial twoparticle system
with zero spin
Two particles
emerging from
initial system with
opposite spins
Now separate them and measure the spin of one particle. Because
they were paired, they have a combined entangled wave function:
Y
1

2
A

B

1

2
A

B
Initial twoparticle
system
Entangled States
But we’re free to choose which
component of the spin we’d
like to measure. Let’s now pick
a perpendicular direction. We
can write the same statement
about that direction also:
Two particles
emerging from
initial system
Y
1

2
A
 B
1

2
A

B
Of course, Quantum Mechanics says we cannot make precise
measurements of both components; making one measurement
perturbs the other.
In any case, making a measurement of either component of one
particle’s spin determines the other. When the measurement is
made, the wave function collapses:
Y 
1

2
A

B

1

2
A

B
or Y 
1

2
A
 B
1

2
A

B
The EPR Paradox
Now do something
really interesting:
Measure the vertical spin component of particle A and the
horizontal spin component of particle B.
Because the particle A measurement determines both particles’
vertical spin components, and the particle B measurement
determines both particles’ horizontal spin components, haven’t we
determined two components of each particle’s spin? And beaten the
Quantum Mechanics?
Initial two-particle
system
Two particles emerging
from initial system
EPR Paradox
Initial twoparticle
system
If this works, then Quantum Mechanics
is incomplete, that is, it’s actually
possible to make precise measurements
if we’re clever, and there’s more going on
than is in Quantum Mechanics.
Two particles
emerging from
initial system
This would be an argument for the
existence of Hidden Variables—
additional quantities that exist and
affect systems, but we just don’t
know about yet and so can’t
control them.
Einstein, perhaps thinking
that he’d nailed Quantum
Mechanics
Alas, Einstein’s trick doesn’t work!
Measuring the vertical-spin component of particle A collapses both
particles’ vertical-spin-component states, as predicted. But, in the
process, it randomizes both particles’ horizontal-spin components!
Measuring A’s vertical spin is just like measuring B’s also!
Even though we never touched particle B!
Quantum Mechanics wins! Quantum Mechanics 1. Einstein 0.
But now you might wonder: Information can’t travel faster than the
speed of light. Suppose we let the particles travel many meters (i.e.,
many nanoseconds for light) apart, and we make the measurements
only picoseconds apart in time, so there isn’t time for the information
from the measurement on particle A to reach particle B in time to
mess up its measurement. That should save Einstein’s idea.
But it doesn’t! This information appears to travel infinitely fast.
So this appears to invalidate Einstein’s beloved Special Relativity!
Quantum Mechanics wins again! Quantum Mechanics 2. Einstein 0.
Implicit assumptions of EPR
The principle of reality: individual
particles possess definite properties
even when they’re not being
observed.
The locality principle: information
from a measurement in one of two
isolated systems cannot produce real
change in the other, especially
superluminally (faster than c).
Taken together, these two seemingly obvious principles imply an
upper limit to the degree of co-ordination possible between
isolated systems or particles.
Interestingly, they both turn out to be wrong.
Local realism is out.
John Bell showed in a 1964 paper
entitled "On the Einstein Podolsky
Rosen paradox,” that local realism
leads to a series of requirements—
known as Bell’s inequalities.
John Bell (1928-1990)
Alain Aspect has performed
numerous beautiful experiments,
proving conclusively that our
universe violates Bell’s Inequalities
big time. And quantum mechanics
explains the effects quite nicely.
Alain Aspect (1947-)
Serious Quantum Weirdness
EPR assumed that the particles had spin in the first place (Reality).
And that such information couldn’t travel infinitely fast (Locality).
It appears that particles simply do not
have properties until we measure
them. It isn’t merely a matter of
ignorance.
And such information can travel
superluminally.
These effects are now known as
nonlocal behavior, quantum
weirdness (and colloquially as spooky
action at a distance).
The EPR paradox
(which isn’t in the
end a paradox) has
deepened our
understanding of
quantum
mechanics by
exposing the
fundamentally nonclassical and
unintuitive
characteristics of
the measurement
process.
Post-EPR Analysis
EPR experiments show that a "measurement" can be performed
on a particle without disturbing it directly by performing a
measurement on a distant entangled particle.
The Copenhagen Interpretation lives!
According to the Copenhagen interpretation, physics depends only
on the outcomes of measurements.
One-slit
pattern
Two-slit
pattern
We can determine where the photon hits the screen by noting a
flash. The Copenhagen interpretation rejects arguments about
where the photon was between the times it was emitted in the
apparatus and when it flashed on the screen.
Alternatives to the
Copenhagen
Interpretation and
its weirdness
Many Worlds Interpretation
The Many Worlds Interpretation says the wavefunction is real, but it denies
the reality of wave-function
collapse. This implies that
all possible alternative
histories and futures are
real—each representing an
actual "world" (or
"universe").
Every possible outcome of every
measurement exists in its own "world“.
So there’s a very large—perhaps
infinite—number of universes, and
everything that could possibly have
happened in our past, but didn't, has
in fact occurred in the past of some
other universe or universes.
Alternatives to the Copenhagen
Interpretation and its weirdness
The Guide Wave Interpretation
In 1927, Louis de Broglie suggested that the Schrodinger wavefunction was a real function that guided real particles along their
paths. In 1952, David Bohm envisioned that the wave-function
included a form of energy not known to classical physics, what he
called the "quantum potential" or "pilot wave." In the two-slit
experiment, the pilot wave would exist through both slits and guide
real particles through the slits to obtain an interference pattern.
Although the de Broglie-Bohm interpretation does state that there is
a real particle following a real path, the statistical nature of the wavefunction and the Heisenberg uncertainty principle remain in effect,
and only probabilities for the location of particles can be determined.
But to account for quantum weirdness, disturbance of the pilot wave
had to propagate instantaneously.
Applications of Quantum Weirdness
Technologies relying on quantum entanglement are now being
developed.
In quantum cryptography, entangled particles are used to
transmit signals that cannot be eavesdropped upon without leaving
a trace.
In quantum computation,
entangled quantum states
are used to perform
computations in parallel,
which may allow certain
calculations to be
performed much more
quickly than they ever
could be with classical
computers.
This presentation was a
project by former GT
Modern Physics student,
Weston Aenchbacher.
I’ve modified it
significantly for
our class.
Copenhagen, Denmark