The EPR Paradox
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Transcript The EPR Paradox
Teaching Quantum Tunneling*
textbook tunneling
the uncertainty principle
Below: The Boston Central
Artery Tunnel, which had
problems unrelated to
quantum mechanics.
wave packet tunneling
tunneling time & velocity
*Special thanks to Neal Anderson (ECE)
for stimulating conversation on this topic.
Blaylock - UMass HEP Seminar 2/12/10
Quantum Tunneling
“Anyone at present in this room has a finite chance of leaving it without
opening the door -- or of course, without being thrown out the window.”
-- R. H. Fowler, after a lecture by George Gamow at the Royal Society
There is a non-zero
probability of finding
oneself located on the
other side of the wall.
Blaylock - UMass HEP Seminar 2/12/10
textbook tunneling
square well potential barrier
time independent solution:
(x 0)
V
Ae ikx Be ikx
(0 x L) Cex De x
(x L)
Fe ikx
reflected wave not shown
No e-ikx term
transmission probability:
Shows E, V, L dependence
Energy eigenstate is time
independent and has infinite extent.
It doesn’t start at one side and move
to the other!
V 2 sinh 2 (L) 1
T 1
4 E(V E)
with
Blaylock - UMass HEP Seminar 2/12/10
1
2mV E
The Heisenberg Uncertainty Principle
x px
2
t E
2
Several ways to use HUP to ‘explain’ tunneling:
• Position is uncertain (the particle can be on other
side of the barrier)
• Momentum is uncertain (the particle may have
enough momentum to make it over the barrier)
• Energy is uncertain (during tunneling the particle
may ‘borrow’ enough energy to surmount the
barrier)
Blaylock - UMass HEP Seminar 2/12/10
Werner Heisenberg as a
young man, chuckling
at the mischief he is
causing with his
uncertainty principle.
Position
leakage
If a professor’s momentum
is even
partially specified (say
he’s going towards a brick wall rather than away from it)
there is an associated non-zero uncertainty on his position.
Professor’s ‘cloud’
representing his position.
Now bring up a
brick wall.
Shows effect of barrier width.
Does not show effect of barrier
height.
Does not explain transition from
one side to the other. Is it
instantaneous?
Blaylock - UMass HEP Seminar 2/12/10
Some of the cloud
overlaps to the
other side.
Ball rolling over a hill
Although classically a particle may not
have enough momentum to make it
over a barrier, quantum mechanically
it’s momentum is uncertain.
Shows effect of barrier height.
Does not show effect of barrier width.
Shows how momentum might be higher than expected.
Does not show why the momentum would be lower again on the other side.
Blaylock - UMass HEP Seminar 2/12/10
Energy
borrowing
HUP says
energy is uncertain over a small
enough time period. In essence, we can
“borrow” energy during the tunneling, as
long as we pay it back soon enough.
Suggests a sensible dependence on height (and width?) of barrier.
Energy-time uncertainty relation is controversial. It can’t be derived from
operator commutation relations since time is a parameter, not an operator.
Energy eigenstate tunneling previously suggested energy doesn’t need to
change in order to leak through the barrier. How do we reconcile this?
In order to tunnel through a fixed width barrier of arbitrary height, we must
pay back the energy in an arbitrarily short time. This suggests the tunneling
velocity can be as large as you like!
Blaylock - UMass HEP Seminar 2/12/10
Not even wrong
“Not only is it not right, it’s not
even wrong!” - Wolfgang Pauli
referring to a colleague’s paper.
Blaylock - UMass HEP Seminar 2/12/10
k0 packet tunneling
Wave
v
m
1.
2.
3.
Wave packet tunneling
is more correct, but also
more subtle.
position
Construct a wave packet out
of many frequencies.
Solve the equation of motion
(e.g. Schroedinger equ.) for
each component.
Numerically integrate to see
how the wave packet
propagates.
(x,t) a(k)e
a(k)
1
2 k
i(kxt )
dk
k k 2
0
exp
2
2 k
- UMass HEP Seminar 2/12/10
Blaylock
Demo at http://phet.colorado.edu/simulations/sims.php?sim=Quantum_Tunneling_and_Wave_Packets
Wave packet tunneling
somefeatures
very interesting features.
Wavereveals
packet
• Each component leaks, even components that
don’t have enough energy classically. They don’t
“borrow” energy.
• The wave packet is altered by dispersion and
interference. The shape of the wave packet (in
position and momentum space) is not the same as
the initial packet; it does not have the same energy
or momentum distribution.
• In certain cases, a significant portion of the wave function is trapped inside
barrier for a while.
• The tunneling time (defined by the peak of the wave packet) can decrease
with increasing barrier height (over some range), leading to superluminal
velocities.
Blaylock - UMass HEP Seminar 2/12/10
Single Photon Tunneling Time
Measurement challenges:
downconverter
• The time it takes for a typical
particle (photon) to traverse a
typical barrier (1 m) is a few
femtoseconds.
• Measuring time before and
after the barrier would change
the energy during the
tunneling.
multilayer
dielectric mirror
Steinberg, Kwiat, Chiao PRL 71 (1993) p. 708-711.
• Produce two photons simultaneously in a parametric downconverter.
• Race them along parallel tracks, one with a barrier, one without.
• Compare finish times via coincidence interference.
Blaylock - UMass HEP Seminar 2/12/10
Photons that tunnel arrive earlier,
Chiaonotresults
later, than photons in air.
expected delay dopt/c
relative delay (avg over 13 runs) t=–1.47±0.21 fs
apparent tunneling velocity = 1.7c
Blaylock - UMass HEP Seminar 2/12/10
Faster than Light!
variable barrier widths
d
Simulation of wave
packet tunneling, base on
Schroedinger equation.
Krenzlin, Budczies, Kehr,
Ann. Physik 7 (1999) 732-736.
Blaylock - UMass HEP Seminar 2/12/10
variable barrier height
dispersion
As it travels, the wave packet
disperses. High frequency (high E)
components move to front of the
packet.
High frequency components have the biggest
transmission coefficients, and tunnel more easily.
The front of the wave packet contributes the most to tunneling!
Blaylock - UMass HEP Seminar 2/12/10
Don’t Phone Home
Group velocities can appear to exceed the speed of light, BUT
no signal travels faster than the speed of light. Signal velocity,
defined by the front edge of the wave packet, never exceeds c.
You still can’t call
Alpha Centauri!
Blaylock - UMass HEP Seminar 2/12/10
Examples
Blaylock - UMass HEP Seminar 2/12/10