Teaching Quantum Tunneling*

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Teaching Quantum Tunneling

• textbook tunneling
• the uncertainty principle
• wave packet tunneling
• tunneling time velocity

Below The Boston Central Artery Tunnel, which
had problems unrelated to quantum mechanics.
Special thanks to Neal Anderson (ECE) for
stimulating conversation on this topic.
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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.
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textbook tunnelingsquare well potential barrier
No e-ikx term

• Shows E, V, L dependence
• Energy eigenstate is time independent and has
  infinite extent. It doesnt start at one side and
  move to the other!

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The Heisenberg Uncertainty Principle
Several ways to use HUP to explain tunneling
Werner Heisenberg as a young man, chuckling at
the mischief he is causing with his uncertainty
principle.

• 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)

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Position leakage
If a professors momentum is even partially
specified (say hes going towards a brick wall
rather than away from it) there is an associated
non-zero uncertainty on his position.
Professors 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?

Some of the cloud overlaps to the other side.
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Ball rolling over a hill
Although classically a particle may not have
enough momentum to make it over a barrier,
quantum mechanically its 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.

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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 cant be derived from operator
  commutation relations since time is a parameter,
  not an operator.
• Energy eigenstate tunneling previously suggested
  energy doesnt 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!

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Not even wrong
Not only is it not right, its not even wrong!
- Wolfgang Pauli referring to a colleagues paper.
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Wave packet tunneling
Wave packet tunneling is more correct, but also
more subtle.

1. Construct a wave packet out of many frequencies.
2. Solve the equation of motion (e.g. Schroedinger
   equ.) for each component.
3. Numerically integrate to see how the wave packet
   propagates.

Demo at http//phet.colorado.edu/simulations/sims
.php?simQuantum_Tunneling_and_Wave_Packets
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Wave packet features
Wave packet tunneling reveals some very
interesting features.

• Each component leaks, even components that dont
  have enough energy classically. They dont
  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.

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Single Photon Tunneling Time

• Measurement challenges
• 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.

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.

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Chiao results
Photons that tunnel arrive earlier, not later,
than photons in air.
relative delay (avg over 13 runs) ?t1.470.21
fs apparent tunneling velocity 1.7c
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Faster than Light!
variable barrier widths
d
variable barrier height
Simulation of wave packet tunneling, base on
Schroedinger equation.
Krenzlin, Budczies, Kehr, Ann. Physik 7 (1999)
732-736.
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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!
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Dont 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 cant call Alpha Centauri!
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Examples