Blaylock - UMass HEP Seminar 2/12/10 Teaching Quantum Tunneling* textbook tunneling the uncertainty principle wave packet tunneling tunneling time & velocity

Blaylock - UMass HEP Seminar 2/12/10

Teaching Quantum Tunneling*

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

*Special thanks to Neal Anderson (ECE) for stimulating conversation on this topic.

Below: The Boston Central Artery Tunnel, which had problems unrelated to quantum mechanics.


Quantum Tunneling

?!

There is a non-zero probability of finding oneself located on the other side of the wall.

“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


V

reflected wave not shown

textbook tunnelingsquare well potential barrier

No e-ikx term

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(x < 0) ψ = Ae ikx + Be−ikx

(0 < x < L) ψ = Ceκx + De−κx

(x > L) ψ = Fe ikx

time independent solution:

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!

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T = 1+V 2 sinh2(κL)

4 E(V − E)

⎛

⎝ ⎜

⎞

⎠ ⎟

−1

transmission probability:

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κ =1

h2m V − E( )wit

h


The Heisenberg Uncertainty Principle

Werner Heisenberg as a young man, chuckling at the mischief he is causing with his uncertainty principle.

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Δx ⋅Δpx ≥h

2

•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)

€

Δt ⋅ΔE ≥h

2

Several ways to use HUP to ‘explain’ tunneling:


Position leakageIf 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?

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.


Energy borrowingHUP 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!


Not even wrong

“Not only is it not right, it’s not even wrong!” - Wolfgang Pauli referring to a colleague’s paper.


Wave packet tunneling

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.

Wave packet tunneling is more correct, but also more subtle.

Wave packet tunneling is more correct, but also more subtle.

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ψ(x, t) = a(k)e i(kx−ωt )

−∞

∞

∫ dk

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a(k) =1

2πσ k πexp −

k − k0( )2

2σ k2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

position

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v =hk0

m

Demo at http://phet.colorado.edu/simulations/sims.php?sim=Quantum_Tunneling_and_Wave_Packets


Wave packet featuresWave packet tunneling reveals some very interesting features.

•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.


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.

multilayer dielectric mirror

downconverter

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.


Chiao results

relative delay (avg over 13 runs) Δt=–1.47±0.21 fs

apparent tunneling velocity = 1.7c

expected delay dopt/c

Photons that tunnel arrive earlier, not later, than photons in air.

Photons that tunnel arrive earlier, not later, than photons in air.


Faster than Light!

Krenzlin, Budczies, Kehr, Ann. Physik 7 (1999) 732-736.

variable barrier widthsvariable barrier widths

d

variable barrier heightvariable barrier heightSimulation of

wave packet tunneling, base on Schroedinger equation.


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!


Don’t Phone Home

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!


Examples

Documents

Blaylock - UMass HEP Seminar 2/12/10 Teaching Quantum Tunneling* textbook tunneling the uncertainty principle wave packet tunneling tunneling time & velocity