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Tunneling through barriers with $T \approx e^{-2\kappa L}$, alpha decay, the momentum kick of an observation, and the quantum limits on precision.
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By the end of this lesson you will be able to estimate tunneling probabilities and the disturbance caused by measuring a particle's position.
From lesson 17 you know that a wave function decays exponentially inside a region where the particle's energy is below the potential. From lesson 11 you know the uncertainty relation. This lesson follows both to their consequences: particles pass through barriers they cannot climb, and every measurement of a particle disturbs it by an amount set by Planck's constant.
| Term | What it means |
|---|---|
| Potential barrier | A region where the potential energy exceeds the particle's energy. |
| Tunneling | Passage through a barrier the particle could not cross classically. |
| Decay constant | $\kappa = \sqrt{2m(U - E)}/\hbar$, the rate at which $\psi$ falls inside a barrier. |
| Transmission probability | $T \approx e^{-2\kappa L}$, the chance of tunneling through thickness $L$. |
| Alpha decay | The emission of a helium nucleus by tunneling out of a heavy nucleus. |
| Measurement back-action | The disturbance a measurement causes to the system measured. |
| Standard quantum limit | The best precision possible when measurement back-action is balanced against imprecision. |
Inside a barrier of height $U$, a particle of energy $E < U$ has a wave function that decays as $e^{-\kappa x}$, with
$$\kappa = \frac{\sqrt{2m(U - E)}}{\hbar}.$$
If the barrier is thin, some of the wave survives to the far side, and the particle can be found there. The tunneling probability is approximately
$$T \approx e^{-2\kappa L}.$$
For an electron, $\kappa = 5.12\sqrt{U - E}$ nm⁻¹ with energies in eV. The exponential makes tunneling extremely sensitive to thickness, height and mass. The flip side of the wave nature is that no measurement can find a particle without disturbing it: locating it with light of wavelength $\lambda$ transfers momentum of order $h/\lambda$, so $\Delta x\,\Delta p \gtrsim h$, as the uncertainty relation requires.
Another way: picture
Picture a ball rolling toward a hill it lacks the energy to climb. Classically it always rolls back. For a quantum particle, the hill is a region where its wave fades rather than stops. If the hill is thin enough, a little wave emerges on the other side, and now and then the particle is found there, having crossed a hill it never climbed.
Another way: steps
In a classically forbidden region, the kinetic energy $E - U$ would be negative, and Schrödinger's equation replaces oscillation with exponential decay. The wave function cannot stop abruptly at the barrier's edge, because it and its slope must be continuous. So it decays through the barrier, and at the far side it joins onto an outgoing wave whose size is set by what remains.
The probability of emerging is roughly the square of the ratio of the wave's size at the far edge to its size at the near edge, $e^{-2\kappa L}$. A more careful calculation multiplies this by a factor of order one that depends on the energies, but the exponential dominates whenever the barrier is thick enough that $\kappa L$ is well above one.
For an electron and a barrier $1$ eV above its energy, $\kappa = 5.1$ nm⁻¹. Through $0.5$ nm, $T = e^{-5.1} = 0.006$; through $1.0$ nm, $3.5 \times 10^{-5}$; through $2.0$ nm, $1.3 \times 10^{-9}$. Each extra tenth of a nanometer cuts the probability by a factor of about $2.8$.
Mass matters just as much. A proton, $1836$ times heavier, has $\kappa$ about $43$ times larger for the same barrier, and through $0.5$ nm its tunneling probability is around $e^{-220}$, effectively zero. That is why electrons tunnel readily through the insulating layers of chips while atoms stay put, and why tunneling is mostly a phenomenon of light particles over atomic distances.
A uranium-238 nucleus holds alpha particles, helium nuclei, behind a barrier formed by the strong nuclear force inside and the electrical repulsion outside. An alpha particle rattling inside with $4$ MeV faces a barrier about $30$ MeV high. Classically it would stay forever.
George Gamow, and independently Ronald Gurney and Edward Condon at Princeton, explained in 1928 that the alpha tunnels out. The alpha strikes the barrier about $10^{21}$ times a second, and the tunneling probability, around $10^{-38}$ for uranium-238, gives a half-life of billions of years. Small differences in alpha energy change the exponent enormously, explaining why half-lives range from microseconds to far longer than the age of the universe.
To see where an electron is, you must bounce something off it, usually a photon. A photon of wavelength $\lambda$ can locate the electron to about $\lambda$, but carries momentum $h/\lambda$, some of which it gives the electron in the collision. Shorter wavelengths give sharper positions and bigger kicks, and the product stays around $h$.
This back-action is not a defect of photons. Every possible measurement of position disturbs momentum by at least as much as the uncertainty relation requires, and the reverse. Precision measurements, like the gravitational-wave detectors LIGO operates in Washington and Louisiana, must balance the imprecision of a gentle measurement against the back-action of a forceful one, reaching what is called the standard quantum limit.
Checking an answer. Tunneling probabilities must fall between 0 and 1, and fall rapidly with thickness and height. Heavier particles must tunnel far less. Measurement kicks must grow as the probe's wavelength shrinks, and $\Delta x\Delta p$ must come out near $h$, never much smaller.
The approximation $T \approx e^{-2\kappa L}$ holds when $\kappa L \gg 1$ and the barrier is roughly flat. For barriers of varying height, the exponent becomes an integral, $2\int\kappa(x)\,dx$, which is how Gamow treated the alpha particle's tapering barrier.
The photon-kick estimate assumes a single photon is enough to locate the electron and that it can transfer up to its full momentum. Using fewer, gentler photons gives worse position information; there is no way around the trade-off, which is the uncertainty relation expressed as an experiment.
Tunneling underlies many devices. Scanning tunneling microscopes image atoms by the tunneling current across a vacuum gap. Flash memory stores bits by pushing electrons through thin oxides. Tunnel diodes, invented by Leo Esaki in 1957, switch in picoseconds. Josephson junctions, where pairs of electrons tunnel between superconductors, are the heart of superconducting qubits and of the SQUID magnetometers that measure the brain's magnetic fields.
Tunneling is also a problem. As transistors shrank, their insulating gate layers became so thin that electrons tunneled through, wasting power. The industry's answer, introduced by Intel in 2007, was to switch from silicon dioxide to hafnium oxide, which can be made thicker for the same electrical effect.
Two protons in the Sun's core, at about $15$ million kelvin, have typical energies of about $1$ keV, while the electrical barrier between them tops out near $500$ keV. Classically they would never get close enough for the strong force to fuse them. They tunnel, with a small probability that makes the Sun burn slowly enough to last ten billion years.
Without tunneling there would be no sunlight, and so no life. The fusion reactors being developed at the National Ignition Facility in California and in tokamaks around the world depend on the same tunneling, with deuterium and tritium, whose lighter barrier makes fusion easier than proton–proton fusion.
Physicists have learned to design measurements around quantum limits. Squeezed light, used at LIGO since 2019, reduces uncertainty in one property of the light at the cost of another, beating the ordinary shot-noise limit in the quantity that matters. Quantum nondemolition measurements probe a quantity without disturbing it, by arranging for all the back-action to land on a quantity no one cares about.
These techniques now improve atomic clocks, magnetometers and gravitational-wave detectors. They do not evade the uncertainty relation; they move its unavoidable disturbance to where it does no harm.
Quantum mechanics also forbids what classical physics allows. A particle with more energy than a barrier's height should, classically, sail over it every time. A quantum particle is sometimes reflected, because its wave changes wavelength abruptly at the barrier's edge, and any abrupt change in a wave's medium reflects part of it, as light reflects partly from a window.
For an electron passing a sudden step down in potential, reflection can be substantial when its energy is only a little above the step. Engineers designing electron devices, from photomultiplier tubes to semiconductor junctions, must account for this reflection as well as for tunneling. Both effects come from the same source: the particle is a wave, and waves partly pass and partly reflect wherever their surroundings change over distances comparable to a wavelength. Smooth, gradual changes reflect little; sharp ones reflect more.
In the transistors of a computer chip, a thin insulating layer separates the gate from the channel. By the early 2000s the silicon dioxide layers in Intel's chips, made in fabs in Hillsboro, Oregon, and Chandler, Arizona, had thinned to about $1.2$ nm, a few atoms thick. Electrons tunneled through, and leakage grew about sixfold for each tenth of a nanometer removed.
Leakage wasted so much power that shrinking further was impossible. In 2007 Intel introduced gates of hafnium oxide, whose higher dielectric constant gives the same control of the channel with a layer several times thicker. The exponential dependence of tunneling on thickness made that small change in material one of the biggest in the history of the chip.
A SQUID is a superconducting loop interrupted by one or two Josephson junctions, thin insulating barriers across which pairs of electrons tunnel. The tunneling current depends on the magnetic flux through the loop in steps of a flux quantum, $h/2e$, making the SQUID the most sensitive magnetic-field detector known.
Hospitals and research centers, including the Massachusetts General Hospital in Boston, use arrays of SQUIDs in magnetoencephalography, recording the femtotesla magnetic fields of electrical currents in the brain to locate the source of epileptic seizures. Geologists and the U.S. military use them to detect buried objects and faint magnetic anomalies.
Classically, a particle facing a hill higher than its energy always turns back. Quantum mechanically, if the barrier is thin enough, it passes through with probability about $e^{-2\kappa L}$, arriving on the other side with its original energy. It does not borrow energy or go over the top.
A related error is to think tunneling probability falls in proportion to thickness. It falls exponentially: a barrier twice as thick gives the square of the probability, which for small probabilities is vastly smaller.
An electron meets a barrier $2.0$ eV above its energy and $0.30$ nm thick. Find $\kappa$.
$\kappa = 5.12\sqrt{2.0} = 7.24\ \text{nm}^{-1}$
Electron mass.
Find the exponent.
$2\kappa L = 2 \times 7.24 \times 0.30 = 4.35$
Dimensionless.
Find the probability.
$T = e^{-4.35} = 0.013$
About one in eighty.
Double the thickness.
$T = 0.013^2 = 1.7 \times 10^{-4}$
Squared.
Quadruple the height instead.
$\kappa \to 14.5, \ T = e^{-8.7} = 1.7 \times 10^{-4}$
The same effect as doubling the thickness.
Compare a proton's $\kappa$ with the electron's for the same barrier.
$\kappa_p = \sqrt{1836}\,\kappa_e = 42.8\kappa_e$
$\kappa \propto \sqrt{m}$.
Find the proton's $\kappa$ for $2.0$ eV.
$\kappa_p = 42.8 \times 7.24 = 310\ \text{nm}^{-1}$
Much larger.
Find its exponent for $0.30$ nm.
$2\kappa L = 186$
Enormous.
Find the probability.
$T = e^{-186} \approx 10^{-81}$
Never in the age of the universe.
Explain why atoms stay put in solids.
$\text{their mass makes tunneling negligible}$
Except for the lightest, hydrogen, in special cases.
Note where protons do tunnel.
$\text{inside stars, over femtometer distances}$
Where $L$ is tiny.
Light of $0.10$ nm locates an electron to about $0.10$ nm. Find the photon's momentum.
$p = \dfrac{6.63 \times 10^{-34}}{1.0 \times 10^{-10}} = 6.6 \times 10^{-24}\ \text{kg·m/s}$
$h/\lambda$.
Find the electron's possible velocity kick.
$\Delta v = \dfrac{6.6 \times 10^{-24}}{9.11 \times 10^{-31}} = 7.3 \times 10^6\ \text{m/s}$
$\Delta p/m$.
Compare with the speed in a hydrogen atom.
$2.2 \times 10^6\ \text{m/s}$
The kick is three times larger.
Find the photon's energy.
$E = \dfrac{1240}{0.10} = 12{,}400\ \text{eV}$
An X-ray.
Compare with hydrogen's binding energy.
$12{,}400 \gg 13.6\ \text{eV}$
Such a look would knock the electron out.
Check the uncertainty product.
$\Delta x\,\Delta p \approx 0.10 \times 10^{-9} \times 6.6 \times 10^{-24} = 6.6 \times 10^{-34}\ \text{J·s}$
About $h$.
Draw the conclusion.
$\text{no gentle way to watch an electron orbit}$
Orbits are not observable paths.
Find the decay constant.
$\kappa = 5.12\ \text{nm}^{-1}$
$5.12\sqrt{1.0}$.
Find the exponent.
$2\kappa L = 4.10$
$2 \times 5.12 \times 0.40$.
Take the exponential.
Electrons tunnel through a thin barrier with probability $0.03$. A barrier of the same height but twice as thick is used instead. What is the tunneling probability now?
Complete the worked solution: to locate an electron to within about $10$ nm, it is struck with a photon of wavelength $10$ nm. With $h = 6.626 \times 10^{-34}$ J·s, $m_e = 9.11 \times 10^{-31}$ kg and $hc = 1240$ eV·nm, find the photon's momentum in units of $10^{-27}$ kg·m/s, the velocity kick it can give the electron in m/s, and the photon's energy in eV.
Divide Planck's constant by the wavelength.
$\Delta p \approx \dfrac{h}{\lambda} =$ p
The photon's momentum.
Divide by the electron's mass.
$\Delta v = \dfrac{\Delta p}{m_e} =$ v
The disturbance to its velocity.
Find the photon energy.
$E = \dfrac{1240}{\lambda} =$ e
Shorter wavelengths hit harder.
Check the uncertainty product.
$\Delta x\,\Delta p \approx \lambda \cdot \dfrac{h}{\lambda} = h$
Sharper position, bigger kick.
Match each quantum limit or effect to its statement.
| $e^{-2\kappa L}$ | $\sqrt{2m(U - E)}/\hbar$ | tunneling out of the nucleus | a momentum kick about $h/\lambda$ | |
|---|---|---|---|---|
| barrier transmission | ||||
| decay constant | ||||
| alpha decay | ||||
| observing with light |
An electron meets a barrier $0.50$ nm thick whose height is $1$ eV above the electron's energy. Fill in the decay constant $\kappa$ in nm⁻¹, the exponent $2\kappa L$, and the tunneling probability $e^{-2\kappa L}$.
| value | |
|---|---|
| κ (nm⁻¹) | |
| 2κL | |
| tunneling probability |
Electrons meet barriers of a fixed height for which the decay constant inside is $\kappa = 7$ nm⁻¹. Write the approximate tunneling probability as a formula in the barrier thickness $L$ (nm).
Answer:
An electron meets a barrier $0.8$ nm thick whose top is $0.5$ eV above the electron's energy. Using $\kappa = 5.123\sqrt{U - E}$ nm⁻¹, what is the tunneling probability, in units of $10^{-3}$?
Answer: × 10⁻³
Chip engineers at an Intel fab in Hillsboro, Oregon, find that electrons leak through a transistor's silicon dioxide gate layer by tunneling, with $\kappa = 9.02$ nm⁻¹. If the layer is made $0.1$ nm thinner, by what factor does the leakage current rise?
Answer: times the leakage
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Electrons meet barriers of a fixed height for which the decay constant inside is $\kappa = 4$ nm⁻¹. Write the approximate tunneling probability as a formula in the barrier thickness $L$ (nm).
Answer:
You can reason about tunneling and measurement. Explain to someone why making an insulating layer slightly thinner can multiply the current leaking through it.
21. Your turn: an electron meets a barrier $1.0$ eV above its energy and $0.40$ nm thick. Find the tunneling probability., step 3
$T = e^{-4.10} = 0.017$
About one in sixty.