By Dr. John Millam1
Part 1 of this series presented an overview of quantum mechanics, and part 2 discussed wave-particle duality. (For a quick review, see Appendix: Overview of Quantum Mechanics.) Now we will go deeper and explore quantum tunneling, a mysterious quantum phenomenon that violates our everyday experience.
What is Quantum Tunneling?
Imagine being able to walk through walls. Wouldn’t that be useful? It sounds like a superpower for a comic book superhero, yet elementary particles regularly perform this handy trick. This quantum phenomenon, called quantum tunneling, enables subatomic particles to cross a barrier even when they lack sufficient energy. This behavior completely defies our traditional understanding of how things work. It is like walking up to a closed door and passing through it to get to the other side without first opening it.
While it sounds impossible, this strange behavior has been experimentally demonstrated. In fact, quantum tunneling occurs commonly in nature and has many valuable technological uses. Some common examples (summarized from “Quantum Tunneling: Applications of Quantum Tunneling”):
- Scanning Tunneling Microscope. Scanning tunneling microscopes can image and manipulate individual atoms by utilizing the connection between quantum tunneling and distance.
- Nuclear Fusion. Quantum tunneling is necessary for nuclear fusion in stars. It allows atomic nuclei to cross the Coulomb barrier (caused by electrostatic repulsion) to initiate thermonuclear fusion even when the star’s average core temperature is not sufficiently high. Without this, our sun would not shine, and life here on Earth would be impossible.
- Electronics. Tunneling is a basic technique employed to set the floating gates in flash memory. Cold emission, tunnel junction, quantum-dot cellular automata, tunnel diode, and tunnel field-effect transistors are some of the main electronic processes or devices that use quantum tunneling. Unfortunately, tunneling is a frequent source of current leakage in very-large-scale integration electronics resulting in substantial power loss and heating effects that can cripple such devices. Because of this, tunneling establishes an effective lower limit on how small functional microelectronic device elements can be.
- Quantum Biology. Electron tunneling is necessary for numerous biochemical redox reactions (cellular respiration, photosynthesis) and enzymatic catalysis. Proton tunneling also has a crucial role in spontaneous DNA mutation.
1 Ph.D. in Theoretical Chemistry from Rice University. Full permission is given to reproduce or distribute this document, or to rearrange/reformat it for other media, as long as credit is given, and no words are added or deleted from the text.
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Theoretical Prediction of Quantum Tunneling
In 1926, Erwin Schrödinger developed a robust mathematical framework for quantum mechanics. Schrödinger’s equation allowed physicists to solve for the wavefunction describing various systems. Unfortunately, many mysteries remained with physicists at that time struggling to understand the implications of certain solutions.
One such example was the theoretical prediction of quantum tunneling, independently described by two groups in 1927.2 First, Friedrich Hund found a curious solution to Schrödinger’s equation for a particle in a symmetric double-well potential (i.e., shaped like a “W”). He noticed that a particle trapped on one side of the hump could spontaneously be found on the other side, even though it lacked enough energy to cross over. Second, Lothar Nordheim used Schrödinger’s equation to study a particle trapped in a finite potential well. Surprisingly, the equation predicted the particles could escape from this trap, even though this was impossible according to classical (Newtonian) mechanics and everyday experience. This result is comparable to placing a marble in a shallow bowl and later finding it spontaneously rolling free.
The following year, Ralph Fowler worked with Nordheim to develop a more realistic version of his model to describe the emission of electrons from metals under strong electric fields. The main difference was they used a triangular rather than a square potential barrier. As with the other two cases, the electron escaped confinement by tunneling through the barrier rather than passing over it. One major prediction was that the electric current (the rate of electrons escaping by quantum tunneling) for what became known as Fowler-Nordheim tunneling was strongly dependent upon the particle’s energy relative to the top of the barrier. In other words, the lower and narrower the barrier that had to be overcome, the greater the likelihood of tunneling out.
These curious theoretical solutions defied common experience and went against of centuries well-established physics. Could they have any basis in reality?
Radioactive Decay by Quantum Tunneling
The first experimental evidence for quantum tunneling was reported independently in 1928 by George Gamow and by the team of Ronald W. Gurney and Edward Condon.3 They were studying radioactive decay (a form of nuclear fission), where an alpha particle (a helium nucleus) splits off from the nuclei of a radioactive element. This process transforms the original radioactive element into a new element. For example, uranium-238 can spontaneously decay into thorium-234 by emitting an alpha particle.
Before considering the role of quantum tunneling in nuclear fission, we need to understand the forces involved. In Figure 1, the horizontal axis describes the distance between a positively charged atomic nucleus (located at 0) and an alpha particle. The vertical axis represents the potential energy resulting from their interaction. Consider an alpha particle moving toward the nuclei (starting from the right side of the diagram and moving left). Because the atomic nucleus and the alpha particle are both positively charged, they repel each other. The electrostatic repulsion is negligible when far apart but as the alpha particle approaches the nucleus, the repulsion rapidly gets stronger, as shown in the diagram. If it is not traveling fast enough, the alpha particle cannot overcome this barrier and will be repelled from the nucleus. But if it is moving at sufficient speed, it will overcome this electrostatic repulsion and come into direct contact with the nucleus. When that happens, the strong nuclear force4 will suddenly dominate overcoming the repulsion and fusing the alpha particle and the nucleus into a new element.
2 Discussion summarized from David D. Nolte, “A Short History of Quantum Tunneling,” Galileo Unbound blog, https://galileo-unbound.blog/2022/11/06/a-short-history-of-quantum-tunneling/.
3 Discussion summarized from https://users.astro.ufl.edu/~guzman/ast7939/glossary/quantum_tunneling.html.
4 The strong nuclear force is one of the four primary forces in nature and serves to bind protons and neutrons together in the nuclei of atoms. It gets is name because it is far stronger than the other forces, however, it is exceptionally short ranged with an effective reach comparable to the diameter of an atomic nucleus. The strong nuclear force was not
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Gamow, Gurney, and Condon were studying radioactive decay, the reverse of the scenario just described. In this case, the nucleus will split into a new element plus an alpha particle. The problem is the alpha particle will have an energy, E, less than the repulsive barrier, thereby preventing it from escaping (see Figure 1). According to classical mechanics, it should remain stuck forever, yet we can readily observe the escaping alpha particles. How do we account for this behavior?
When these three scientists learned about Fowler-Nordheim tunneling, they immediately realized it could be applied to their work on radioactive decay. By tailoring the tunneling model to use the known nuclear potential, the theory predicted the alpha particles could escape by tunneling through the potential barrier.
This was a bold proposal, but how can we be certain this is the correct explanation? Their model, as with Fowler-Nordheim tunneling, predicted the time it takes for the alpha particle to escape should depend strongly on the energy of the escaping particle, E. Experimental results showed that across a wide range of isotopes of various elements, the energies of the emitted alpha particles only varied slightly, between 2 to 8 MeV. Yet, the experimentally measured half-lives (the time it takes for half of the atoms to decay) varied from 1011 years down to 10-6 seconds. This result was astonishing. For a small range of emission energies (a factor of 4), the emission times varied by an enormous factor of 1024 (i.e., 1 million billion billion)! The three scientists recognized this sensitivity to the alpha particle energy would have been difficult to explain by any mechanism other than quantum tunneling.
Quantum Tunneling is Needed for Fusion in Stars
While not understood until later, the opposite process, nuclear fusion, was also shown to be enabled by quantum tunneling.5 The fusion of hydrogen into helium in stars releases enormous amounts of energy in the form of light and heat. This process is needed to keep our sun burning brightly for billions of years. Without the warming light from the sun, Earth would be a frozen, barren rock with no chance for life. We would not even exist to contemplate this dilemma.
The interiors of stars possess high temperatures and densities, resulting in high-energy collisions between protons (hydrogen nuclei). Only if the interior temperature of the star is high enough (about 10 billion K) will the particles crash together with enough force for the hydrogen to fuse into helium. Our sun has a core temperature of around 15 million K, far short of the 10 billion K needed for direct fusion. How, then, do we explain our bright sun? We are saved by the
described until 1971. Prior to that, scientists simply knew that there was a force capable of holding the nucleus together by overcoming the repulsion between positively charged protons.
5 Summarized from Ethan Siegal, “The Sun Wouldn’t Shine Without Quantum Physics,” Starts with a Bang!, https://medium.com/starts-with-a-bang/the-sun-wouldnt-shine-without-quantum-physics-9a8b5b8bca39.
Figure 1: Potential energy diagram for alpha particle decay.
Credit: John Millam based on footnote 3.
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fact that quantum tunneling enables fusion to occur at much lower temperatures. Particles can tunnel through the repulsive barrier without requiring the energy needed to go over it.
Quantum tunneling is critical for stars and life in yet another way. Tunneling is probabilistic (as will be explained later), so under stellar conditions only a minuscule fraction of those collisions will result in fusion. This slows down the fusion process so that it happens gradually (like a wood fire) rather than explosively (like a hydrogen bomb) enabling stars like our sun to burn steadily for billions of years.
Potential Energy Diagrams
To best understand quantum tunneling, it is essential to understand a physics notion: potential energy diagrams. We have already encountered one in our discussion on radioactivity (see Figure 1). While primarily intended for physicists, they can be insightful for non-specialists. Anyone can visualize a particle’s response to a particular potential energy curve by imagining a marble rolling on a hill or valley of equivalent shape. As a simple example, many science centers have an exhibit known as a gravity well—a large funnel-shaped surface representing the gravitational potential energy of a black hole (see Figure 2). People can roll coins around the surface of the well and watch them slowly spiraling inward, simulating a star being pulled into a black hole. In the same way, one can visualize the behavior associated with a wide variety of potential energy diagrams.
To keep our discussion simple, we will initially consider the case of a single particle using classical (pre-quantum) physics. In classical physics, a particle’s total energy (TE) can be factored into two parts: kinetic energy (KE) and potential energy (PE):
TE = KE + PE, Equation 1
where TE is total energy, KE is kinetic energy, and PE is potential energy.
Kinetic energy is a measure of the particle’s motion, while potential energy involves its response to external forces. We will discuss both in more detail in just a moment, but first let us introduce a few simplifying assumptions into our idealized scenario: - The particle is a simple point with mass, m.
- There is no friction.
- The potential energy does not vary with time.
- The total energy is conserved (i.e., TE = KE + PE = constant).
Now let us take a closer look at the kinetic and potential energies.
a) b)
Figure 2: a) Gravity well exhibit and b) shape of gravitational potential well.
Credit: a) Steve Matsumoto and b) AllenMcC.
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Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. We can easily calculate
it for our particle using the formula:
KE = ½ mv2, Equation 2
where KE is the kinetic energy of the particle, m is its mass, and v is its velocity.
From this we can see that the kinetic energy must be positive (because mass is always positive,
and the velocity is squared). Flipping this relationship around, for a given amount of kinetic
energy, we can calculate the associated velocity:
v = 2*KE m , Equation 3
where v is the velocity of the particle, KE is its kinetic energy, and m is its mass.
This equation provides us another insight into why kinetic energy must be positive. If it became
negative, then the associated velocity would be an imaginary number.6 That is absolutely
forbidden in classical physics because if it occurred, the particle would travel into imaginary space.
The importance of this restriction on a particle’s kinetic energy will be discussed later.
Potential Energy
Potential energy represents the particle’s response to all external forces acting on it. In the
interests of generality, we will only consider the most basic distinction: potentials can either be
repulsive or attractive. A simple example of a repulsive potential in real life is a hill. That’s
because it takes energy to climb an incline (see Figure 3a). Anything that only makes it partway
up the slope will roll back down rather than go over the top. An example of an attractive potential
is something rolling downhill gaining energy as it goes (see Figure 3b).
Quantum Golf
Using the framework just described, we can now discuss how quantum tunneling operates
and how it differs from ordinary classical mechanics. To make it easy to visualize, we will use
miniature golf for our illustration. Imagine the situation where you are must putt a golf ball up a
6 Imaginary here refers to a mathematically complex number, not a denial of their reality.
a) b)
Figure 3: Repulsive and attractive potentials.
Credit: John Millam
Figure 4: Putting uphill in
miniature golf.
Credit: John Millam
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ramp (see Figure 4). For simplicity, we will use a rounded hill instead of a ramp. Now let us consider three scenarios using the same potential energy diagram.
Initial Putt
Placing the ball at the base of the hill, you hit it with your putter, imparting some initial amount of kinetic energy and consequently giving it an initial velocity (see Figure 5). The initial potential energy is zero, so the total energy of the ball equals the initial kinetic energy. Because the total energy is conserved in this idealized example, it is represented by a horizontal dashed line. As the ball ascends the hill, it gains potential energy while correspondingly losing kinetic energy. This progression is shown in Figure 5: KE(initial) → KE1 → KE2. And as the kinetic energy decreases, so does its forward velocity (see Equation 3).
Putt #1: Total energy greater than the maximum potential energy (TE > PEmax)
In our first scenario, let us consider the case where you hit the ball hard enough (give it enough kinetic energy) to get over the top (see Figure 6). In this case, the total energy is greater than the potential energy. As before, the ball will be propelled uphill, progressively losing kinetic energy, and slowing down. When it reaches the top, it has some excess kinetic energy. This means its remaining velocity will carry it over the top, where it will then roll down the far side of the hill.
Figure 5: Initial putt imparts kinetic energy to the golf ball.
Credit: John Millam.
Figure 6: Total energy greater than the maximum potential energy.
Credit: John Millam.
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Putt #2: Total energy less than the maximum potential energy (TE < PEmax)
For our second scenario, you hit the ball weakly, such that the total energy is less than the maximum potential energy (see Figure 7). Once again, the ball is propelled uphill, but this time the kinetic energy goes to zero before it can reach the summit. At that point, its forward velocity is also zero, meaning the ball has stopped, after which it will begin to retreat down the hill returning the way it came.
As shown in the figure, the peak of the potential extends above the total energy, representing a classically forbidden region because it corresponds to the kinetic energy being negative. As previously explained, the ball cannot reach this area because classical mechanics forbids a particle from having an imaginary velocity.
Putt #3: Quantum tunneling with a total energy less than the maximum potential energy
For our third scenario, let us revisit the previous case but applying quantum mechanics. If we solve Schrödinger’s equation for this case, it shows that the wavefunction will somehow be present (i.e., is non-zero) on the far side of the hill. This means that in some cases, the ball will somehow appear on the other side, where it will continue its journey onward (see Figure 8).
How does quantum mechanics explain this? To begin, we need to think of quantum entities as waves representing a range of positions and velocities. According to Heisenberg’s uncertainty principle, this uncertainty about the exact location of particles allows them to break the rules of classical mechanics and move in space past the potential energy barrier instead of passing over it.
Figure 7: Total energy less than the maximum potential energy.
Credit: John Millam.
Figure 8: Quantum tunneling.
Credit: John Millam.
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An even more mysterious feature of this explanation is that the ball will only tunnel through some of the time. This result was unexpected because in our classical scenarios (putts 1 and 2), the ball either always goes over or always rolls back. The difference depends on whether the total energy is greater than or less than the maximum potential energy. But with quantum tunneling for a given initial kinetic energy, a certain percentage of balls will successfully tunnel through while the remainder will roll back. Thus, for identical initial conditions, we can observe different outcomes. This probabilistic behavior is intrinsic to quantum mechanics.
To summarize what has been presented so far, an excellent 1 ½ minutes long animated video visually summarizes the essentials of quantum tunneling and its application to scanning tunneling microscopy.
Mystery of Quantum Tunneling
We know that quantum tunneling is real because we can observe it across a wide range of fields (e.g., nuclear physics, chemistry, and electronics). So far, we have described what we know about quantum tunneling and how it operates, yet some disturbing mysteries remain.
Big Mystery #1: Probabilistic Outcome
Classically, a particle will or will not overcome the barrier depending on its total energy. In contrast, quantum tunneling is probabilistic—each particle has a well-defined chance of tunneling through the barrier. Solving Schrödinger’s equation reveals parts of the wavefunction on both sides of the obstacle. The probability density (i.e., the square of the wavefunction) defines the likelihood of finding the particle at a given location. This allows us to calculate the probability that a particle will tunnel across the barrier. While the predicted probabilities are consistent with experimental results, this does not answer the more profound question of how an individual particle determines whether it makes it through.
Part of the difficulty in understanding tunneling is we naturally view particles as point-like objects with well-defined positions and velocities. That was the picture in classical mechanics, but according to quantum mechanics we must view them as waves. The wavefunction described by Schrödinger’s equation represents a range of positions and velocities, each with an associate probability. The wavefunction, therefore simultaneously describes particles that tunnel through and those that do not. The actual outcome is purely random according to the quantum probabilities.
Big Mystery #2: Non-locality
Tunneling can only occur if there is a location on the other side of the barrier that the particle can reach. When a particle encounters a barrier, it will either tunnel through or rebound depending on conditions that exist on the opposite side of the obstacle. This is surprising because this means the particle’s behavior is determined by conditions it has not yet encountered. To anthropomorphize this situation, how does the particle “sense” what is on the other side of the barrier to determine if it can tunnel through?
The apparent dilemma comes from thinking of the particle as being located at a single point in space. However, as we have discussed, classical concepts do not apply here (see Figure 9a). Quantum mechanics, in contrast, treats particles as waves or wave packets. This wave packet is spread over a small region of space, thus allowing it to be on both sides of the barrier at the same time (see Figure 9b). The particle can therefore “know” what is on the opposite side of the barrier because part of the wavefunction is located there.
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Big Mystery #3: How Fast is Tunneling?
One of the most contentious aspects of quantum tunneling is the amount of time required for a particle to tunnel through a barrier.7 This question has perplexed scientists since quantum tunneling was discovered in 1927-28. At that time, it was not possible to experimentally measure tunneling time with sufficient precision, forcing physicists to rely on theoretical predictions. The problem is that the results did not seem to make sense.
The first published analysis of tunneling time came in 1932, but it was Thomas Hartman in 1962 who fully embraced the shocking truth—the particle seems to leap across the forbidden region. It takes time for the particle to both enter and leave the barrier region, but almost no time passes while it is in the barrier. This means that the tunneling time is nearly independent of barrier thickness. According to the Hartman effect, a particle traversing a sufficiently thick barrier could hop from one side to the other faster than light traveling the same distance through empty space.
This prediction was just a theoretical curiosity until 2014, when scientists could finally make ultraprecise timings. These measurements support Hartman’s prediction that tunneling through a barrier can take less time than it takes to pass through empty space. Could this be used for superluminal (faster-than-light) signaling? If so, it would be possible to send and receive a signal before the original signal was emitted, therefore, the effect will precede the cause. This would conflict with Einstein’s theory of relativity that dictates that no signal or information can be communicated faster than the speed of light (preserving causality).
Researchers have not fully resolved this issue but generally agree this is a mystery, not an actual conflict. Superluminal tunneling does not violate relativity because it does not allow superluminal signaling for statistical reasons. Even though tunneling through an extremely thick barrier happens very fast, the chance of a tunneling event happening through such a barrier is extraordinarily low. A signaler would always prefer to send the signal through free space where it is more certain to get through. Not everyone agrees with this solution, so more research needs to be done.
7 Discussion summarized from: https://www.quantamagazine.org/quantum-tunnel-shows-particles-can-break-the-speed-of-light-20201020/
a) b)
Figure 9: a) Classical and b) quantum description of an electron encountering a repulsive electric field.
Credit: John Millam.
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Appendix: Resources on Quantum Mechanics
Robert Gilmore, Alice in Quantumland: An Allegory of Quantum Physics.
Erica W. Carlson, The Great Courses: Understanding the Quantum World.
James Kakalios, The Amazing Story of Quantum Mechanics.
Alex Montwill and Ann Breslin, The Quantum Adventure: Does God Play Dice?
Brian Clegg, The Quantum Physics Bible: The Definitive Guide to 200 Years of Subatomic Science.
Banesh Hoffmann, The Strange Story of the Quantum.
Amir D. Aczel, Entanglement: The Greatest Mystery in Physics.
George Musser, Spooky Action at a Distance.