Quantum Tunneling

Quantum tunneling is a phenomenon in which a quantum particle has a probability of passing through an energy barrier even when, according to classical physics, it does not have enough energy to cross that barrier.

Classically:

particle energy < barrier energy → particle cannot cross

Quantum mechanically:

particle energy < barrier energy → particle may still appear on the other side

Tunneling is possible because particles such as electrons and alpha particles are described by wavefunctions rather than simply as tiny classical objects following definite paths.

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The Classical Energy Barrier

Imagine a ball approaching a hill.

If the ball has enough kinetic energy, it can travel over the hill.

If:

KE > energy needed to cross

the ball crosses.

But if:

KE < energy needed to cross

the ball rolls back.

This is the classical prediction.

A classical particle cannot simply appear on the opposite side of a barrier that it lacks the energy to cross.

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The Quantum Picture

Quantum particles behave differently.

An electron, proton, or other quantum particle is described by a:

wavefunction, ψ

The probability of detecting the particle is related to:

|ψ|²

When the particle encounters a barrier, its wavefunction does not necessarily stop suddenly at the edge.

Instead, the wavefunction can:

extend into the barrier

and, if the barrier is sufficiently thin, extend through to the other side.

Therefore there can be a non-zero probability of detecting the particle beyond the barrier.

This is:

quantum tunneling

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Probability and the Wavefunction

Quantum mechanics does not generally say exactly where an individual particle will be detected.

Instead, it predicts:

probabilities

The probability density is related to:

|ψ|²

If the wavefunction has a non-zero value on the far side of a barrier, there is a non-zero probability of detecting the particle there.

This does not mean scientists simply do not know how the particle crossed.

The probabilistic behaviour is a fundamental part of the quantum description.

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What Happens at a Barrier?

Suppose a quantum particle approaches a potential-energy barrier.

There are two possible outcomes:

reflection

or:

transmission

Reflection means the particle remains on the original side.

Transmission means the particle is detected on the opposite side.

The probability of transmission is called the:

tunneling probability

or:

transmission probability

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The Wavefunction Inside the Barrier

For a particle with energy below the barrier height, the wavefunction generally decreases rapidly inside the classically forbidden region.

This decrease is approximately:

exponential

The wavefunction does not normally become zero immediately.

Instead:

larger distance into barrier → smaller wavefunction amplitude

Therefore, a thin barrier may still leave significant wavefunction amplitude on the other side.

A thick barrier may reduce the amplitude to an extremely small value.

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Tunneling Probability

For a simple rectangular barrier, the tunneling probability has approximately the form:

T ∝ e⁻²κL

where:

• T = transmission probability
• L = barrier width
• κ depends on the particle mass and the difference between barrier energy and particle energy

A useful expression is:

κ = √[2m(U − E)]/ħ

where:

• m = particle mass
• U = barrier potential energy
• E = particle energy
• ħ = h/2π

The exact mathematics can become more advanced, but the physical trends are extremely important.

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What Makes Tunneling More Likely?

Tunneling becomes more likely when the barrier is:

thinner

lower

and when the particle is:

lighter

or has energy closer to the barrier energy.

Therefore:

thin barrier → greater tunneling probability

thick barrier → smaller tunneling probability

low barrier → greater tunneling probability

high barrier → smaller tunneling probability

lighter particle → generally greater tunneling probability

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Why Barrier Width Matters So Much

Because tunneling probability depends approximately exponentially on barrier width, even a tiny change in distance can produce a very large change in tunneling probability.

For example, increasing the barrier width slightly might reduce the tunneling probability by:

• a factor of 10
• a factor of 100
• a factor of 1000

depending on the system.

This extreme sensitivity to distance is used in important technologies.

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Tunneling Does Not Violate Conservation of Energy

A common misunderstanding is that a tunneling particle temporarily "borrows" energy to cross the barrier.

That is not the correct basic explanation.

If an electron approaches with energy E and tunnels through:

energy before = E

and:

energy after = E

The particle does not emerge with extra energy simply because it tunneled.

Energy conservation is maintained.

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A Better Picture

Avoid imagining tunneling as:

tiny particle digs a tunnel through a wall

Instead, think in terms of the quantum state.

The wavefunction extends into a region that would be forbidden to a classical particle.

If it extends through the entire barrier:

there is a probability of detecting the particle beyond it

The "tunnel" is therefore not a physical hole through the barrier.

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Electron Tunneling

Electrons are extremely light particles.

They can therefore exhibit significant tunneling across very small barriers.

Electron tunneling occurs in:

• semiconductor devices
• scanning tunneling microscopes
• tunnel diodes
• flash memory
• Josephson junction-related quantum systems
• nanoscale electronics

At microscopic scales, tunneling is not a rare curiosity.

It is an important physical process.

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Tunneling in Radioactive Decay

One of the most important natural examples of quantum tunneling is:

alpha decay

An unstable atomic nucleus can emit an:

alpha particle

An alpha particle contains:

• 2 protons
• 2 neutrons

It is therefore essentially a:

helium-4 nucleus

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The Alpha Particle Inside the Nucleus

Inside a heavy nucleus, an alpha particle can be thought of approximately as being trapped by nuclear interactions.

The strong nuclear interaction helps hold nuclear particles together.

Outside the nucleus, the positively charged alpha particle experiences electromagnetic repulsion from the positively charged daughter nucleus.

This creates an effective:

potential-energy barrier

Classically, an alpha particle whose energy is below the top of this barrier should remain trapped.

Yet alpha decay occurs.

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Alpha Decay Through Tunneling

Quantum mechanics provides the explanation.

The alpha particle has a wavefunction.

Its wavefunction extends into the potential barrier.

There is a small probability that the alpha particle will:

tunnel through the barrier

Once outside, electromagnetic repulsion pushes the positively charged alpha particle away from the daughter nucleus.

Thus:

alpha decay is a quantum tunneling process

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Why Alpha Decay Is Random

We cannot normally predict exactly when one particular unstable nucleus will decay.

Instead, quantum mechanics gives a:

probability of decay per unit time

For a large number of identical radioactive nuclei, this produces predictable statistical behaviour.

This is why radioactive decay is:

random for an individual nucleus

but:

predictable statistically for a large sample

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Tunneling and Half-Life

Different radioactive isotopes can have enormously different half-lives.

One important reason is that tunneling probability is extremely sensitive to:

• barrier height
• barrier width
• alpha-particle energy

A small change in these properties can cause a huge change in the probability that an alpha particle escapes.

Therefore:

slightly different nuclear conditions → dramatically different half-lives

This was a major success of the quantum theory of alpha decay.

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Gamow's Explanation

In 1928, George Gamow and, independently, Ronald Gurney and Edward Condon applied quantum mechanics to alpha decay.

They showed that an alpha particle could escape a nucleus by:

quantum tunneling

This successfully connected nuclear decay rates with quantum theory.

It was one of the earliest major applications of quantum mechanics to nuclear physics.

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Tunneling in Nuclear Fusion

Quantum tunneling also plays an important role in:

nuclear fusion

Two positively charged nuclei repel each other because of the electromagnetic force.

This repulsion creates a:

Coulomb barrier

Classically, many nuclei inside stars do not have enough energy to overcome this barrier directly.

Quantum tunneling gives them a small probability of penetrating it.

Once the nuclei become sufficiently close, the strong nuclear interaction can bind them.

Therefore, tunneling helps make:

stellar fusion possible at temperatures lower than a simple classical calculation would suggest

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Quantum Tunneling in the Sun

The Sun's core is extremely hot.

Even so, the average proton does not have enough classical kinetic energy to simply overcome the full electrostatic barrier between two protons.

Quantum tunneling provides a probability that protons can approach sufficiently closely for nuclear processes to occur.

Thus tunneling contributes to the reactions ultimately responsible for:

solar energy

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The Scanning Tunneling Microscope

One of the most remarkable technological applications of quantum tunneling is the:

scanning tunneling microscope (STM)

The STM can investigate surfaces with extremely high spatial resolution, reaching the atomic scale under suitable conditions.

It works by measuring:

electron tunneling

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How an STM Works

An STM uses an extremely sharp conducting tip.

The tip is positioned extremely close to a conducting or semiconducting surface.

The gap may be on the scale of:

nanometres or less

Classically, electrons should not readily cross the gap under the relevant conditions.

Quantum mechanically, their wavefunctions extend across the gap.

Electrons can:

tunnel between the tip and the surface

This produces a measurable:

tunneling current

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Distance and Tunneling Current

The tunneling current is extremely sensitive to the distance between:

tip and sample

A very small change in separation can cause a large change in current.

Therefore, by scanning the tip across a surface and monitoring the tunneling current, scientists can map extremely small variations in the surface's electronic structure and topography.

This makes STM extraordinarily sensitive.

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Imaging Atomic-Scale Surfaces

An STM can produce images that reveal features associated with individual atoms on suitable surfaces.

The basic process is:

tip moves across surface

↓

tunneling current changes

↓

computer records current/position

↓

surface map is constructed

The STM became an important tool in:

• nanotechnology
• surface physics
• materials science
• semiconductor research

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Tunnel Diodes

A tunnel diode is a semiconductor device designed to make significant use of quantum tunneling.

Electrons tunnel through a narrow energy barrier within the semiconductor structure.

Tunnel diodes can respond extremely rapidly.

They have been used in:

• high-frequency electronics
• oscillators
• microwave circuits
• specialized electronic systems

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Flash Memory

Quantum tunneling is also important in some forms of:

flash memory

Electrons can be moved through very thin insulating barriers during programming or erasing processes.

The presence or absence of stored charge can represent digital information.

Thus a phenomenon that seems strange from a classical perspective contributes to everyday technologies such as:

• solid-state storage
• memory devices
• electronic systems

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Modern Transistors and Tunneling

As electronic components become extremely small, quantum tunneling becomes increasingly important.

Very thin insulating layers may allow electrons to tunnel through when engineers do not want them to.

This can produce:

leakage current

Therefore, tunneling can be:

useful when deliberately engineered

but also:

a problem when unwanted

Quantum mechanics places important constraints on how small conventional electronic devices can become.

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Josephson Tunneling

Quantum tunneling can also occur in superconducting systems.

Two superconductors separated by a very thin insulating layer can allow paired electrons to tunnel through the barrier.

This is associated with the:

Josephson effect

Josephson junctions are important in:

• extremely sensitive magnetic-field measurements
• voltage standards
• superconducting electronics
• some quantum computers

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Tunneling and Quantum Computing

Some quantum-computing technologies rely on circuits containing:

Josephson junctions

These systems exploit quantum effects that have no classical equivalent.

Quantum tunneling therefore contributes to research into:

quantum information technology

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Comparing Classical and Quantum Predictions

The major difference is:

quantum mechanics permits transmission through classically forbidden regions

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Probability Does Not Mean Anything Can Happen Easily

Quantum tunneling does not mean a person could realistically tunnel through a wall.

Although quantum mechanics technically applies to all matter, tunneling probability decreases dramatically for:

• large masses
• thick barriers
• high barriers

For a macroscopic object, the tunneling probability through an ordinary wall is so extraordinarily tiny that it is effectively:

zero for practical purposes

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Why Electrons Tunnel More Easily

The tunneling factor depends on particle mass.

Very roughly:

T ∝ e⁻²κL

with:

κ ∝ √m

A larger mass makes the exponential suppression stronger.

Electrons have extremely small mass:

mₑ ≈ 9.11 × 10⁻³¹ kg

Therefore electron tunneling can be significant across nanoscale barriers.

Macroscopic objects contain enormously greater mass and do not show observable tunneling through everyday barriers.

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Worked Example: Comparing Barrier Widths

Suppose a particle has a tunneling probability through a particular barrier.

Now imagine making the barrier wider while keeping everything else unchanged.

Because approximately:

T ∝ e⁻²κL

increasing L causes T to decrease exponentially.

Therefore:

wider barrier → dramatically smaller tunneling probability

This explains why tunneling is mainly important across extremely small distances.

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Worked Example: Comparing Particle Energies

Two identical particles approach the same barrier.

Particle A has energy much lower than the barrier.

Particle B has energy only slightly below the barrier.

Particle B has the greater tunneling probability.

Why?

The difference:

U − E

is smaller.

Therefore the wavefunction decreases less rapidly inside the barrier.

So:

energy closer to barrier height → greater tunneling probability

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Worked Example: STM

Suppose an STM tip moves slightly closer to a surface.

The barrier width decreases.

Because tunneling probability depends exponentially on distance:

tunneling probability increases strongly

Therefore:

tunneling current increases

The STM can use this large change in current to detect extremely small changes in surface height.

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Worked Example: Alpha Decay

Consider an alpha particle trapped inside a nucleus.

Its energy is below the Coulomb barrier.

Classically:

escape impossible

Quantum mechanically:

the alpha-particle wavefunction penetrates the barrier.

There is a small probability of transmission.

If transmission occurs:

alpha particle appears outside nucleus

and is repelled away.

Therefore:

radioactive alpha decay occurs through quantum tunneling

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Interpreting Probability

Suppose a tunneling probability is:

1 × 10⁻⁶ per attempt

This does not mean every millionth attempt must succeed exactly.

It means that over a very large number of similar attempts, approximately:

1 in 1,000,000

would be expected to succeed on average.

Quantum probabilities describe statistical outcomes.

Individual events remain unpredictable.

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Repeated Attempts

A trapped particle may interact with a barrier extremely frequently.

Even if the tunneling probability per encounter is very small, repeated attempts can make escape significant over time.

This idea is important in:

alpha decay

A tiny probability per encounter can still produce measurable radioactive decay because the quantum system continually evolves and interacts with the barrier.

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Tunneling and Uncertainty

Quantum tunneling is sometimes incorrectly explained entirely using the Heisenberg uncertainty principle.

The uncertainty principle is an important part of quantum mechanics, but tunneling is more directly understood by solving the quantum wave equation for a potential barrier.

The essential idea is:

the wavefunction penetrates into classically forbidden regions

not:

the particle temporarily violates energy conservation

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Tunneling and the Schrödinger Equation

Quantum tunneling emerges naturally from the:

Schrödinger equation

When the equation is solved for a potential barrier, the wavefunction inside a classically forbidden region generally decreases exponentially rather than immediately becoming zero.

If the barrier is finite in width, some wavefunction can exist beyond it.

Therefore:

non-zero wavefunction beyond barrier → non-zero transmission probability

Quantum tunneling is not an extra rule added to quantum mechanics.

It follows directly from its fundamental equations.

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Real-World Importance

Quantum tunneling is important across many areas of physics.

Nuclear physics

• alpha decay
• nuclear fusion

Electronics

• tunnel diodes
• flash memory
• nanoscale transistor behaviour

Microscopy

• scanning tunneling microscopy

Superconductivity

• Josephson junctions

Quantum technology

• superconducting quantum circuits
• quantum computing research

It connects fundamental quantum theory directly to practical technology.

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From Nuclei to Electronics

The same fundamental principle explains seemingly unrelated phenomena:

alpha particle escapes nucleus

↓

quantum tunneling

protons approach during stellar fusion

↓

quantum tunneling

electron crosses nanoscale barrier

↓

quantum tunneling

STM measures atomic-scale surfaces

↓

quantum tunneling

This demonstrates the power of quantum mechanics: one theory can explain phenomena across very different physical systems.

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Common Mistakes

Mistake: Saying a tunneling particle breaks through the barrier physically.
The barrier is not mechanically broken; tunneling arises from the quantum wavefunction.

Mistake: Saying the particle must have more energy than the barrier.
The defining feature of tunneling is that transmission can occur even when E < U.

Mistake: Saying tunneling violates conservation of energy.
Energy remains conserved.

Mistake: Saying the particle borrows energy from nowhere.
This is not an accurate explanation of quantum tunneling.

Mistake: Saying every particle will eventually tunnel immediately.
Tunneling is probabilistic, and the probability can be extremely small.

Mistake: Saying barrier width has little effect.
Tunneling probability depends extremely strongly on barrier width.

Mistake: Saying heavier particles tunnel more easily.
Other things equal, greater particle mass generally reduces tunneling probability.

Mistake: Saying alpha decay occurs because the alpha particle simply has enough classical energy to escape.
Quantum tunneling allows escape through the nuclear/Coulomb barrier.

Mistake: Saying an STM directly photographs atoms with visible light.
An STM measures tunneling current between a sharp tip and a surface.

Mistake: Saying tunneling only occurs in laboratories.
It occurs naturally in radioactive decay and contributes to nuclear processes in stars.

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Key Terms

Quantum tunneling — quantum phenomenon in which a particle has a non-zero probability of passing through a barrier even when its energy is below the classical barrier energy.

Potential-energy barrier — region in which a particle would require greater energy to enter or cross according to classical physics.

Classically forbidden region — region that a classical particle with the given energy could not occupy.

Wavefunction (ψ) — mathematical description of a quantum state.

Probability density — quantity related to |ψ|² that determines probabilities of particle detection.

Transmission probability — probability that a quantum particle passes through a barrier.

Reflection probability — probability that a particle is reflected by a barrier.

Alpha decay — radioactive process in which an unstable nucleus emits an alpha particle.

Alpha particle — particle consisting of two protons and two neutrons.

Coulomb barrier — electrostatic energy barrier produced by repulsion between positively charged nuclei or particles.

Scanning tunneling microscope (STM) — instrument that uses electron tunneling to investigate conducting surfaces at extremely small scales.

Tunneling current — electric current produced by electrons tunneling through a barrier.

Tunnel diode — semiconductor device that deliberately uses quantum tunneling.

Josephson junction — two superconductors separated by a thin barrier through which quantum tunneling can occur.

Classical limit — situation in which quantum effects become so small that classical physics provides an excellent approximation.

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Key Takeaways

• Quantum tunneling allows a particle to pass through a barrier even when its energy is below the classical barrier height.
• Classical physics predicts that a particle with insufficient energy cannot cross such a barrier.
• Quantum particles are described by wavefunctions.
• The probability density is related to |ψ|².
• A quantum wavefunction can extend into a classically forbidden region.
• For a finite barrier, the wavefunction may extend to the other side.
• A non-zero wavefunction beyond the barrier gives a non-zero probability of transmission.
• Tunneling is probabilistic.
• Tunneling does not violate conservation of energy.
• Tunneling probability generally decreases exponentially as barrier width increases.
• Lower barriers generally produce greater tunneling probabilities.
• Particles with energies closer to the barrier height generally tunnel more easily.
• Lighter particles generally show greater tunneling probabilities than heavier particles under similar conditions.
• Alpha decay can be explained by quantum tunneling through a nuclear potential barrier.
• Tunneling helps explain why radioactive decay is probabilistic.
• Differences in tunneling probability help explain large differences in radioactive half-lives.
• Quantum tunneling contributes to nuclear fusion in stars.
• A scanning tunneling microscope measures electron tunneling between a sharp tip and a surface.
• STM tunneling current is extremely sensitive to tip-surface distance.
• Tunnel diodes deliberately exploit quantum tunneling.
• Tunneling is important in some semiconductor memory and nanoscale electronic devices.
• Josephson junctions use quantum tunneling in superconducting systems.
• Tunneling can be useful in technology but can also produce unwanted leakage in very small electronic components.
• Quantum tunneling follows naturally from quantum mechanics and the behaviour of the wavefunction.
• It is one of the clearest examples of a phenomenon that cannot be adequately explained using classical physics alone.

Check Your Understanding

1. What is quantum tunneling?
2. What would classical physics predict for a particle whose energy is lower than a potential barrier?
3. How does the quantum prediction differ from the classical prediction?
4. What does |ψ|² represent?
5. What happens to the wavefunction inside a classically forbidden barrier?
6. Why can a particle sometimes be detected on the far side of a barrier?
7. What happens to tunneling probability when barrier width increases?
8. What happens to tunneling probability when barrier height decreases?
9. Why do light particles such as electrons generally tunnel more easily than massive particles?
10. Explain why quantum tunneling does not violate conservation of energy.
11. Describe how quantum tunneling explains alpha decay.
12. What is the Coulomb barrier in alpha decay?
13. Why is radioactive decay unpredictable for an individual nucleus but predictable for a large sample?
14. Explain why small changes in nuclear conditions can produce large differences in half-life.
15. How does quantum tunneling contribute to nuclear fusion in stars?
16. Explain how a scanning tunneling microscope uses quantum tunneling.
17. Why is an STM extremely sensitive to the distance between its tip and the surface?
18. Give two technological applications of quantum tunneling other than STM.
19. Explain why a macroscopic object does not have a realistic chance of tunneling through an ordinary wall.
20. Explain how quantum tunneling demonstrates the difference between classical and quantum descriptions of matter.