Quantum Tunneling through Potential Barriers

Page Claimed By: Haripratiik Arunkumar Malarkodi, Spring 2026

Quantum tunneling is a quantum mechanics effect where a particle can be found on the other side of a potential barrier even when, classically, it does not have enough energy to cross it. This sounds impossible from a classical point of view, but it makes sense once the particle is treated as a wavefunction instead of as a tiny ball following one exact path.

The main idea is that the wavefunction does not suddenly become zero when it reaches a finite barrier. Instead, it can extend into the barrier and decay as it moves through it. If the barrier is thin enough, some of the wavefunction can still exist on the far side. That gives a nonzero probability of finding the particle past the barrier.

This matters because tunneling is not just a strange theoretical effect. It helps explain alpha decay, fusion in stars, scanning tunneling microscopes, and tunneling effects in semiconductor devices.

Overview

A simple way to picture tunneling is to compare a quantum particle to a ball rolling toward a hill. In classical mechanics, if the ball does not have enough energy to get over the hill, it rolls back. There is no chance that it appears on the other side.

Quantum mechanics gives a different result. A particle is described by a wavefunction, and the wavefunction can enter regions that would be forbidden in classical mechanics. For a rectangular barrier of height [math]\displaystyle{ V_0 }[/math] and width [math]\displaystyle{ L }[/math], a particle with energy [math]\displaystyle{ E\lt V_0 }[/math] cannot cross the barrier classically. However, its wavefunction can decay inside the barrier. If the barrier is not too wide, part of the wavefunction reaches the other side and becomes a transmitted wave.

The important point is that tunneling is probabilistic. We cannot say exactly when one particle will tunnel, but we can calculate the probability that it will.

Quantum tunneling through a square potential barrier. The diagram shows an incoming wave encountering a finite barrier and a smaller transmitted wave appearing on the far side. Image by MikeRun, licensed under Creative Commons Attribution-ShareAlike 4.0 International, via Wikimedia Commons.

Classical Expectations

Under classical mechanics, a particle can cross a potential barrier only if its total energy is greater than the barrier's potential energy. Therefore:

If [math]\displaystyle{ E\gt V_0 }[/math], the particle can pass over the barrier.
If [math]\displaystyle{ E\lt V_0 }[/math], the particle cannot pass through or over the barrier.
The classical transmission probability for [math]\displaystyle{ E\lt V_0 }[/math] is exactly zero.

This classical model works well for macroscopic objects, but it fails for microscopic particles such as electrons, protons, and alpha particles.

Quantum Theory

Wave Behavior of Particles

Quantum mechanics describes particles using wavefunctions. A wavefunction, usually written as [math]\displaystyle{ \psi(x) }[/math], contains information about the probability of finding a particle at different positions. The probability density is

[math]\displaystyle{ |\psi(x)|^2 }[/math].

When a quantum particle reaches a finite potential barrier, its wavefunction must satisfy boundary conditions at the edges of the barrier. Since waves generally do not suddenly disappear at a boundary, the wavefunction extends into the barrier. Inside the barrier, the wavefunction has an exponentially decaying form. If the barrier is thin enough, the wavefunction can still have a nonzero value on the far side.

This is the origin of tunneling.

Important Intuition

The tunneling probability is very sensitive to the barrier. Three changes matter the most:

Increasing the barrier width [math]\displaystyle{ L }[/math] makes tunneling much less likely.
Increasing the barrier height [math]\displaystyle{ V_0 }[/math] makes tunneling much less likely when the particle energy stays the same.
Increasing the particle mass [math]\displaystyle{ m }[/math] makes tunneling much less likely.

This is why tunneling is noticeable for very small particles, like electrons, but not for everyday objects. A baseball technically has a quantum wavefunction, but its mass is so large that the tunneling probability through an ordinary wall is effectively zero.

Heisenberg Uncertainty Principle

The uncertainty principle is sometimes used to give intuition for tunneling, but it should not be treated as the main explanation. A common oversimplification is to say that the particle "borrows energy" for a short time. That picture can be helpful at first, but it is not how tunneling is actually calculated.

The more reliable explanation comes from the Schrödinger equation. When the wavefunction meets a finite barrier, the boundary conditions force part of the wavefunction to extend into the barrier. Inside the barrier, the wavefunction decays exponentially instead of disappearing instantly.

The Schrödinger Equation

The time-independent Schrödinger equation determines the allowed wavefunctions for a particle in a given potential energy function:

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x)=E\psi(x) }[/math].

For a finite potential barrier, the wavefunction must be continuous at each boundary. Its derivative must also be continuous as long as the potential is finite. These conditions allow us to solve for the reflected and transmitted parts of the wave.

Mathematical Model

Consider a rectangular potential barrier:

[math]\displaystyle{ V(x)= \begin{cases} 0, & x\lt 0 \\ V_0, & 0\le x\le L \\ 0, & x\gt L \end{cases} }[/math]

Assume the particle energy satisfies

[math]\displaystyle{ 0\lt E\lt V_0 }[/math].

This means the particle does not have enough energy to cross the barrier classically.

Wave Equations

In each region, the Schrödinger equation has a different form.

Region I: Before the barrier, [math]\displaystyle{ x\lt 0 }[/math]

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi_I}{dx^2}=E\psi_I }[/math]

The general solution is

[math]\displaystyle{ \psi_I(x)=Ae^{ikx}+Be^{-ikx} }[/math].

Here:

[math]\displaystyle{ Ae^{ikx} }[/math] is the incident wave moving toward the barrier.
[math]\displaystyle{ Be^{-ikx} }[/math] is the reflected wave moving away from the barrier.

Region II: Inside the barrier, [math]\displaystyle{ 0\le x\le L }[/math]

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi_{II}}{dx^2}+V_0\psi_{II}=E\psi_{II} }[/math]

Since [math]\displaystyle{ E\lt V_0 }[/math], the solution is exponential rather than sinusoidal:

[math]\displaystyle{ \psi_{II}(x)=Ce^{-\kappa x}+De^{\kappa x} }[/math].

This shows that the wavefunction decays inside the barrier instead of oscillating like a free traveling wave.

Region III: After the barrier, [math]\displaystyle{ x\gt L }[/math]

[math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2\psi_{III}}{dx^2}=E\psi_{III} }[/math]

Since there is no incoming wave from the far right, the transmitted wave is

[math]\displaystyle{ \psi_{III}(x)=Fe^{ikx} }[/math].

The constants are

[math]\displaystyle{ k=\frac{\sqrt{2mE}}{\hbar} }[/math]

and

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math].

The quantity [math]\displaystyle{ k }[/math] describes the oscillating wave outside the barrier. The quantity [math]\displaystyle{ \kappa }[/math] describes the exponential decay inside the barrier.

Boundary Conditions

The wavefunction and its derivative must match at both edges of the barrier.

At [math]\displaystyle{ x=0 }[/math]:

[math]\displaystyle{ A+B=C+D }[/math]

[math]\displaystyle{ ik(A-B)=-\kappa C+\kappa D }[/math]

At [math]\displaystyle{ x=L }[/math]:

[math]\displaystyle{ Ce^{-\kappa L}+De^{\kappa L}=Fe^{ikL} }[/math]

[math]\displaystyle{ -\kappa Ce^{-\kappa L}+\kappa De^{\kappa L}=ikFe^{ikL} }[/math]

These four equations can be solved for the unknown amplitudes [math]\displaystyle{ B }[/math], [math]\displaystyle{ C }[/math], [math]\displaystyle{ D }[/math], and [math]\displaystyle{ F }[/math] in terms of the incoming amplitude [math]\displaystyle{ A }[/math].

Tunneling Probability

The tunneling probability is also called the transmission coefficient. It is written as

[math]\displaystyle{ T=\frac{\text{transmitted intensity}}{\text{incident intensity}} }[/math].

Since intensity is proportional to the squared amplitude,

[math]\displaystyle{ T=\frac{|F|^2}{|A|^2} }[/math].

For a rectangular barrier with [math]\displaystyle{ E\lt V_0 }[/math], the exact transmission coefficient is

[math]\displaystyle{ T=\left[1+\frac{V_0^2\sinh^2(\kappa L)}{4E(V_0-E)}\right]^{-1} }[/math].

For a thick or high barrier, where [math]\displaystyle{ \kappa L }[/math] is large, a useful approximation is

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math].

This approximation shows the most important idea:

[math]\displaystyle{ T }[/math] decreases exponentially as the barrier becomes wider or harder to penetrate.

Since the exponential term is [math]\displaystyle{ e^{-2\kappa L} }[/math], even a small increase in barrier width can cause a huge decrease in tunneling probability.

Computational Model

The model below is an original GlowScript/Trinket visualization of quantum tunneling through a rectangular potential barrier. It lets the user change the particle energy [math]\displaystyle{ E }[/math], barrier height [math]\displaystyle{ V_0 }[/math], and barrier width [math]\displaystyle{ L }[/math]. The model shows the oscillating wave before the barrier, the decaying wave inside the barrier, and the smaller transmitted wave after the barrier.

Original GlowScript/Trinket model: Quantum Tunneling through a Rectangular Potential Barrier

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If the embedded model does not load inside the wiki, open the linked Trinket model above in a new tab. This model was created for this page and is meant to show the main qualitative idea: the wavefunction does not instantly become zero inside the barrier. It decays, and the amount left on the far side depends strongly on the barrier height, barrier width, and particle energy.

The PhET quantum tunneling simulation below is also useful because it lets users compare plane waves and wave packets for different potential shapes.

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When using the simulations, try the following:

Set the particle energy below the barrier height.
Increase the barrier width and observe how the transmitted wave decreases.
Increase the barrier height and observe how the transmitted wave decreases.
Compare a plane wave with a wave packet.

Common Exam Mistakes

Thinking tunneling is classically allowed. If [math]\displaystyle{ E\lt V_0 }[/math], classical mechanics predicts no transmission. Tunneling is a quantum effect.

Forgetting that the probability drops exponentially. The factor [math]\displaystyle{ e^{-2\kappa L} }[/math] means that even a small increase in barrier width can make tunneling much less likely.

Mixing up [math]\displaystyle{ k }[/math] and [math]\displaystyle{ \kappa }[/math]. The symbol [math]\displaystyle{ k }[/math] describes the oscillating wave outside the barrier. The symbol [math]\displaystyle{ \kappa }[/math] describes the decaying wave inside the barrier.

Saying the particle climbs over the barrier. In tunneling, the particle is not following a classical path over the barrier. The wavefunction extends through the barrier.

Ignoring mass. Since [math]\displaystyle{ \kappa }[/math] depends on mass, heavier particles tunnel much less easily than lighter particles.

Examples

Simple

An electron has energy [math]\displaystyle{ 5.0\text{ eV} }[/math]. What is the approximate probability that it will tunnel through a potential barrier of height [math]\displaystyle{ 10.0\text{ eV} }[/math] and width [math]\displaystyle{ 1.00\text{ nm} }[/math]?

Solution

Given:

[math]\displaystyle{ E=5.0\text{ eV} }[/math]
[math]\displaystyle{ V_0=10.0\text{ eV} }[/math]
[math]\displaystyle{ L=1.00\text{ nm}=1.00\times10^{-9}\text{ m} }[/math]
[math]\displaystyle{ m_e=9.11\times10^{-31}\text{ kg} }[/math]
[math]\displaystyle{ \hbar=1.055\times10^{-34}\text{ J}\cdot\text{s} }[/math]
[math]\displaystyle{ 1\text{ eV}=1.602\times10^{-19}\text{ J} }[/math]

First calculate [math]\displaystyle{ \kappa }[/math]:

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math]

[math]\displaystyle{ \kappa= \frac{\sqrt{2(9.11\times10^{-31})(5.0)(1.602\times10^{-19})}}{1.055\times10^{-34}} }[/math]

[math]\displaystyle{ \kappa\approx1.15\times10^{10}\text{ m}^{-1} }[/math]

Now use the approximation:

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math]

[math]\displaystyle{ T\approx 16\left(\frac{5.0}{10.0}\right)\left(1-\frac{5.0}{10.0}\right)e^{-2(1.15\times10^{10})(1.00\times10^{-9})} }[/math]

[math]\displaystyle{ T\approx4.5\times10^{-10} }[/math]

So the tunneling probability is approximately

[math]\displaystyle{ \boxed{4.5\times10^{-10}} }[/math].

This is a very small probability, but it is not zero.

Middling

An electron with energy [math]\displaystyle{ 4.0\text{ eV} }[/math] approaches a rectangular barrier of height [math]\displaystyle{ 8.0\text{ eV} }[/math]. Compare the tunneling probability when the barrier width is [math]\displaystyle{ 0.50\text{ nm} }[/math] and when the barrier width is [math]\displaystyle{ 1.00\text{ nm} }[/math].

Solution

Given:

[math]\displaystyle{ E=4.0\text{ eV} }[/math]
[math]\displaystyle{ V_0=8.0\text{ eV} }[/math]
[math]\displaystyle{ V_0-E=4.0\text{ eV} }[/math]

Calculate [math]\displaystyle{ \kappa }[/math]:

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math]

[math]\displaystyle{ \kappa\approx1.02\times10^{10}\text{ m}^{-1} }[/math]

The approximate tunneling probability is

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math].

Since [math]\displaystyle{ E/V_0=4.0/8.0=0.50 }[/math],

[math]\displaystyle{ 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)=16(0.50)(0.50)=4 }[/math].

For [math]\displaystyle{ L=0.50\text{ nm} }[/math]:

[math]\displaystyle{ T\approx 4e^{-2(1.02\times10^{10})(0.50\times10^{-9})} }[/math]

[math]\displaystyle{ T\approx1.4\times10^{-4} }[/math].

For [math]\displaystyle{ L=1.00\text{ nm} }[/math]:

[math]\displaystyle{ T\approx 4e^{-2(1.02\times10^{10})(1.00\times10^{-9})} }[/math]

[math]\displaystyle{ T\approx5.0\times10^{-9} }[/math].

Doubling the width does not merely cut the probability in half. It reduces the tunneling probability by many orders of magnitude because the width appears inside an exponential.

Difficult

An electron with energy [math]\displaystyle{ 2.0\text{ eV} }[/math] approaches a rectangular barrier of height [math]\displaystyle{ 5.0\text{ eV} }[/math]. What barrier width gives an approximate tunneling probability of [math]\displaystyle{ 1.0\times10^{-6} }[/math]?

Solution

Given:

[math]\displaystyle{ E=2.0\text{ eV} }[/math]
[math]\displaystyle{ V_0=5.0\text{ eV} }[/math]
[math]\displaystyle{ T=1.0\times10^{-6} }[/math]
[math]\displaystyle{ V_0-E=3.0\text{ eV} }[/math]

Start with the approximate tunneling equation:

[math]\displaystyle{ T\approx 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right)e^{-2\kappa L} }[/math].

Calculate the coefficient:

[math]\displaystyle{ 16\frac{E}{V_0}\left(1-\frac{E}{V_0}\right) = 16\left(\frac{2.0}{5.0}\right)\left(1-\frac{2.0}{5.0}\right) }[/math]

[math]\displaystyle{ =16(0.40)(0.60)=3.84 }[/math].

So

[math]\displaystyle{ 1.0\times10^{-6}=3.84e^{-2\kappa L} }[/math].

Divide both sides by [math]\displaystyle{ 3.84 }[/math]:

[math]\displaystyle{ e^{-2\kappa L}=2.60\times10^{-7} }[/math].

Take the natural logarithm of both sides:

[math]\displaystyle{ -2\kappa L=\ln(2.60\times10^{-7}) }[/math]

[math]\displaystyle{ -2\kappa L\approx -15.16 }[/math].

Now solve for [math]\displaystyle{ L }[/math]:

[math]\displaystyle{ L=\frac{15.16}{2\kappa} }[/math].

Calculate [math]\displaystyle{ \kappa }[/math]:

[math]\displaystyle{ \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar} }[/math]

[math]\displaystyle{ \kappa= \frac{\sqrt{2(9.11\times10^{-31})(3.0)(1.602\times10^{-19})}}{1.055\times10^{-34}} }[/math]

[math]\displaystyle{ \kappa\approx8.87\times10^9\text{ m}^{-1} }[/math].

Therefore,

[math]\displaystyle{ L=\frac{15.16}{2(8.87\times10^9)} }[/math]

[math]\displaystyle{ L\approx8.54\times10^{-10}\text{ m} }[/math].

So the barrier width is approximately

[math]\displaystyle{ \boxed{0.854\text{ nm}} }[/math].

Applications

Nuclear Fusion in Stars

Quantum tunneling helps explain how nuclear fusion can occur inside stars. Protons repel each other because they are positively charged. Classically, many protons in a star's core would not have enough energy to overcome this electric repulsion. Quantum tunneling gives them a small probability of getting close enough for the strong nuclear force to bind them together.

Alpha Decay

Alpha decay occurs when an unstable nucleus emits an alpha particle. The alpha particle is trapped inside the nucleus by a potential barrier. Classically, it may not have enough energy to escape. Quantum mechanically, it can tunnel through the barrier and leave the nucleus.

Scanning Tunneling Microscopy

A scanning tunneling microscope, or STM, uses tunneling current to image surfaces at the atomic scale. When a sharp conducting tip is brought very close to a surface, electrons can tunnel between the tip and the surface. The tunneling current is extremely sensitive to distance, which allows the STM to detect tiny changes in surface height.

Semiconductor Devices

Some semiconductor devices rely on tunneling. Tunnel diodes, flash memory, and very small transistors can involve tunneling effects. As electronic devices become smaller, tunneling becomes more important because electrons interact with barriers only a few nanometers wide.

Connection to Matter and Interactions

In Matter and Interactions, energy is often used to decide what motion is possible. For a classical particle, if [math]\displaystyle{ E\lt V_0 }[/math], the particle cannot cross the barrier. Quantum tunneling is interesting because it shows where that classical energy reasoning stops being complete.

The particle still does not have enough classical energy to go over the barrier. The difference is that quantum mechanics describes the particle with a wavefunction. That wavefunction can extend into the barrier and sometimes reach the other side. So instead of asking only whether the particle has enough energy to cross, we ask what probability the wavefunction gives for finding the particle past the barrier.

Historical Background

Quantum tunneling was developed during the early years of quantum mechanics. Friedrich Hund discussed wavefunction penetration in molecular systems in the 1920s. George Gamow later applied tunneling to alpha decay, showing that quantum mechanics could explain how alpha particles escape from atomic nuclei. Tunneling soon became one of the clearest examples of how quantum mechanics differs from classical mechanics.

External Resources

References

MikeRun, Quantum-tunneling.svg, Wikimedia Commons. Licensed under Creative Commons Attribution-ShareAlike 4.0 International.
OpenStax, University Physics Volume 3, Chapter 7: Quantum Mechanics. Licensed under Creative Commons Attribution 4.0 International.
LibreTexts, Quantum Tunneling of Particles through Potential Barriers.
HyperPhysics, Barrier Penetration.
PhET Interactive Simulations, Quantum Tunneling and Wave Packets.
Original GlowScript/Trinket model by Haripratiik Arunkumar Malarkodi, Spring 2026.

Quantum Tunneling through Potential Barriers

Contents

Overview

Classical Expectations

Quantum Theory

Wave Behavior of Particles

Important Intuition

Heisenberg Uncertainty Principle

The Schrödinger Equation

Mathematical Model

Wave Equations

Boundary Conditions

Tunneling Probability

Computational Model

Common Exam Mistakes

Examples

Simple

Middling

Difficult

Applications

Nuclear Fusion in Stars

Alpha Decay

Scanning Tunneling Microscopy

Semiconductor Devices

Connection to Matter and Interactions

Historical Background

See Also

External Resources

References

Navigation menu

Quantum Tunneling through Potential Barriers

Overview

Classical Expectations

Quantum Theory

Wave Behavior of Particles

Important Intuition

Heisenberg Uncertainty Principle

The Schrödinger Equation

Mathematical Model

Wave Equations

Boundary Conditions

Tunneling Probability

Computational Model

Common Exam Mistakes

Examples

Simple

Middling

Difficult

Applications

Nuclear Fusion in Stars

Alpha Decay

Scanning Tunneling Microscopy

Semiconductor Devices

Connection to Matter and Interactions

Historical Background

See Also

External Resources

References

Navigation menu

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