Quantum Tunneling through Potential Barriers

From Physics Book
Revision as of 12:25, 27 April 2026 by Hari (talk | contribs)
Jump to navigation Jump to search

Page Claimed By: Haripratiik Arunkumar Malarkodi Spring '26

Quantum tunneling is a fundamental effect in quantum mechanics in which a particle has a nonzero probability of passing through a potential barrier even when its classical energy is too small to go over the barrier. In classical mechanics, this would be impossible. In quantum mechanics, particles are described by wavefunctions, and those wavefunctions can extend into regions that would be classically forbidden.

Quantum tunneling is important because it explains physical processes that classical physics cannot explain, including alpha decay, nuclear fusion in stars, scanning tunneling microscopy, and tunneling effects in semiconductor devices.

Overview

In classical physics, a particle with energy lower than a barrier height cannot cross that barrier. For example, if a ball does not have enough kinetic energy to roll over a hill, it rolls back down. Quantum particles behave differently because they are represented by wavefunctions rather than by perfectly localized objects.

A typical tunneling scenario involves a finite potential barrier with height [math]\displaystyle{ V_0 }[/math] and width [math]\displaystyle{ L }[/math]. If a particle with energy [math]\displaystyle{ E }[/math] approaches the barrier and [math]\displaystyle{ E \lt V_0 }[/math], classical mechanics predicts total reflection. Quantum mechanics predicts that part of the particle's wavefunction decays inside the barrier. If the barrier is not too wide, a small portion of the wavefunction can survive to the far side and become a transmitted wave.

In simple terms, tunneling does not mean that the particle "borrows energy" in a literal way. The more precise explanation is that the particle's wavefunction does not instantly become zero inside the barrier. Instead, it decreases exponentially. If enough of the wavefunction remains after crossing the barrier region, there is a nonzero probability of detecting the particle on the other side.

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 given by

[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

Tunneling probability depends strongly on three main quantities:

  • Barrier width [math]\displaystyle{ L }[/math]: Wider barriers greatly reduce tunneling.
  • Barrier height [math]\displaystyle{ V_0 }[/math]: Taller barriers reduce tunneling when the particle energy is fixed.
  • Particle mass [math]\displaystyle{ m }[/math]: Heavier particles tunnel much less easily than lighter particles.

This is why tunneling is common for electrons and other microscopic particles, but not noticeable for everyday objects.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle is sometimes used as a qualitative way to think about tunneling, but it should not be treated as the main calculation method. Tunneling is not best understood as a particle simply "borrowing energy" and then paying it back. The more accurate explanation comes from solving the Schrödinger equation with the correct boundary conditions.

The uncertainty principle is still useful conceptually because it reminds us that quantum particles are not perfectly localized objects with perfectly definite classical trajectories.

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{ \psi_I(0)=\psi_{II}(0) }[/math]

[math]\displaystyle{ \frac{d\psi_I}{dx}\bigg|_{x=0}=\frac{d\psi_{II}}{dx}\bigg|_{x=0} }[/math]

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

[math]\displaystyle{ \psi_{II}(L)=\psi_{III}(L) }[/math]

[math]\displaystyle{ \frac{d\psi_{II}}{dx}\bigg|_{x=L}=\frac{d\psi_{III}}{dx}\bigg|_{x=L} }[/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].

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 PhET quantum tunneling simulation lets users adjust the energy of a particle and the shape of the potential barrier. This is useful because tunneling is easier to understand when the wavefunction can be seen visually.

<iframe src="https://phet.colorado.edu/sims/cheerpj/quantum-tunneling/latest/quantum-tunneling.html?simulation=quantum-tunneling" 

       width="800"
       height="600"
       allowfullscreen>

</iframe>

When using the simulation, 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

  • Using classical reasoning only. In classical physics, [math]\displaystyle{ E\lt V_0 }[/math] means zero transmission. In quantum mechanics, the transmission probability can be small but nonzero.
  • Forgetting the exponential dependence. Tunneling probability is extremely sensitive to barrier width because of the factor [math]\displaystyle{ e^{-2\kappa L} }[/math].
  • Confusing [math]\displaystyle{ k }[/math] and [math]\displaystyle{ \kappa }[/math]. The quantity [math]\displaystyle{ k }[/math] describes oscillation outside the barrier. The quantity [math]\displaystyle{ \kappa }[/math] describes exponential decay inside the barrier.
  • Thinking tunneling means the particle literally climbs over the barrier. The particle is not following a classical path over the barrier. The wavefunction extends through the barrier.
  • Ignoring mass. Heavier particles have larger [math]\displaystyle{ \kappa }[/math], so they tunnel much less easily.

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\approx8.55\times10^{-10}\text{ m} }[/math].

So the barrier width is approximately

[math]\displaystyle{ \boxed{0.855\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, many problems focus on the relationship between energy, force, and motion. Quantum tunneling is a case where the classical energy model is not enough by itself. Classically, a particle with [math]\displaystyle{ E\lt V_0 }[/math] cannot pass the barrier. Quantum mechanically, the wavefunction still exists inside the barrier and may continue to the far side.

This makes tunneling a useful example of how quantum mechanics changes the meaning of motion. Instead of asking only whether a particle has enough energy to cross a barrier, we ask what probability the wavefunction gives for finding the particle beyond 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.

See Also

External Resources

References

  • OpenStax, University Physics Volume 3, Chapter 7: Quantum Mechanics.
  • LibreTexts, Quantum Tunneling of Particles through Potential Barriers.
  • HyperPhysics, Barrier Penetration.
  • PhET Interactive Simulations, Quantum Tunneling and Wave Packets.
Retrieved from "http://www.physicsbook.gatech.edu/index.php?title=Quantum_Tunneling_through_Potential_Barriers&oldid=48173"