Quantum Tunneling through Potential Barriers

Page Claimed By: Joshua Kim Fall '25

Quantum tunneling is a fundamental effect in quantum mechanics in which a particle has a nonzero probability of appearing on the far side of a potential barrier, even when its classical energy is insufficient to cross that barrier. This behavior arises from the wave-like nature of matter and is described mathematically through the Schrodinger equation and related quantum principles, such as the Heisenberg uncertainty principle.

Overview

In classical physics, a particle with energy lower than a barrier height cannot enter the barrier at all. Quantum mechanics, however, treats particles as wavefunctions rather than solid objects. These wavefunctions can extend into and beyond regions that would be forbidden in classical mechanics, giving rise to a small--but finite--chance that the particle will emerge on the opposite side. A typical tunneling scenario involves a finite potential barrier of Height V_0 and width L. When a particle of energy E < V_0 approaches the barrier, part of its wavefunction decays inside the barrier yet continues far enough to "leak through", creating a transmitted wave.

Classical Expectations

Under classical mechanics, the criterion for overcoming a barrier is simple: the particle's energy must exceed the potential energy of the barrier. If E < V_0, the particle should reflect entirely. No motion is predicted within the barrier, and transmission probability is strictly zero.

Quantum Theory

Wave Behavior of Particles

Quantum mechanics models particles with wavefunctions that satisfy the time-independent Schrodinger equation. These wavefunctions can be decomposed into incident, reflected, and transmitted components. Because waves generally do not vanish abruptly at boundaries, the wavefunction must extend into the barrier. Within this region, the wave amplitude decays exponentially. When the decaying wave reaches the far side of the barrier, a small fraction survives and becomes a traveling wave in the output region. This survival mechanism produces tunneling.

Heisenberg Uncertainty Principle

A qualitative way to appreciate tunneling is through the energy-time variant of the Heisenberg uncertainty principle. Over extremely short time intervals, a particle's energy cannot be known exactly. This uncertainty permits temporary fluctuations that may allow the particle to explore regions classically inaccessible to it. While not a precise calculation tool, this principle offers intuitive support for why penetration into forbidden regions is possible.

The Schrödinger Equation

The Schrödinger equation tells us that a wave function must be continuous at each boundary it encounters, and the derivative of the wave function must also be continuous except when the boundary height is infinite. Applying this to a finite potential barrier, we can find the probability a particle can tunnel through a potential barrier. Notably, the wave functions and calculations most resemble the solution for a single particle in a semi-infinite well when [math]\displaystyle{ V_0 \gt E }[/math] in regions II and III.

Mathematical Model

Beyond the conceptual understanding of how quantum tunneling works, calculations can be made to determine the probability a wave function can tunnel through a specific finite potential barrier. This is known as transmission probability and to understand this equation, we must look at the wave equations associated with the potential barrier.

Wave Equations

The time independent Schrodinger equation is given by the following equation.

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

As mentioned before, the finite potential barrier creates three different regions with their own unique wave equations. Let's assume that the potential barrier is from [math]\displaystyle{ 0 \le x \le L }[/math]. Given that, we can define the potential as follows:

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

Thus, we can write the Schrodinger equations for the three regions.

• Region I : [math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} = E \psi(x) }[/math]

• Region II : [math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} + V_0 \psi (x) = E \psi(x) }[/math]

• Region III : [math]\displaystyle{ -\frac{\hbar^2}{2m}\frac{d^2 \psi(x)}{dx^2} = E \psi(x) }[/math]

Boundary conditions require that the wave functions and their derivatives are continuous on each boundary i.e [math]\displaystyle{ x = 0 }[/math] and [math]\displaystyle{ x = L }[/math]. The general solution for each region,

• Region I: [math]\displaystyle{ \psi(x)_I = Ae^{ikx} + Be^{-ikx} }[/math]

• Region II: [math]\displaystyle{ \psi(x)_{II} = Ce^{-\alpha x} + De^{\alpha x } }[/math]

• Region III: [math]\displaystyle{ \psi(x)_{III} = Fe^{ikx} + Ge^{-ikx} }[/math]

where [math]\displaystyle{ \alpha = \sqrt {\frac{2m}{\hbar^2} (V_0 - E)} }[/math] and [math]\displaystyle{ k = \sqrt {\frac{2m}{\hbar^2} E} }[/math].

There is only one direction in which the wave in region III is traveling ([math]\displaystyle{ +\infty }[/math]). Thus the wave functions are now,

• Region I: [math]\displaystyle{ \psi(x)_I = Ae^{ikx} + Be^{-ikx} }[/math]

• Region II: [math]\displaystyle{ \psi(x)_{II} = Ce^{-\alpha x} + De^{\alpha x } }[/math]

• Region III: [math]\displaystyle{ \psi(x)_{III} = Fe^{ikx} }[/math]

It is important to note that the constants A, B, C, D, and F refer to the amplitude of the five waves in this problem:

• A: incident wave at [math]\displaystyle{ x = 0 }[/math]
• B: reflected wave at [math]\displaystyle{ x = 0 }[/math]
• C: incident wave at [math]\displaystyle{ x = L }[/math]
• D: reflected wave at [math]\displaystyle{ x = L }[/math]
• F: transmitted wave

Tunneling Probability

The probability that a particle can tunnel through a potential barrier utilizes the wave functions discussed in the previous section. Specifically, tunneling probability is the ratio of the transmitted wave intensity [math]\displaystyle{ |F|^2 }[/math] and the incident wave intensity [math]\displaystyle{ |A|^2 }[/math]. To solve for these constants, we must apply boundary conditions.

At [math]\displaystyle{ x = 0 }[/math], we know region I and II must agree and create a continuous wave function.

[math]\displaystyle{ Ae^{ikx} + Be^{-ikx} = Ce^{-\alpha x } + De^{\alpha x} }[/math]

[math]\displaystyle{ \frac{d}{dx}(Ae^{ikx} + Be^{-ikx} = Ce^{-\alpha x } + De^{\alpha x}) }[/math]

[math]\displaystyle{ ik(Ae^{ikx} - Be^{-ikx}) = \alpha(-Ce^{-\alpha x } + De^{\alpha x}) }[/math]

The same applies at [math]\displaystyle{ x = L }[/math].

[math]\displaystyle{ Ce^{-\alpha x } + De^{\alpha x} = Fe^{ikx} }[/math]

[math]\displaystyle{ \frac{d}{dx}(Ce^{-\alpha x } + De^{\alpha x} = Fe^{ikx}) }[/math]

[math]\displaystyle{ \alpha(-Ce^{-\alpha x } + De^{\alpha x}) = ikFe^{ikx} }[/math]

After some lengthy algebra and calculations, we know that the ratio of the transmitted wave amplitude and incident wave amplitude is this messy expression.

[math]\displaystyle{ \frac{F}{A} = \frac{e^{-ikL}}{cosh(\alpha L) +i(\gamma /2)sinh(\alpha L)} }[/math][1]

where [math]\displaystyle{ \gamma = \frac{\alpha}{k} - \frac{k}{\alpha} }[/math].

We know that tunneling probability is a ratio of intensity. So when we multiply the ratio of amplitudes by the conjugate, we get the following equation for tunneling probability.

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

where, [math]\displaystyle{ k = \sqrt {\frac{2m(V_0-E)}{\hbar^2}} }[/math] and L is the width of the barrier.

Computational Model

The following simulation illustrates a wave function for various potentials in both plane wave and wave packet form.

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

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

</iframe> [2]

Examples

Simple

A particle has 5.0 eV of energy. What is the probability that it will tunnel through a potential barrier of 10.0 eV with a width/thickness of 1.00 nm?

Solution

First, let's define the given variables and equation we must use

• [math]\displaystyle{ E = 5.0 eV }[/math]
• [math]\displaystyle{ V_0 = 10.0 eV }[/math]
• [math]\displaystyle{ L = 1.00 nm = 1.00 \times 10^{-9} m }[/math]

• [math]\displaystyle{ T \approx 16\frac {E}{V_0} (1-\frac{E}{V_0})e^{-2kL} }[/math]
• [math]\displaystyle{ k = \sqrt{\frac {2m(V_0 -E)}{\hbar^2}} }[/math]

Next, plug in our given values into the equation and solve.

[math]\displaystyle{ k = \sqrt {\frac {2(9.11 \times 10^{-31} kg)(10.0-5.0 eV)(1.602 \times 10^{-19} J)}{(1.055 \times 10^{-34})^2}} = 1.145 \times 10^{10} m^{-1} }[/math]

[math]\displaystyle{ T \approx 16\frac {5.0}{10.0} (1-\frac{5.0}{10.0})e^{-2(1.145 \times 10^{10} m^{-1})(1.00 \times 10^{-9})} = 4.52 \times 10^{-10} }[/math]

Applications

Nuclear Fusion in Stars

Quantum tunneling enables protons to overcome their mutual electrostatic repulsion, allowing fusion reactions to occur at stellar core temperatures.

Scanning Tunneling Microscopy (STM)

Tunneling current between a sharp metallic tip and a surface allows imaging of materials with atomic resolution.

Semiconductor Devices

Tunnel diodes and modern transistors exploit controlled tunneling currents.

Historical Background

The notion of barrier penetration was introduced in the late 1920s by physicist Friedrich Hund, who noticed that quantum wavefunctions could leak between adjacent molecular wells. Soon after, George Gamow applied tunneling to explain alpha decay, firmly establishing it as a central quantum phenomenon.

See Also

Heisenberg Uncertainty Principle

Wave-Particle Duality

Solution for a Single Particle in a Semi-Infinite Quantum Well

Application of Statistics in Physics

Related Topics

• Heisenberg Uncertainty Principle
• Wave-Particle Duality
• Schrodinger Equation
• Quantum Wells and Bound States

External Resources

• [3]
• [4]
• [5]

References

• https://openstax.org/details/books/university-physics-volume-3
• http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
• https://physicstoday.scitation.org/doi/10.1063/1.1510281
• https://phet.colorado.edu/en/simulations/quantum-tunneling