Quantum Tunneling: When Particles Walk Through Walls

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Here is something that would make any classical physicist scratch their head: imagine throwing a ball at a wall and having it occasionally appear on the other side, without making a hole. In classical physics, that is impossible. In quantum mechanics, it happens all the time.

This phenomenon, called quantum tunneling, lets particles pass through energy barriers they classically should not be able to overcome. It is not science fiction: it is happening right now in the Sun (enabling nuclear fusion), in your flash drive (allowing data storage), and potentially in your body (in enzyme reactions).

What makes quantum tunneling work is the wave nature of matter. When you solve Schrodinger's equation for a particle approaching a potential barrier, the wave function does not simply bounce off. Instead, it penetrates into the barrier region, decaying exponentially. If the barrier is thin enough, some of that wave function amplitude makes it through to the other side. And because the wave function squared gives probability, there is a non-zero chance of finding the particle beyond the barrier.

How to Use This Simulation

Here's what happens when you actually try this: set the particle energy below the barrier height and watch the wave packet approach. Most of it bounces back (the red reflected wave), but some amplitude leaks through to the other side (the green transmitted wave). That leakage is quantum tunneling.

Main Controls

Presets

Barrier Parameters

If you could slow this down enough to watch the wave function inside the barrier, you'd see exponential decay: the amplitude falls off as e^(-kappa x). Double the width, and transmission doesn't halve, it squares.

Particle Parameters

Results Display

Bottom Tabs

Keyboard Shortcuts

Tips for Exploration

1. Start with the Thin preset: Notice how even with E < V0, significant transmission occurs. This is the essence of tunneling.

2. Double the barrier width: Watch transmission plummet. The relationship is exponential, not linear. A 2 nm barrier transmits roughly the square of what a 1 nm barrier does.

3. Increase energy toward V0: As E approaches V0, kappa approaches zero and tunneling probability increases dramatically. What happens when E exceeds V0?

4. Try the Alpha Decay preset: Set mass to 7294 (alpha particle relative to electron). See why nuclear decay takes billions of years: massive particles tunnel with vanishingly small probability.

5. Use the Export Report button: Generate a full analysis with your parameters. Useful for comparing different configurations or documenting your exploration.

The Physics of Quantum Tunneling

The Rectangular Barrier Problem

Consider a particle with energy E approaching a rectangular potential barrier of height V0 and width d. Classically, if E < V0, the particle bounces back every time. Quantum mechanically, things get interesting.

The wave vector k and decay constant kappa are given by:

Outside barrier: k = sqrt(2mE) / hbar

Inside barrier (E < V0): kappa = sqrt(2m(V0 - E)) / hbar

Transmission Coefficient

For a thick barrier (kappa times d >> 1), the transmission coefficient simplifies to:

T = exp(-2 kappa d)

This exponential dependence is crucial. Doubling the barrier width does not halve the transmission; it squares it. A 1 nm barrier might have 10% transmission, but a 2 nm barrier of the same height would have roughly 1%.

The exact formula for a rectangular barrier is:

T = [1 + (V0^2 sinh^2(kappa d)) / (4E(V0 - E))]^(-1)

Key Parameters

The mass dependence explains why electrons tunnel readily but protons and heavier particles do not. An alpha particle is about 7,300 times heavier than an electron, making its tunneling probability vastly smaller for the same barrier.

Exploration Activities

Activity 1: Barrier Width Dependence

Objective: Verify the exponential dependence of transmission on barrier width.

Steps:

1. Set V0 = 5 eV, E = 3 eV, m = 1 m_e
2. Record transmission T for d = 0.5, 1.0, 1.5, 2.0 nm
3. Plot ln(T) vs d
4. Verify the slope is approximately -2 kappa
5. Calculate kappa from the slope and compare to the theoretical value

Activity 2: Energy Threshold Behavior

Objective: Observe the transition from tunneling to classical transmission.

Steps:

1. Set V0 = 5 eV, d = 1 nm, m = 1 m_e
2. Start with E = 1 eV and increase gradually
3. Note how T increases as E approaches V0
4. What happens when E slightly exceeds V0?
5. Continue increasing E and observe transmission oscillations

Activity 3: Mass Scaling

Objective: Understand why heavy particles do not tunnel effectively.

Steps:

1. Set V0 = 5 eV, d = 1 nm, E = 3 eV
2. Start with m = 1 (electron) and note T
3. Increase to m = 4 (helium-4 equivalent) and observe
4. Try m = 7294 (alpha particle) to see why alpha decay timescales are long
5. Calculate the ratio of transmission coefficients

Activity 4: STM Sensitivity

Objective: Explore why STM achieves atomic resolution.

Steps:

1. Set parameters typical for STM: V0 = 4 eV, E near 0
2. Vary d from 0.5 nm to 1.5 nm in 0.1 nm steps
3. Calculate how much the tunneling current changes
4. Verify that a 0.1 nm gap change causes approximately 10x current change
5. This extreme sensitivity enables atomic-resolution imaging

Real-World Applications

Scanning Tunneling Microscope (STM)

The STM, invented by Binnig and Rohrer in 1981 (Nobel Prize 1986), exploits quantum tunneling to image surfaces with atomic resolution. A sharp metal tip is brought within about 1 nm of a conductive surface. When a voltage is applied, electrons tunnel between the tip and sample.

The tunneling current I follows:

I proportional to exp(-2 kappa z)

where z is the tip-surface distance. For typical work functions (~4 eV), a 0.1 nm change in z causes roughly a factor of 10 change in current. This extreme sensitivity allows detection of individual atoms.

Alpha Decay

Radioactive alpha decay puzzled physicists for years. How could alpha particles escape from nuclei when they did not have enough energy to overcome the Coulomb barrier? The answer is quantum tunneling.

George Gamow showed in 1928 that tunneling explains alpha decay rates. The longer the tunneling distance (for heavier nuclei), the lower the probability, explaining the enormous range of half-lives from microseconds to billions of years.

Tunnel Diode

Tunnel diodes (Esaki diodes, Nobel Prize 1973) use heavily doped p-n junctions to create a very thin depletion region. Electrons can tunnel through this thin barrier.

The key feature is "negative differential resistance": there is a voltage range where increasing voltage decreases current. Applications include high-frequency oscillators and fast switching circuits.

Nuclear Fusion in Stars

The Sun's core temperature (~15 million K) gives protons kinetic energies around 1 keV. But the Coulomb barrier between two protons is about 550 keV. Classical physics says fusion should not happen. Yet the Sun has been fusing hydrogen for 4.6 billion years.

Quantum tunneling makes stellar fusion possible. Even though individual tunneling probabilities are tiny, there are so many protons colliding so frequently that fusion occurs at the rate needed to power stars.

Flash Memory

Modern flash memory stores data by trapping electrons on a floating gate surrounded by insulating oxide. Writing and erasing data involves electrons tunneling through this oxide barrier, called Fowler-Nordheim tunneling.

Physical Constants Reference

Challenge Questions

Level 1: Conceptual

1. Why does transmission probability depend exponentially on barrier width rather than linearly?

2. If you double the particle mass while keeping all other parameters the same, what happens to kappa and the transmission coefficient?

Level 2: Analytical

1. An electron approaches a 3 eV barrier with energy 2 eV. If the barrier is 0.5 nm wide, calculate the transmission coefficient using the thick-barrier approximation.

2. For an STM with a 4 eV work function, calculate how much the tunneling current changes when the tip-sample distance increases from 0.8 nm to 0.9 nm.

Level 3: Advanced

1. The half-life of Po-212 alpha decay is 0.3 microseconds, while U-238 is 4.5 billion years. Using tunneling theory, explain this enormous range.

2. Calculate the temperature at which thermal energy kT equals 1 meV. Why is this relevant for understanding when classical vs quantum effects dominate?

3. In flash memory, the oxide barrier is about 8 nm thick with an effective height of 3.2 eV. At what electric field strength does significant Fowler-Nordheim tunneling begin?

Common Misconceptions

Frequently Asked Questions

Why can I not tunnel through walls in everyday life?

The transmission probability depends on exp(-2 kappa d). For a person trying to tunnel through a wall (d ~ 0.1 m), the probability is approximately 10^(-10^30). Even if you ran into walls a billion times per second for the age of the universe, you would never tunnel through [1].

Does the particle actually travel through the barrier?

This is a subtle question that touches on interpretation of quantum mechanics. The wave function exists inside the barrier with exponentially decreasing amplitude, but we never detect the particle inside a barrier with E < V0. Whether it "traveled through" depends on your interpretation [2].

How fast does tunneling happen?

The "tunneling time" is controversial. Different theoretical approaches give different answers. Experiments show that the process is very fast, perhaps instantaneous, but this does not allow faster-than-light signaling [3].

Why does the Sun need tunneling to fuse hydrogen?

At the Sun's core temperature, protons have kinetic energies around 1 keV. The Coulomb barrier between two protons is about 550 keV. Without tunneling, fusion would be impossibly rare [4].

Can tunneling be turned off?

No, tunneling is a fundamental consequence of quantum mechanics. However, making barriers thicker, higher, or using heavier particles effectively suppresses tunneling to negligible levels [5].

References

1. Griffiths, D.J. "Introduction to Quantum Mechanics," 3rd Edition, Cambridge University Press, 2018. Chapter 2.

2. MIT OpenCourseWare 8.04: Quantum Physics I. Available at: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/

3. Razavy, M. "Quantum Theory of Tunneling," 2nd Edition, World Scientific, 2014.

4. Binnig, G. and Rohrer, H. "Scanning Tunneling Microscopy," IBM Journal of Research and Development, 30(4), 355-369, 1986.

5. Gamow, G. "Zur Quantentheorie des Atomkernes," Zeitschrift fur Physik, 51, 204-212, 1928.

6. NIST: Fundamental Physical Constants. Available at: https://physics.nist.gov/cuu/Constants/

7. HyperPhysics: Quantum Tunneling. Georgia State University. Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html

8. The Feynman Lectures on Physics: Quantum Mechanics, Volume III. Available at: https://www.feynmanlectures.caltech.edu/III_toc.html

9. Esaki, L. "New Phenomenon in Narrow Germanium p-n Junctions," Physical Review, 109, 603, 1958.

10. OpenStax University Physics: Quantum Tunneling. Available at: https://openstax.org/books/university-physics-volume-3/pages/7-6-the-quantum-tunneling-of-particles-through-potential-barriers

About the Data

Physical constants used in this simulation are from NIST CODATA 2022 values:

• Reduced Planck constant: hbar = 1.054571817 x 10^-34 J s
• Electron mass: m_e = 9.1093837015 x 10^-31 kg
• Elementary charge: e = 1.602176634 x 10^-19 C

The transmission coefficient calculation uses the exact analytical formula for a rectangular potential barrier, derived from matching wave function amplitudes and derivatives at the barrier boundaries.

How to Cite

Simulations4All Team. "Quantum Tunneling Simulator: Wave Packet Propagation Through Potential Barriers." Simulations4All, 2025. https://simulations4all.com/simulations/quantum-tunneling-simulator

For academic use:

@misc{simulations4all_quantum_tunneling, title = {Quantum Tunneling Simulator}, author = {Simulations4All Team}, year = {2025}, url = {https://simulations4all.com/simulations/quantum-tunneling-simulator} }

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