Quantum Tunneling Lab

Investigate how quantum particles penetrate energy barriers that classical physics forbids. Adjust particle energy, barrier height, and barrier width to see how the transmission coefficient changes, and visualize the wave function across incident, barrier, and transmitted regions.

Guided Experiment: Effect of Barrier Width

Hypothesis

Setup

Run Experiment

Analyze

Conclude

How does increasing the barrier width affect the transmission probability when the particle energy is below the barrier height? What mathematical relationship do you expect?

Write your hypothesis in the Lab Report panel, then click Next.

Controls

View

Presets

Particle Energy (E)3.0 eV

0.120 eV

Barrier Height (V₀)5.0 eV

0.540 eV

Barrier Width (L)1.0 nm

0.120 nm

Results

Tunneling (E < V₀)

Transmission (T)

0.204061

20.406%

Reflection (R)

0.795939

79.594%

Wave number (k)1.7321 nm⁻¹

Decay constant (κ)1.4142 nm⁻¹

Approximate T0.059106

T + R check1.0000000000

Key Formulas

Data Table

(0 rows)

#	E (eV)	V₀ (eV)	L (nm)	T	R	κ (nm⁻¹)	Regime

Hypothesis

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Observations

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Conclusions

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Reference Guide

Quantum Tunneling Basics

In classical physics, a particle with energy E cannot pass through a potential barrier of height V₀ when E < V₀. Quantum mechanics tells a different story. The particle's wave function does not abruptly stop at the barrier but instead decays exponentially inside it. If the barrier is thin enough, a non-zero portion of the wave function emerges on the other side.

The probability of transmission depends on three factors: particle energy, barrier height, and barrier width. Even when tunneling is extremely unlikely (such as in alpha decay), the sheer number of attempts per second means that tunneling events still occur at measurable rates.

Transmission Coefficient

The exact transmission coefficient for a rectangular barrier in the tunneling regime (E < V₀) is given by:

T = \frac{1}{1 + \frac{V_0^2 \sinh^2(\kappa L)}{4E(V_0 - E)}}

where the decay constant is $\kappa = \sqrt{2m(V_0 - E)/\hbar^2}$ . For thick barriers, this simplifies to the well-known approximation $T \approx e^{-2\kappa L}$ , showing the exponential sensitivity to barrier width.

Wave Function Behavior

The wave function divides into three regions. In Region I (before the barrier), the incident wave travels right and a reflected wave travels left, producing a standing-wave pattern with |ψ|² > 1 at some points.

Inside the barrier (Region II), the wave function decays exponentially for E < V₀ or oscillates with a different wavelength for E > V₀. In Region III (after the barrier), a pure transmitted wave travels right with reduced amplitude |F|² = T.

\psi(x) = \begin{cases} Ae^{ik_1 x} + Be^{-ik_1 x} & x < 0 \\ Ce^{-\kappa x} + De^{\kappa x} & 0 \le x \le L \\ Fe^{ik_1 x} & x > L \end{cases}

Applications

Scanning Tunneling Microscope (STM) — A sharp tip held nanometers from a surface detects tunneling current that is exponentially sensitive to the tip-surface gap. Moving the tip across the surface maps out individual atoms.

Alpha Decay — An alpha particle inside a nucleus faces a Coulomb barrier tens of MeV high. Despite having energy far below the barrier, the particle tunnels out at a rate that explains radioactive half-lives from microseconds to billions of years.

Semiconductor Devices — Tunnel diodes, flash memory, and quantum-well structures all rely on electrons tunneling through thin potential barriers, enabling technologies from fast switching to non-volatile data storage.