Quantum Tunneling Simulator

Visualize a particle's wave function encountering a potential energy barrier. Adjust energy, barrier height, and width to see how much of the wave transmits through and how much reflects back. The exact transmission coefficient is calculated, along with step-by-step math and an animated time-dependent wave.

Wave Function Visualization

Controls

Presets

Particle Energy (E)3.00 eV

Barrier Height (V₀)5.00 eV

Barrier Width (L)1.00 nm

Show |ψ|² probability densityShow classical comparison

Results

Regime:Tunneling (E < V₀)

Transmission (T)

20.41%

0.204061

Reflection (R)

79.59%

0.795939

Wave Number (k)

1.73205 nm\u207B\u00B9

Decay Constant (κ)

1.41421 nm\u207B\u00B9

Approximate T (thick barrier)

0.0591057

Step-by-Step

\kappa = \sqrt{V_0 - E} = \sqrt{5.00000 - 3.00000} = 1.41421 \text{ nm}^{-1}

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

T = 0.204061

R = 1 - T = 0.795939

T_{\text{approx}} = e^{-2\kappa L} = e^{-2(1.41421)(1.00000)} = 0.0591057

Reference Guide

Quantum Tunneling

In quantum mechanics, a particle can pass through a potential barrier even when its energy is less than the barrier height. This has no classical analogue and arises because the wave function does not drop to zero inside the barrier but instead decays exponentially.

The probability of tunneling depends on the barrier height, width, and the particle's energy. Thinner and lower barriers allow more transmission.

Wave-Particle Duality

Particles like electrons behave as waves described by the Schrödinger equation. The wave function $\psi(x)$ encodes the probability amplitude of finding the particle at position x.

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi

The probability density $|\psi|^2$ tells us where the particle is likely to be found. Inside a barrier, $|\psi|^2$ decays but never reaches zero, allowing tunneling.

Transmission Coefficient

For a rectangular barrier of height $V_0$ and width $L$ , the exact transmission coefficient when $E < V_0$ is

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

where $\kappa = \sqrt{2m(V_0 - E)}/\hbar$ . For thick barriers, this simplifies to the WKB approximation $T \approx e^{-2\kappa L}$ . Conservation requires $T + R = 1$ .

Applications

Scanning Tunneling Microscope (STM)

A sharp tip is brought within ~1 nm of a surface. Electrons tunnel across the vacuum gap, and the tunneling current is exponentially sensitive to the tip-surface distance, enabling atomic-resolution imaging.

Nuclear Alpha Decay

Alpha particles are trapped inside nuclei by the nuclear potential. They tunnel through the Coulomb barrier, which explains why decay rates vary enormously with small changes in energy (Geiger-Nuttall law).

Semiconductor Devices

Tunnel diodes and flash memory rely on electrons tunneling through thin insulating barriers. The tunnel junction is a fundamental building block in modern electronics.