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Quantum Tunneling Calculator

Calculate tunneling probabilities and transmission coefficients for quantum particles

Parameters

eVⓘ
mⓘ
eVⓘ
kgⓘ
Show Trail

Controls

xⓘ

Calculated Values

Transmission Coefficient:
0.00;0.00;
Reflection Coefficient:
1.00;1.00;
Decay Constant:
36226262844.04;m−136226262844.04;m⁻¹
Tunneling Length:
0.00;m0.00;m
Approximate T:
0.00;0.00;

Examples

Example 1: Electron Tunneling

An electron with 50 eV energy tunneling through a 100 eV, 1 nm barrier.

  • Transmission Coefficient: 0.140.14
  • Reflection Coefficient: 0.860.86
  • Decay Constant: 11500000000.0011500000000.00
  • Tunneling Length: 0.000.00

Example 2: Alpha Particle Decay

An alpha particle with 5 MeV energy tunneling through a 20 MeV, 10 fm barrier.

  • Transmission Coefficient: 0.000.00
  • Reflection Coefficient: 1.001.00
  • Decay Constant: 1730000000000000.001730000000000000.00
  • Tunneling Length: 0.000.00

Example 3: Proton Tunneling

A proton with 1 keV energy tunneling through a 2 keV, 0.1 nm barrier.

  • Transmission Coefficient: 0.040.04
  • Reflection Coefficient: 0.960.96
  • Decay Constant: 231000000000.00231000000000.00
  • Tunneling Length: 0.000.00

Visualization

Quantum Tunneling

Quantum tunneling is a fundamental quantum mechanical phenomenon where particles can pass through potential energy barriers that would be impenetrable according to classical physics. This effect is a direct consequence of the wave-like nature of matter and the uncertainty principle.

The tunneling probability depends on the barrier height, barrier width, particle energy, and particle mass. For a rectangular barrier, the transmission coefficient T is approximately given by T ≈ e^(-2αL), where α = √(2m(V₀-E))/ℏ is the decay constant, L is the barrier width, V₀ is the barrier height, and E is the particle energy.

Key factors affecting tunneling include: (1) Barrier height - higher barriers reduce tunneling probability exponentially; (2) Barrier width - wider barriers dramatically reduce tunneling; (3) Particle energy - higher energy increases tunneling probability; (4) Particle mass - lighter particles tunnel more easily.

Quantum tunneling has numerous applications in physics and technology, including: (1) Nuclear fusion in stars (proton-proton fusion); (2) Scanning tunneling microscopy (STM) for atomic imaging; (3) Tunnel diodes in electronics; (4) Alpha decay in radioactive nuclei; (5) Quantum computing with superconducting qubits.

The phenomenon demonstrates fundamental quantum concepts like wave-particle duality, the uncertainty principle, and the probabilistic nature of quantum mechanics. It's impossible in classical physics but essential for understanding many quantum systems.

Key Concepts

  • Tunneling Probability: T ≈ e^(-2αL) - exponential decay with barrier width
  • Decay Constant: α = √(2m(V₀-E))/ℏ - determines tunneling rate
  • Transmission Coefficient: T - probability of particle passing through barrier
  • Reflection Coefficient: R = 1 - T - probability of particle being reflected
  • Classically Forbidden: E < V₀ - region where classical physics predicts no transmission
  • Wave Function: Exponentially decaying inside barrier

Real-World Applications

  • Nuclear Physics: Alpha decay and nuclear fusion
  • Solid State Physics: Tunnel diodes and quantum dots
  • Surface Science: Scanning tunneling microscopy
  • Quantum Computing: Superconducting qubits
  • Molecular Biology: Enzyme catalysis and electron transfer

Explore Further

More quantum mechanics tools

Physics Equations

Tunneling Probability:
T≈e−2αLT \approx e^{-2\alpha L}
Decay Constant:
α=2m(V0−E)ℏ2\alpha = \sqrt{\frac{2m(V_0-E)}{\hbar^2}}
Transmission Coefficient:
T=11+V02sinh⁡2(αL)4E(V0−E)T = \frac{1}{1 + \frac{V_0^2\sinh^2(\alpha L)}{4E(V_0-E)}}
Reflection Coefficient:
R=1−TR = 1 - T
Tunneling Current:
I∝T×incident fluxI \propto T \times \text{incident flux}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Decay Constant

The decay constant determines how quickly the wavefunction decays inside the barrier:

Equation:

α=2m(V0−E)ℏ2\alpha = \sqrt{\frac{2m(V_0-E)}{\hbar^2}}

Calculation:

α=2(9.11e−31)(1.60e−17−8.01e−18)(1.05e−34)2=3.62e+10 m−1\alpha = \sqrt{\frac{2(9.11e-31)(1.60e-17 - 8.01e-18)}{(1.05e-34)^2}} = 3.62e+10 \text{ m}^{-1}

Explanation:

This gives the rate at which the wavefunction decays inside the barrier.

2

Step 2: Calculate Tunneling Length

The tunneling length is the characteristic distance for tunneling:

Equation:

Tunneling Length=1α\text{Tunneling Length} = \frac{1}{\alpha}

Calculation:

Tunneling Length=13.62e+10=2.76e−11 m\text{Tunneling Length} = \frac{1}{3.62e+10} = 2.76e-11 \text{ m}

Explanation:

This is the distance over which the wavefunction decays significantly.

3

Step 3: Calculate Transmission Coefficient

The transmission coefficient gives the probability of tunneling:

Equation:

T=11+V02sinh⁡2(αL)4E(V0−E)T = \frac{1}{1 + \frac{V_0^2\sinh^2(\alpha L)}{4E(V_0-E)}}

Calculation:

T=11+(1.60e−17)2sinh⁡2(3.62e+10×1.00e−10)4(8.01e−18)(1.60e−17−8.01e−18)=2.85e−3T = \frac{1}{1 + \frac{(1.60e-17)^2\sinh^2(3.62e+10 \times 1.00e-10)}{4(8.01e-18)(1.60e-17 - 8.01e-18)}} = 2.85e-3

Explanation:

This gives the exact probability that the particle will tunnel through the barrier.

4

Step 4: Calculate Reflection Coefficient

The reflection coefficient is the probability of being reflected:

Equation:

R=1−TR = 1 - T

Calculation:

R=1−2.85e−3=9.97e−1R = 1 - 2.85e-3 = 9.97e-1

Explanation:

This gives the probability that the particle will be reflected by the barrier.

Frequently Asked Questions (FAQ)

What is quantum tunneling?

Quantum tunneling is a quantum mechanical phenomenon where particles can pass through potential energy barriers that would be impenetrable according to classical physics. It's a direct consequence of the wave-like nature of matter.

Why does tunneling occur?

Tunneling occurs because of the wave-like nature of particles and the uncertainty principle. The wavefunction can extend into classically forbidden regions, allowing particles to 'tunnel' through barriers.

How does barrier height affect tunneling?

Higher barriers reduce tunneling probability exponentially. The transmission coefficient decreases as T ≈ e^(-2αL), where α increases with barrier height.

How does particle mass affect tunneling?

Lighter particles tunnel more easily because the decay constant α = √(2m(V₀-E))/ℏ increases with mass. This is why electrons tunnel more readily than protons or alpha particles.

What are some applications of quantum tunneling?

Quantum tunneling is used in scanning tunneling microscopy, tunnel diodes, nuclear fusion, alpha decay, and quantum computing. It's essential for understanding many quantum systems.

Practice MCQs

  1. The tunneling probability depends exponentially on:
  2. For a given barrier, which particle would tunnel most easily?
  3. The transmission coefficient T and reflection coefficient R are related by:
  4. As the barrier width increases, the tunneling probability:
  5. Quantum tunneling is impossible in:

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