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Quantum Tunneling Calculator

Calculate quantum tunneling properties including transmission probability, barrier penetration, and tunneling current

Parameters

kgⓘ
eVⓘ
nmⓘ
eVⓘ
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Animation

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Calculated Values

Decay Constant:
3622748827.35;1/m3622748827.35;1/m
Transmission Probability:
2.8494e−3;2.8494e-3;
Reflection Probability:
1.00;1.00;
Barrier Penetration:
7.1338e−4;7.1338e-4;
De Broglie Wavelength:
2.7603e−10;m2.7603e-10;m

Visualization

Quantum Tunneling

Quantum tunneling is a fundamental quantum mechanical phenomenon where particles can pass through potential energy barriers that would be classically forbidden. This effect is crucial for understanding many physical processes and technological applications.

The transmission probability T through a rectangular barrier is given by T = 1/(1 + (V₀²/4E(V₀-E))sinh²(κa)), where V₀ is the barrier height, E is the particle energy, a is the barrier width, and κ = √(2m(V₀-E)/ℏ²) is the decay constant.

When the particle energy is less than the barrier height (E < V₀), the particle can still tunnel through with a probability that decreases exponentially with barrier width and the square root of the energy difference.

The wave function inside the barrier decays exponentially as ψ(x) ∝ e^(-κx), where κ is the decay constant. This exponential decay is characteristic of quantum tunneling.

Quantum tunneling has important applications in scanning tunneling microscopy, nuclear fusion, semiconductor devices, and radioactive decay. It's also fundamental to understanding chemical reactions and biological processes.

Key Concepts

  • Transmission Probability: T = 1/(1 + (V₀²/4E(V₀-E))sinh²(κa))
  • Decay Constant: κ = √(2m(V₀-E)/ℏ²)
  • Wave Function Decay: ψ(x) ∝ e^(-κx)
  • Tunneling Current: I ∝ T
  • Reflection Probability: R = 1 - T
  • Barrier Penetration: P ∝ e^(-2κa)

Real-World Applications

  • Scanning Tunneling Microscopy (STM)
  • Nuclear Fusion: Alpha decay
  • Semiconductor Devices: Tunnel diodes
  • Chemical Reactions: Electron transfer
  • Biological Systems: Enzyme catalysis

Physics Equations

Decay Constant:
κ=2m(V0−E)ℏ2\kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}
Transmission Probability:
T=11+V024E(V0−E)sinh⁡2(κa)T = \frac{1}{1 + \frac{V_0^2}{4E(V_0-E)}\sinh^2(\kappa a)}
Reflection Probability:
R=1−TR = 1 - T
Barrier Penetration:
P∝e−2κaP \propto e^{-2\kappa a}
Tunneling Current:
I∝TI \propto T

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Decay Constant

The decay constant determines how rapidly the wave function decays inside the barrier:

Equation:

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

Calculation:

κ=2×9.110e−31×(8.011e−20)(1.055e−34)2=3.623e+9 1/m\kappa = \sqrt{\frac{2 \times 9.110e-31 \times (8.011e-20)}{(1.055e-34)^2}} = 3.623e+9 \text{ 1/m}

Explanation:

This gives the rate at which the wave function decays inside the barrier.

2

Step 2: Calculate Transmission Probability

The transmission probability through the barrier:

Equation:

T=11+V024E(V0−E)sinh⁡2(κa)T = \frac{1}{1 + \frac{V_0^2}{4E(V_0-E)}\sinh^2(\kappa a)}

Calculation:

T=11+2.567e−384×8.011e−20×8.011e−20×(1.871e+1)2=2.849e−3T = \frac{1}{1 + \frac{2.567e-38}{4 \times 8.011e-20 \times 8.011e-20} \times (1.871e+1)^2} = 2.849e-3

Explanation:

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

3

Step 3: Calculate Reflection Probability

The reflection probability is the complement of transmission:

Equation:

R=1−TR = 1 - T

Calculation:

R=1−2.849e−3=9.972e−1R = 1 - 2.849e-3 = 9.972e-1

Explanation:

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

4

Step 4: Calculate Barrier Penetration

The barrier penetration factor:

Equation:

P∝e−2κaP \propto e^{-2\kappa a}

Calculation:

P∝e−2×3.623e+9×1.000e−9=7.134e−4P \propto e^{-2 \times 3.623e+9 \times 1.000e-9} = 7.134e-4

Explanation:

This shows how much the wave function penetrates into the barrier.

Frequently Asked Questions

What is quantum tunneling?

Quantum tunneling is the phenomenon where particles can pass through potential energy barriers that would be classically forbidden, due to the wave-like nature of quantum particles.

How does barrier width affect tunneling?

The transmission probability decreases exponentially with barrier width. Wider barriers make tunneling much less likely due to the exponential decay of the wave function inside the barrier.

What is the decay constant κ?

The decay constant κ = √(2m(V₀-E)/ℏ²) determines how rapidly the wave function decays inside the barrier. Larger κ means faster decay and lower tunneling probability.

Can particles tunnel when E > V₀?

When particle energy exceeds barrier height, tunneling still occurs but the transmission probability is different. The particle can be reflected even when classically it should pass through.

What are some applications of quantum tunneling?

Applications include scanning tunneling microscopy, nuclear fusion, semiconductor devices like tunnel diodes, and understanding chemical reactions and biological processes.

Practice MCQs

  1. The transmission probability through a barrier decreases:
  2. What happens to the decay constant κ when the energy difference (V₀-E) increases?
  3. The wave function inside the barrier decays as:
  4. What is the relationship between transmission and reflection probabilities?
  5. Which device relies heavily on quantum tunneling?