Quantum Tunneling Probability Calculator

Quantum Tunneling Probability Calculator computes tunneling probability for a particle through a barrier, aiding quantum mechanics and nanotechnology studies.

Particle Energy (eV):

Barrier Height (eV):

Barrier Width (nm):

Particle Mass (kg):

Formulas Used in Quantum Tunneling Probability Calculator

The calculator uses the following formulas for a rectangular potential barrier:

Tunneling Probability (E < V_0):

\\[ T = \left[ 1 + \frac{V_0^2}{4 E (V_0 – E)} \sinh^2 \left( \sqrt{\frac{2 m (V_0 – E)}{\hbar^2}} L \right) \right]^{-1} \\]

Wavevector inside Barrier:

\\[ \kappa = \sqrt{\frac{2 m (V_0 – E)}{\hbar^2}} \\]

Where:

\\( T \\): Tunneling probability (0 to 1)
\\( E \\): Particle energy (eV)
\\( V_0 \\): Barrier height (eV)
\\( L \\): Barrier width (nm)
\\( m \\): Particle mass (kg)
\\( \hbar \\): Reduced Planck’s constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
\\( \kappa \\): Wavevector inside barrier (m\\(^{-1}\\))

Example Calculations

Example 1: Electron, Low Barrier

Input: Particle Energy = 1 eV, Barrier Height = 2 eV, Barrier Width = 1 nm, Particle Mass = 9.109e-31 kg

\\[ E = 1 \cdot 1.60217662 \times 10^{-19} = 1.60217662 \times 10^{-19} \, \text{J}, \quad V_0 = 2 \cdot 1.60217662 \times 10^{-19} = 3.20435324 \times 10^{-19} \, \text{J} \\] \\[ \kappa = \sqrt{\frac{2 \cdot 9.1093837 \times 10^{-31} \cdot (3.20435324 \times 10^{-19} – 1.60217662 \times 10^{-19})}{(1.0545718 \times 10^{-34})^2}} \approx 5.124 \times 10^9 \, \text{m}^{-1} \\] \\[ T = \left[ 1 + \frac{(3.20435324 \times 10^{-19})^2}{4 \cdot 1.60217662 \times 10^{-19} \cdot (3.20435324 \times 10^{-19} – 1.60217662 \times 10^{-19})} \sinh^2 (5.124 \times 10^9 \cdot 10^{-9}) \right]^{-1} \approx 0.135 \\]

Result: Tunneling Probability: 13.5%, Wavevector: 5.12e9 m\\(^{-1}\\)

Example 2: Electron, High Barrier

Input: Particle Energy = 1 eV, Barrier Height = 5 eV, Barrier Width = 1 nm, Particle Mass = 9.109e-31 kg

\\[ E = 1.60217662 \times 10^{-19} \, \text{J}, \quad V_0 = 5 \cdot 1.60217662 \times 10^{-19} = 8.0108831 \times 10^{-19} \, \text{J} \\] \\[ \kappa \approx 8.858 \times 10^9 \, \text{m}^{-1} \\] \\[ T \approx \left[ 1 + \frac{(8.0108831 \times 10^{-19})^2}{4 \cdot 1.60217662 \times 10^{-19} \cdot 6.40870648 \times 10^{-19}} \sinh^2 (8.858 \times 10^9 \cdot 10^{-9}) \right]^{-1} \approx 0.001 \\]

Result: Tunneling Probability: 0.1%, Wavevector: 8.86e9 m\\(^{-1}\\)

Example 3: Electron, Wide Barrier

Input: Particle Energy = 1 eV, Barrier Height = 2 eV, Barrier Width = 2 nm, Particle Mass = 9.109e-31 kg

\\[ E = 1.60217662 \times 10^{-19} \, \text{J}, \quad V_0 = 3.20435324 \times 10^{-19} \, \text{J} \\] \\[ \kappa \approx 5.124 \times 10^9 \, \text{m}^{-1} \\] \\[ T \approx \left[ 1 + \frac{(3.20435324 \times 10^{-19})^2}{4 \cdot 1.60217662 \times 10^{-19} \cdot 1.60217662 \times 10^{-19}} \sinh^2 (5.124 \times 10^9 \cdot 2 \times 10^{-9}) \right]^{-1} \approx 0.003 \\]

Result: Tunneling Probability: 0.3%, Wavevector: 5.12e9 m\\(^{-1}\\)

Formulas Used in Quantum Tunneling Probability Calculator

Example Calculations

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