Determine the probability of an electron tunneling through a 1 nm wide potential barrier when it has an energy of 10 eV and the barrier height is 5 eV.
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To determine the probability of an electron tunneling through a potential barrier, we can use the tunneling probability formula given by the Wentzel-Kramers-Brillouin (WKB) approximation:
P = exp(-2κd)
Where:
- P is the probability of tunneling
- κ is the wave number inside the barrier
- d is the width of the potential barrier
The wave number inside the barrier can be calculated using the formula:
κ = sqrt((2m/h^2)(V0 - E))
Where:
- m is the mass of the electron
- h is Planck's constant
- V0 is the barrier height
- E is the energy of the electron
Using the given values:
- m = 9.10938356 x 10^-31 kg (mass of the electron)
- h = 6.62607015 × 10^-34 J·s (Planck's constant)
- V0 = 5 eV = 5.0 x 1.602176634 × 10^−19 J (barrier height in joules)
- E = 10 eV = 10.0 x 1.602176634 × 10^−19 J (energy of the electron in joules)
- d = 1 nm = 1 x 10^-9 m (width of the potential barrier in meters)
We can now calculate the probability of tunneling:
κ = sqrt((2 * 9.10938356 x 10^-31 kg / (6.62607015 × 10^-34 J·s)^2) * (5.0 x 1.602176634 × 10^−19 J - 10.0 x 1.602176634 × 10^−19 J)) = 6.342 x 10^9 m^(-1)
P = exp(-2 * 6.342 x 10^9 m^(-1) * 1 x 10^-9 m) ≈ exp(-1.268) ≈ 0.281
Therefore, the probability of an electron tunneling through a 1 nm wide potential barrier, when it has an energy of 10 eV and the barrier height is 5 eV, is approximately 0.281 or 28.1%.
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