Answer to Question #257961 in Physics for Anand

Question

What is quantum tunneling? Calculate (i) the tunneling length and (ii) the tunneling probability when an electron of energy 4.0 eV is incident on a potential barrier of height 10 eV and width of 0.2 nm.

Answer 1

Quantum tunnelling is the quantum mechanical phenomenon where a particle (more accurately, its wavefunction) can propagate through a potential barrier.

Let tunneling lenght be $a = 0.2nm = 0.2\times 10^{-9}m$, the energy of the particle $E = 4eV \approx 6.409 \times 10^{-19}J$ and the potential height $U = 10eV\approx 16.02\times 10^{-19}J$. The tunneling propability is then given as follows (see https://courses.physics.illinois.edu/phys485/fa2015/web/tunneling.pdf, page 13):

$$T = \left( 1 + \dfrac{\sinh^2(\beta a)}{4(E/V)(1-E/V)} \right)^{-1}$$

where $\beta=\dfrac{\sqrt{2m(V-E)}}{\hbar}$, $m = 9.1\times 10^{-31}kg$ is the mass of electron and $\hbar=1.055\times 10^{-34}Js$. Thus, obtain:

$$\beta=\dfrac{\sqrt{2\cdot 9.1\times 10^{-31}kg\cdot (10-4)eV}}{1.055\times 10^{-34}Js} \approx12.54\times 10^{9}m^{-1}$$

$$T = \left( 1 + \dfrac{\sinh^2(12.54\times 10^{9}m^{-1}\cdot 0.2\times 10^{-9}m)}{4(4eV/10eV)(1-4eV/10eV)} \right)^{-1} \approx 0.025$$

Answer. (i) 0.2nm, (ii) 0.025.