The quantum tunneling phenomenon is negligible in the classical mechanics.

Show that the tunneling probability equals 0 for an object of mass 0.007 kg moving at a speed of 0.1 m/s against a solid obstacle of height 0.05 m and width 0.01 m.

###### Solution.

The transmission coefficient T (the tunneling probability) for a particle tunneling through a single potential barrier is

T = e^{(-2w/ħ)√(2m(U-Ek))} (1),

where U is the potential energy of the barrier, E_{k} is the kinetic energy of the object, w is the width of the barrier, m is the mass of the object and ħ is the Planck constant divided by 2π.

The potential energy of the barrier can be written as mgH, where H is the height of the barrier.

The kinetic energy of the object is mv^{2}/2.

Then, we can rewrite the equation (1) as

T = e^{(-2w/ħ)√(2m(mgH-mv2/2))} (2).

We substitute the values and we get

T ≈ e^{-1.31*1030} ≈ 0.