The quantum tunneling phenomenon

Tunneleffekt3d

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, Ek 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 mv2/2.

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.

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