Tunnel Effect in Quantum

Tunnel effect is one of the most important manifestations of quantum mechanics

Classic analogy: full internal reflection
In the geometric optics sin(r) starts to exceed 1 when n2 is low enough. Thus, refraction becomes impossible.
Actually, electromagnetic wave penetrates into the optically more dense medium at a distance of the wave length λ. It can be found by the indicated set-up when δ ≤ λ.
Hence, a massive particle (electron, proton, etc.) directly reveals its wave properties!

Classic analogy: full internal reflection

Examples of tunnel phenomena: α-decay of heavy nuclei
Potential energy of the particle inside and outside the atomic nucleus:
γ is the probability of the α-particle escape during the time unit

Scientists, who discovered the tunnel effect

Scientists, who discovered the tunnel effect

Cold emission of metal electrons
Cold emission current
E is the external electrostatic field

Cold emission of metal electrons

Oscillation of a particle between two potential wells
Separate wells
Coupled wells
Initially the particle is in the left well

Oscillation of a particle between two potential wells
Results of calculation:
W(t) is a probability of the particle to occur in the left well at the moment t
Limiting cases:
The particle spends equal times in both wells
The particle is predominately in the left well

Tunneling in the periodic lattice; electron band formation
One-dimensional periodic lattice potential
ΔEk0 is the bandwidth

Franz-Keldysh effect
Tunneling probability W(B→C) from the valence band AB into the conductance band CD is proportional to exp{-c[εg]3/2/E}, where εg is the forbidden-gap width. This is Zener effect. It changes the coefficient αE of the light absorption in a semiconductor in the homogeneous electric field E. This is Franz-Keldysh effect.

Tunneling in chemistry

Tunneling in chemistry

Tunneling in chemistry

Tunneling in chemistry

Single-electron tunneling

Single-electron tunneling
(4)
(2)
ΔE = e(e/2±Q)/C
Correct formula:

Single-electron tunneling