The Tunneling Phenomenon

Let's consider a particle with mass m which is flying with energy E (from left side) up to potential barrier having width a. It's immediately clear what's going on here in classical case: if particle energy E is smaller than barrier height V then the particle will be reflected. Only in the case when E is greater than V the particle could pass this barrier. The quantum mechanics gives us ambiguous answer. The Schroedinger equation is then divided into 3 single equations each of them is true only for one of the three zones. The solutions of the three equations in the transition regions must continuously change from one to another. It must also be true for steps (first derivatives) of wavefunctions. The ratio of squared probability amplitude for rightwards transmitted particle and that for leftwards falling particle gives the transmission probability T (Derivation):

T = {1 + sinh2(ka)/[4 E/V (1-E/V)]}-1 where k = (2m(V-E)/h²)½

Although E < V the transmission probability T isn't equal to 0, i.e. a particle can cross the potential barrier that isn't allowed in classical physics. The wavefunction isn't equal to 0 at point x=0 since V is finite in height and width. One can write down the following approximation for ka > 1:

T ≈ 16E/V² (V − E) e−2ka

One can see the tunneling probability T against different energies E and constant potential barrier V on the next figure. Where the idealized fragmentation potential of H in H-N3 was chosen in a way that H-N3 would be in excited state.

The tunneling probability (idealized case) for H-N3 and D-N3 in the excited states. Classically T = 0 for E/V<1 and T = 1 for E/V>1.

The potential width is 30 pm and its height is V = 0,39 eV ≡ 37,6 kJ/mol. The tunneling probability strongly depends on mass as one can see from replacement of H into D.

For E > V T can be < 1 , which means that particles are reflected back even if the particle energy is greater than the barrier height. Here one can write down (sinh is replaced by sin):

T = [1 - sin² k'a/4 E/V (1 − E/V)]-1 where k' = (2m(E-V)/h²)½

P.S. For E = V:

T = [1 −ma²V/2h²]-1

The technical application of tunneling phenomenon is Rastertunnelmicroscop.