Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 29

Finally, the particle in a box with a finite energy barrier, V₀, will be discussed qualitatively. This is a situation where the particle is no longer strictly forbidden outside the confinement box and leads to the concept of tunneling, that is, the probability of the electron found outside the box. The shape of the potential function is shown in Figure 2.5b.

The potential energy for this case is written as

(2.55)

and

(2.56)

(Notice that the boundaries of the box were shifted from 0 to L to −L/2 to +L/2 for symmetry reasons that will be taken up again in Section 3.2.) The Schrödinger equation is written in two parts: Inside the box, where the potential energy is zero, the same equation holds that was used earlier:

(2.23)

Outside the box, the Schrödinger equation is

(2.57)

Figure 2.5 (a) Particle in a box with infinite potential energy barrier. (b) Particle in a box with infinite potential energy barrier.

The solutions of this equation will be of the form

(2.58)

(2.59)

where

(2.60)

For is an exponential decay function, and for is an exponential growth function. This is shown in Figure 2.5b to the right and left of the potential energy box, respectively. This represents the probability of finding the electron outside the box, a process that is known as “tunneling.” Inside the box, the solutions of Eq. (2.23) resemble the bound wavefunctions of the particle in a box, except that the amplitude at the boundary is no longer zero, but must meet with the wavefunction outside the box. This is depicted in Figure 2.5. Bound states exist for energies E(n) < V₀ only; for E(n) > V₀, the electron exists as a traveling wave as discussed before for unbound states.

The concept of tunneling may seem esoteric at first, but it has interesting consequences. For example, a technique exists that is known as “tunneling electron microscopy (TEM)” where a very sharp metal tip is moved very close (within fractions of a nanometer) to the surface of the analyte, which is at a positive potential with respect to the metal tip. A tunneling current is observed between the tip and the analyte that is due to electrons tunneling from the tip to the analyte. As the substrate is moved laterally under the tip, the tip is lowered or raised to keep the tunneling current constant. In this way, an “image” of the morphology of the analyte can be obtained. Tunneling may also play a role in certain chemical reactions that depend on electron transfer from a donor to a receptor; some of these reactions are faster than expected from computations of the reaction rate from the activation energy. It is thought that in these reactions, the electron may tunnel from donor to receptor at a very fast rate. Finally, in the last example of “real‐world PIBs” in the section below, tunneling plays a major role.