Tunneling
For the particle in a rectangular well (Section 2.4), Fig. 2.5 and the equations for $\psi{\mathrm{I}}$ and $\psi{\text {III }}$ show that for the bound states there is a nonzero probability of finding the particle in regions I and III, where its total energy $E$ is less than its potential energy $V=V_{0}$. Classically, this behavior is not allowed. The classical equations $E=T+V$ and $T \geq 0$, where $T$ is the kinetic energy, mean that $E$ cannot be less than $V$ in classical mechanics.
Consider a particle in a one-dimensional box with walls of finite height and finite thickness (Fig. 2.6). Classically, the particle cannot escape from the box unless its energy is greater than the potential-energy barrier $V{0}$. However, a quantum-mechanical treatment (which is omitted) shows that there is a finite probability for a particle of total energy less than $V{0}$ to be found outside the box.
The term tunneling denotes the penetration of a particle into a classically forbidden region (as in Fig. 2.5) or the passage of a particle through a potential-energy barrier whose height exceeds the particle's energy. Since tunneling is a quantum effect, its probability of occurrence is greater the less classical is the behavior of the particle. Therefore, tunneling is most prevalent with particles of small mass. (Note that the greater the mass $m$, the more rapidly the functions $\psi{\mathrm{I}}$ and $\psi{\text {III }}$ of Section 2.4 die away to zero.) Electrons tunnel quite readily. Hydrogen atoms and ions tunnel more readily than heavier atoms.
The emission of alpha particles from a radioactive nucleus involves tunneling of the alpha particles through the potential-energy barrier produced by the short-range attractive nuclear forces and the Coulombic repulsive force between the daughter nucleus and the alpha particle. The $\mathrm{NH}{3}$ molecule is pyramidal. There is a potential-energy barrier to inversion of the molecule, with the potential-energy maximum occurring at the planar configuration. The hydrogen atoms can tunnel through this barrier, thereby inverting the molecule. In $\mathrm{CH}{3} \mathrm{CH}_{3}$ there is a barrier to internal rotation, with a potential-energy
FIGURE 2.6 Potential energy for a particle in a onedimensional box of finite height and thickness. 
maximum at the eclipsed position of the hydrogens. The hydrogens can tunnel through this barrier from one staggered position to the next. Tunneling of electrons is important in oxidation-reduction reactions and in electrode processes. Tunneling usually contributes significantly to the rate of chemical reactions that involve transfer of hydrogen atoms. See R. P. Bell, The Tunnel Effect in Chemistry, Chapman \& Hall, 1980.
Tunneling of H atoms occurs in some enzyme-catalyzed reactions; see Quantum Tunnelling in Enzyme-Catalyzed Reactions, R. Allemann and N. Scrutton (eds.), RSC Publishing, 2009.
The scanning tunneling microscope, invented in 1981, uses the tunneling of electrons through the space between the extremely fine tip of a metal wire and the surface of an electrically conducting solid to produce images of individual atoms on the solid's surface. A small voltage is applied between the solid and the wire, and as the tip is moved across the surface at a height of a few angstroms, the tip height is adjusted to keep the current flow constant. A plot of tip height versus position gives an image of the surface.