Simple Nature - Light and Matter

The detailed study of the solution of the Schrödinger equationis beyond the scope of this book, but we can gain some importantinsights by considering the easiest version of the Schrödinger equation,in which the interaction energy U is constant. We can thenrearrange the Schrödinger equation as follows:d 2 Ψ 2m(U − E)=dx2 2 Ψ ,which boils down tod 2 Ψdx 2 = aΨ ,where, according to our assumptions, a is independent of x. We needto find a function whose second derivative is the same as the originalfunction except for a multiplicative constant. The only functionswith this property are sine waves and exponentials:d 2dx 2 [ q sin(rx + s) ] = −qr2 sin(rx + s)d 2dx 2 [qerx+s ] = qr 2 e rx+sThe sine wave gives negative values of a, a = −r 2 , and theexponential gives positive ones, a = r 2 . The former applies to theclassically allowed region with U < E.This leads us to a quantitative calculation of the tunneling effectdiscussed briefly in the preceding subsection. The wavefunctionevidently tails off exponentially in the classically forbidden region.Suppose, as shown in figure m, a wave-particle traveling to the rightencounters a barrier that it is classically forbidden to enter. Althoughthe form of the Schrödinger equation we’re using technicallydoes not apply to traveling waves (because it makes no referenceto time), it turns out that we can still use it to make a reasonablecalculation of the probability that the particle will make it throughthe barrier. If we let the barrier’s width be w, then the ratio of thewavefunction on the left side of the barrier to the wavefunction onthe right isqe rx+sqe r(x+w)+s = e−rw .m / Tunneling through a barrier.Probabilities are proportional to the squares of wavefunctions, soSection 13.3 Matter As a Wave 869