Simple Nature - Light and Matter

A classically allowed region with constant U example 21In a classically allowed region with constant U, we expect thesolutions to the Schrödinger equation to be sine waves. A sinewave in three dimensions has the formΨ = sin ( k x x + k y y + k z z ) .When we compute ∂ 2 Ψ/∂x 2 , double differentiation of sin gives− sin, and the chain rule brings out a factor of k 2 x . Applying allthree second derivative operators, we get()∇ 2 Ψ = −kx 2 − ky 2 − kz2 sin ( k x x + k y y + k z z )()= − kx 2 + ky 2 + kz2 Ψ .The Schrödinger equation givesE · Ψ = − 22m ∇2 Ψ + U · ΨE − U = 22m= − 2 (2m · − kx 2 + ky 2 + kz2)(k 2 x + k 2 y + k 2 z)Ψ + U · Ψwhich can be satisfied since we’re in a classically allowed regionwith E − U > 0, and the right-hand side is manifestly positive.,Use of complex numbersIn a classically forbidden region, a particle’s total energy, U +K,is less than its U, so its K must be negative. If we want to keep believingin the equation K = p 2 /2m, then apparently the momentumof the particle is the square root of a negative number. This is asymptom of the fact that the Schrödinger equation fails to describeall of nature unless the wavefunction and various other quantitiesare allowed to be complex numbers. In particular it is not possibleto describe traveling waves correctly without using complex wavefunctions.Complex numbers were reviewed in subsection 10.5.5,p. 603.This may seem like nonsense, since real numbers are the onlyones that are, well, real! Quantum mechanics can always be relatedto the real world, however, because its structure is such thatthe results of measurements always come out to be real numbers.For example, we may describe an electron as having non-real momentumin classically forbidden regions, but its average momentumwill always come out to be real (the imaginary parts average out tozero), and it can never transfer a non-real quantity of momentumto another particle.Section 13.3 Matter As a Wave 873