Quantum Mechanics/Complex Waves

In quantum mechanics we are interested in solutions to the Schrödinger equation that are renormalizable. One class of functions that does this is the complex exponentials, and we can write a solution to the Schrödinger equation as the sum of two complex exponentials

${\displaystyle \Psi (x)=Ae^{ikx}+Be^{-ikx}}$

which is the superposition of waves, one traveling from the left, the other from the right. k is the wave number which is in units of 1/m and is usually given by ${\displaystyle 2*\pi /L}$ where L is determined by boundary conditions.

Let us consider a situation in which there is a wave traveling from the left to the right. On the interval (-${\displaystyle \infty }$,0) V(x)<E(x)and from [0, ${\displaystyle \infty }$) V(x)>E(x) and the solutions of the Schrödinger Equation are of the form

${\displaystyle \Psi (x)=Ae^{i{\sqrt {{\frac {2m}{\hbar ^{2}}}(E(x)-V(x))}}x}+Be^{-i{\sqrt {{\frac {2m}{\hbar ^{2}}}(E(x)-V(x))}}x}}$

on the first interval and

${\displaystyle \Psi (x)=Ce^{{\sqrt {{\frac {2m}{\hbar ^{2}}}(E(x)-V(x))}}x}+De^{-{\sqrt {{\frac {2m}{\hbar ^{2}}}(E(x)-V(x))}}x}}$

on the second interval

Even though at first glance our wave equation looks like a superposition of normal exponentials, it is still a complex wave though because the term inside of the square root is negative and this allows us to always have renormalizable solutions.