Quantum Mechanics/Complex Waves

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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

\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 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 (-\infty,0) V(x)<E(x)and from [0, \infty) V(x)>E(x) and the solutions of the Schrödinger Equation are of the form

\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

\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.