I am reading 'Particle in 1 dimensional box' and 'Potential step' in Quantum Mechanics. In the figure shown we have two same looking schrodinger equations, equation 1 and equation 2. Why solutions of two identical equations are different?
Where symbol $E$ means energy of particle, $V$ means potential energy and other symbols have their usual meaning as they do in QM.
Best Answer
They aren't different. Note that $e^{\pm ikx}=\cos(kx) \pm i\sin(kx)$, so $$A\cos(kx)+B\sin(kx) = C e^{ikx} + D e^{-ikx}$$
for appropriate choices of the constants. They are just different ways of writing the same thing.
A particular choice might make more sense depending on the interpretation of the problem and the boundary conditions you're going to apply, but you could write either one in either way and it would be correct.