Simple exponential multiplication (electron interfernce)

[Moonspex]

Homework Statement

Briefly, the question asks to prove how the interference of 2 electrons (travelling in opposite directions as 1-D waves) would affect the probability of finding each electron in free space. My issue has to do with the first step in the solution.

Homework Equations

\Psi_{1} = \Psi_{0} e^{jkx}
\Psi_{2} = \Psi_{0} e^{-jkx} (Note change in direction)

Hence the interference of these two functions will be given by their sum:
\Psi_{total} = \Psi_{0} e^{jkx} \Psi_{0} · e^{-jkx} (i)
\Psi_{total} = 2\Psi_{0} cos (kx) (ii)

The Attempt at a Solution

I just don't understand how to get (ii) from (i)... thanks for looking!

 

[RoyalCat]

A correction (You put a multiplication sign instead of an addition one. I think that was your mistake):

\Psi_{total}=\Psi_0(e^{jkx}+e^{-jkx})
Use Euler's formula, and just see that the imaginary sine terms cancel, while the real cosine terms add up.

e^{i\theta}=\cos \theta + i\sin \theta

 

[Moonspex]

Ah I see! I tend to forget simple steps like that which involve such simple identities. Thanks!
(PS: That was a typo on my part)