# Simple exponential multiplication (electron interfernce)

1. Jan 8, 2010

### Moonspex

1. The problem statement, all variables and given/known data

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.

2. Relevant 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)

3. The attempt at a solution

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

2. Jan 8, 2010

### RoyalCat

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

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

3. Jan 8, 2010

### 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)