Derivation related to Euler’s Formula

[mysearch]

Hi,
I wasn’t sure whether to post this issue in a physics forum or here in this maths forum, because although it relates to physics is appears to be grounded in maths, i.e. Euler theorem. Therefore, I was wondering if anybody could help me resolve some issues with the following derivation, taken from a textbook, for 2 waves in superposition. The derivation is linked to the Young’s double-slit experiment, which acts as a precursor to quantum theory

[1] $$\Psi = A exp \; i \left[ k \left( r- \frac{d}{2}sin \theta\right) - wt \right] + A exp \; i \left[ k \left( r+ \frac{d}{2}sin \theta\right) - wt \right]$$

[2] $$\Psi = A exp \; i \left( kr-wt\right) \left( exp \left[ ik \left( \frac{d}{2} \right)sin \theta \right] + exp \left[ -ik \left( \frac{d}{2} \right) sin \theta \right] \right)$$

[3] $$\Psi = 2A exp \; i \left( kr-wt\right) cos \left[ k \left( \frac{d}{2} \right)sin \theta \right]$$

Purely, by way of background, equation [1] represents 2 waves in superposition, hence the 2 parts, separated by the distance between the 2 slits [d]. The distance [r] is the mean distance to the screen where the interference pattern is seen. However, my first maths question relates to the premise of equation [1] being based on the following relationship and whether equation [1] has to consider both the real and imaginary components of the wave function? (I accept this might not be a pure maths issue)

[4] $$e^{i \theta} = cos \theta + i.sin \theta$$

While I think I follow the basic steps from [1] to [3], there appears to be an assumption when going from [2] to [3] that:

[5] $$\left( exp \left[ ik \left( \frac{d}{2} \right)sin \theta \right] + exp \left[ -ik \left( \frac{d}{2} \right) sin \theta \right] \right) = 2 cos \left[ k \left( \frac{d}{2} \right)sin \theta \right]$$

So my second question is whether this based on the mathematical relationship?

[6] $$\Re (e^{i \theta}) = cos \theta = (e^{i \theta}+ e^{-i \theta})/2$$

In part, this goes back to the scope of whether equation [1] is intended to include the real and imaginary parts of the wave function. Would appreciate any knowledgeable insights to the maths and/or physics. Thanks

 

[yyat]

The formula

$$\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}$$

is true for all $$\theta\in\mathbb{C}$$, so [2] is equivalent to [3]. Note though that

$$\Re{e^{i\theta}}=\cos(\theta)$$

is only true for real $$\theta$$.

 

[mysearch]

Note though that
$$\Re{e^{i\theta}}=\cos(\theta)$$
is only true for real $$\theta$$.

Thanks. I agree with your note, but wouldn’t this suggest that the whole of equation [1] has to be based on the real part, i.e. this equation is not considering the imaginary (+i.sin) component? I wasn't sure of the implication of this statement, but the issue may be more relevant to a physics forum addressing the wave function for mechanical and quantum waves.