Complex Wave Orientation Correction

[Nick.]

The configuration and dimensions of any experiment are important in determining wave amplitudes. Then why are the orientations of complex waves not considered when they are added?

For example in two dimensions;

To find a resulting wave at a point P1 from two paths R1 & R2 we have Ψ=e^I(kR1-ωt)+e^I(kR2-ωt) so the amplitude being |Ψ|².

However, paths R1 and R2 are not parallel they are separated at the arrival point P1 by an angle θ.

Why are the two waves not corrected to suit the arrival orientation so;
|Ψ²|=(isin(kR2-ωt)+isin(kR1-ωt))²+(cos(kR2-ωt)cosθ+cos(kR1-ωt))²+(cos(kR2-ωt)sinθ)²

This translates the second wave into the co-ordinates of the first wave and provides some small corrections to the first wave. I.e. On a two slit experiment set up this is unlikely to make much of a difference as the length of R1 & R2 are so large that the angle θ will be tiny so it this tweak could be virtually ignored - although as the screen come close to the slits the effect would become considerable as θ becomes larger.

Since the example is 2D the complex planes are still additive without any adjustment but in 3D it would also require some re-orientation (sure the complex plane cannot be aligned to any I, j, k coordinates as it is in the complex Z plane - but I am sure someone will have thoughts...)

Any thoughts??

 

[Nick.]

Given the enormous response to this one I thought I might add a few more details.

Typical Amplitude Calculation
|Ψ|²=(e^i(kR1-ωt)+e^i(kR1-ωt))(e^i(kR1-ωt)+e^i(kR1-ωt))

Amplitude Calculation for adjusted orientation
|Ψ|₂=(isin(kR2-ωt)+isin(kR1-ωt))²+(cos(kR2-ωt)cosθ+cos(kR1-ωt))²+(cos(kR2-ωt)sinθ)2²

Where isin(kR2-ωt)+isin(kR1-ωt) is sum of the two imaginary parts of e^i(kR1-ωt) at point P1 = |z|.

|y| = cos(kR2-ωt)sinθ
This is a whole new component not seen when adding the waves in a straight orientation. This small when θ is small and reaches it maxima at π/2.

|x| = cos(kR2-ωt)cosθ+cos(kR1-ωt)
This is subtly adjusted by cosθ to account for part of the wave being in the y axis.

then with a bit of Pythagoras; |x|²+|y|²+|z|² = |Ψ|²

What would it yield?
Sticking with the 2D double slit example. It means when the screen is very close the interference fringes would distort or disappear...say if the screen is at the same distance as the two slits are separated so that θ is π/2 it means two at values (where |x|=1) that any value of the second wave e^i(kR2-ωt) would yield the same |Ψ|² value - ordinarily this would be sinusoidal.

What is the motivation?
As the waves have a relationship with space and time then I would have expected orientation to be important also - i.e. a vector quantity within some sort of configuration space. If the value of |Ψ| is merely a scalar value (which I guess is what is typically implied) then adding say two energy level values of the same type makes sense regardless of orientation. However, if the waves contain a momentum then it shouldn't it be some interesting vector quantity - so why isn't aligned to give the portion related to direction?

Any ideas...?