- #1
Nick.
- 15
- 0
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=eI(kR1-ωt)+eI(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??
For example in two dimensions;
To find a resulting wave at a point P1 from two paths R1 & R2 we have Ψ=eI(kR1-ωt)+eI(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??