Double Slit Probability Amplitude Orientation

[Nick.]

Why are the orientations of waves arriving at the screen not considered when adding amplitudes?

For example a double slit in 2D has two radial lengths of R1 and R2, one from each source Slit 1 and Slit 2, arriving at point P1. (See attached) The provides a probability amplitude of;

|Ψ|²=(e^i(kR1-ωt)+e^i(kR2-ωt))(e^i(kR1-ωt)+e^i(kR2-ωt))

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;

|Ψ|²=(isin(kR2-ωt)+isin(kR1-ωt))²+(cos(kR2-ωt)cosθ+cos(kR1-ωt))²+(cos(kR2-ωt)sinθ)²

Where I have used R1 as the y axis, and x is perpendicular so in the above function;

isin(kR2-ωt)+isin(kR1-ωt) is sum of the two imaginary parts of ei(kR1-ωt) at point P1 = |z| - this is the same as normal in this 2D example.

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

cos(kR2-ωt)cosθ+cos(kR1-ωt)=|x| 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 so;

|Ψ|²=(isin(kR2-ωt)+isin(kR1-ωt))²+(cos(kR2-ωt)cosθ+cos(kR1-ωt))²+(cos(kR2-ωt)sinθ)²

On a typical two slit experiment set up this adjustment 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.

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.

As the wave paths have a relationship with space and time then I would have expected orientation to be important also - i.e. as 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, that is not how the space is being configured – hence the orientation question.

Any ideas...?

 

[jtbell]

Why are the orientations of waves arriving at the screen not considered when adding amplitudes?

It's a matter of which approximation you are using for diffraction/interference calculations. Introductory treatments use the Fraunhofer approximation, which is the limit as the screen goes very far away from the slits/sources and the rays arriving at the screen become effectively parallel. Closer to the slits/sources, the Fresnel approximation gives better results. A Google search for "Fresnel diffraction" and "Fraunhofer diffraction" should turn up some discussion of this. It's also covered in intermediate and advanced optics textbooks, e.g. Hecht.

By the way, it's not a good idea to post Word documents as attachments here. Some people don't have Word, and some people that do are reluctant to download and open such documents because of possible security issues. I opened it, and some of the symbols (Greek letters?) didn't display properly because of mismatched fonts. (I'm on a Macintosh).

Put the diagram in a .gif or .jpg image, and attach that. The text belongs in the body of the post, where you can use the editor's palette for Greek letters, or even use LaTeX for more complicated equations.

 

[Nick.]

Good point on the word doc - I have attached a more usable PDF file...(note the text in the word file was the same as the post so loss if it can't be read - I typed it in word first).

I have read up on both Fraunhofer and Fresnel diffraction and both appear to be fundamentally calculated in the same way (happy to shown otherwise as that will answer the question - in this case I referred to http://www.hyperphysics.phy-astr.gsu.edu ). In fact it was in reading these two effects which made me ask the question.

Essentially, Fraunhofer is ignoring the effects of the slit widths (sticking with the double slit example) due to the large distance R1 & R2 from the slits. Whereas, Fresnel accounts for each path within each slit opening to account for its effects when the distance to the screen is short. However, neither of these look at the orientation of the wave relative to each path - which is the question - why are they not taken into account?