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I was wondering if anybody could help me understand a derivation connected to the double-slit experiment that I came across within an introduction to quantum theory paper. I was interested in understanding this approach because it seems to provide a useful correlation of the meaning of the square of the amplitude for an electromagnetic wave [tex][Intensity= \Psi^2][/tex] in comparison to a mechanical wave [tex][Energy=A^2][/tex]. The derivation starts off explaining that it intends to use complex notation:

[1] [tex] \Psi = \Re (e^{i (kx-wt)}) [/tex]

As far as I can see, only the `real part` of this equation is being referenced throughout the derivation, which I assume corresponds to the following relationship?

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

I have attached a diagram showing all the basic parameters used:

[3] [tex] r_1 \approx r - d/2(sin \theta); \; r_2 \approx r + d/2(sin \theta) [/tex]

This approximation is based on R>>d and the derivation continues based on Huygen’s theory, where the slits are 2 sources of light having equal amplitude and phase, which meet in superposition at a given point [P] defined by [tex] [ \theta][/tex]

[4] [tex] \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] [/tex]

[5] [tex]\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)[/tex]

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

As far as I can see, the step from [5] to [6] is based on [2], which seems to suggest that we are only dealing with the`real part`of the complex wave function? However, the derivation then goes on to point out that the square of this wave function, i.e. [6], corresponds to the intensity (I) of the resulting superposition wave, but then only equates (I) via proportionality to part of [6]:

[7] [tex]I \propto 4A^2 cos^2[k(d/2)sin \theta][/tex]

If we are only dealing with the `real part` of the complex notation, why can whole of [6] be converted by into a trigonometric form via applying [2] as follows?

[8] [tex] I = \left( Acos(kr-wt) * 2 cos \left[ k \left( \frac{d}{2} \right)sin \theta \right] \right)^2 [/tex]

On the basis that all the variable parameters are driven by the position [P], then the main parameter is [tex][ \theta][/tex] and we should be able to plot the intensity across the screen by setting the other parameters in [8] to relative values as follows:

[tex]A=1; R=1; r = Rsin \theta; d=0.001; k=2 \pi/500nm; t=0[/tex]?

Therefore [8] becomes:

[9] [tex] I = \left( cos[(2 \pi/500*10^{-9})Rsin \theta] * 2 cos \left[(2 \pi/500*10^{-9})) (0.001/2)sin \theta \right] \right)^2 [/tex]

Unfortunately, plotting [9] against [tex] \theta [/tex] doesn’t seem to produce the expected alternating constructive/destructive interference intensity, so I must assume that I have made an error or wrong assumption. I realise that this post may be too involved to interest anybody, but any insights by somebody who understands both the maths and physics would be appreciated. Thanks

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# Double-Slit Wave Equation?

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