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The equation for the double slit experiment requires an approximation in naming the two angles used to both be theta, correct? I feel like the majority of equations are exact relations, and it just seems strange that the equation for calculating the maxima an minima doesn't have an exact form. Is the solution of d*sin(theta)=m*lambda really just an approximation. I know that it is a very accurate one, given the size of wavelengths, however it just seems so strange to be using an approximation, and to never see the real solution.

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Yes, there is an approximation in calling both angles $\theta$. Usually (at least in a good text) this approximation should be motivated by

  1. The distance of the slits to the screen is much larger than the distance between the slits themselves (else the result returns to two individual slits at a distance from each other)
  2. A small angle approximation, as the actual relation between the difference in distance for the two rays $\Delta s$ and the position on the screen $x$ is $$\arcsin \frac{\Delta s}{a} = \arctan \frac{x}{d}$$ where $a$ is the distance between the slits and $d$ is the distance from the slits to the screen. This reduces to the better known $$\frac{\Delta s}{a} = \frac{x}{d}$$ on $\Delta s \ll a$ and $x \ll d$ (from the german Wikipedia).

The real relation is (as you can see above, where the angles at both slits have already been set equal) really messy and will lead to transcendental equations. Moreover, you simply don't need it to prove the principle behind double slit experiments and there is little need to solve the situation where any of those approximations break down analytically (I guess if it appeared in an application, a numerical solution would suffice).

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