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In using the superposition principle to calculate intensities in interference patterns, can we add the intensities of the waves instead of their amplitudes? I think that amplitude account for the intensities so that both are the same thing and so it doesn't matter.

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Intensities are the squares of amplitudes. They are definitely not the same thing. – Michael Brown Apr 6 '13 at 5:44
This is basically the physics version of the Freshman's dream! – zkf Apr 6 '13 at 7:20
up vote 5 down vote accepted

No, it is amplitude. Amplitude is $\Psi$, intensity is $|\Psi| ^2$.

Schrödinger's equation (where $\hat H$ is linear) is: $$\hat H\,\Psi=E\,\Psi.$$ So, if you have two possible states $\Psi_1,\Psi_2$, then $$\hat H\,\Psi_1=E\,\Psi_1,\\\hat H\,\Psi_2=E\,\Psi_2.$$

We can add these and get $$\hat H (\Psi_1+\Psi_2)=E(\Psi_2+\Psi_2).$$

This shows us that the superposition of amplitudes still satisfies Schrödinger's wave equation. On the other hand, there is no guarantee that $\sqrt{\Psi_1^2+\Psi_2^2}$ (the amplitude that you get if you superpose intensities) will satisfy the wave equation.

Besides, intensities are positive; you'd never get the chance for destructive interference if you were only superposing intensities.

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