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I can't understand why the sum of two real waves result in a time dependent wave, but not so for the complex waves. In details, I can't get this passage on p.38-39 in A.C. Phillips, Introduction to Quantum Mechanics:

$$\tag{3.9}\Psi ~=~ A\cos (kR_1 - \omega t) + A\cos (kR_2 - \omega t)$$


$$\tag{3.10}\Psi^2 ~=~ 2A^2\cos^2\left(\frac{k(R_1-R_2)}{2}\right)\cos^2\omega t.$$

This is driving me crazy!

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What part of the equations don't you understand? – Mew Feb 13 '13 at 23:23
Hi Chris, I can't get why you get this function if you square Fi. I've tried to get this with basic sine\cosine transformation but I failed.I tried to ask wolframalpha and it gave me the same formula but with additional terms in the argument of cos^2(wt). It seems easy, but I can't see how the second follows the first, and it's a pity because it's a fundamental concept I guess. Thank you for your reply! – Lorenzo Feb 14 '13 at 8:56

Yes, OP is right. The book contains an error. The formula (3.10) for the classical real wave$^1$ should have been

$$\tag{3.10'}\Psi^2 ~=~ 4A^2\cos^2\left(\frac{k(R_1-R_2)}{2}\right)\cos^2\left(\omega t-\frac{k(R_1+R_2)}{2}\right).$$

Note however, that the qualitative physical conclusions that Phillips draws from eq. (3.10) in subsequent paragraphs remain valid.


$^1$ Note that Phillips is here (on p.38-39) for pedagogical reasons considering a real classical wave (3.9) rather than the correct quantum mechanical wave function.

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A, thank you very much! I'm such a pedant. I think I've fount two other errors while reading: pg40 (3.13) should be 4A^2 and not 2A^2 pg50 should be e^-(x^2/2a) otherwise you can't normalize it – Lorenzo Feb 14 '13 at 16:07

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