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I have been reading a online-book/blog/material on Quantum Mechanics, when I encountered a notation on a page and I have no idea what it means. See if you can help.

Here's the link and follows the paragraph where I am stuck.

Observe that exchanging either the incoming or the out­going par­ti­cles is tan­ta­mount to exchanging the two alter­na­tives and, cor­re­spond­ingly, the two ampli­tudes, so that A2 takes the place of A1 and vice versa. Since the two ampli­tudes have the same mag­ni­tude, there is a com­plex number c of unit mag­ni­tude such that A2 = A1 c. In other words, mul­ti­pli­ca­tion by c = [1:β] rep­re­sents an exchange of the incoming or out­going particles.

If the incoming or out­going par­ti­cles are exchanged twice, then (i) A1 gets mul­ti­plied by $c^2$ and (ii) the orig­inal sit­u­a­tion is restored. Thus A1 = A1 $c^2$, whence it fol­lows that $c^2$ = [1:2β] = 1. This means that 2β must be equal to an inte­gral mul­tiple of 360°, and this leaves us with two pos­si­bil­i­ties: β = 0°, in which case A2 = A1, or β = 180°, in which case A2 = −A1.

I have put the notation in bold. What is it that the writer means exactly by $[1:\beta] \ \ and \ [1: 2 \beta]$ ?


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1 Answer 1

up vote 6 down vote accepted

The author uses this weird notation $[c:\gamma]$ to represent complex numbers. It means: c is short for the mag­ni­tude $|c|$ of c, $\gamma$ is the phase of c.

I have never seen this before either ;-).

The author explains it earlier in his book, check out this link.

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The colon is interesting but the square brackets are somewhat standard for polar coordinates and the complex numbers in polar form, see e.g. pages 1 and 66-67 of – Luboš Motl Mar 9 '13 at 7:24
I've never seen it before. I've been studying physics and math for 5 years now. – seb Mar 9 '13 at 7:43

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