It is said that if there is already n bosons in a particular quantum state, the probability of another boson joining them is (n+1) times larger than it would have been otherwise. But if we apply this rule to calculate probability for one horizontally (H) polarized photon to join a bunch of n=99 diagonally (D) polarized photons, we'll get (99+1) 1/2 = 50. What's wrong?
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In general photon number is not conserved and the word 'otherwise' is unclear. We could say there is an infinite set of photons for you to steal one to add to the collection of $n$ photons. Anyway, there exists a set of probabilities $p_i$ such that $p_i / p_{i+1} = i + 1$. You wanted the inverse ratio but we can easily label any set of probabilities in reverse, and shift the index. For example, this rule is satisfied for $p_i = e^{-1} (n!)^{-1}$. The factor of
$$e = 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = \sum_i \frac{1}{i!}$$
normalizes the sum so that it satisfies the probability rule $\sum_i p_i = 1$.
BrianWa
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