Probabilities for system of two bosons

[ajl1989]

If there are two indistinguishable bosons that can either be in the |0> or |1> state, what is the probability that both will be in the |0> state? (ie the system will be in the |0>|0> state)

I know there are only three possibilities for the total state of the system: |0>|0>, |1>|1>, and (1/sqrt2)(|0>|1>+|1>|0>), but are these states equally probable? (I'm assuming that a single boson has an equal probability of being in either |0> or |1>) Would the probability of being in |0>|0> just be 1/3? or is it more complicated than that?

 

[Avodyne]

Call the three states

$$|00\rangle,\;|11\rangle,\;\hbox{and}\;|S\rangle \equiv{\textstyle{1\over\sqrt2}}\bigl(|01\rangle+|10\rangle\bigr).$$

Then the most general state of the two particles is

$$a|00\rangle+b|11\rangle+c|S\rangle$$

with $|a|^2+|b|^2+|c|^2=1$.

What can you deduce about $a$, $b$, and $c$ from the requirement that each particle be equally likely to be in $|0\rangle$ or $|1\rangle$?

 

[ajl1989]

Well, I guess if a=b=c, then they'd each have to be 1/sqrt3... unless we also have to take into account the 1/sqrt2 factor in front of |S>?