I have a question that I sort of feel should be easy to answer, but I haven't figured out how yet... Hopefully someone can either show me the easy answer or tell me that it's a little more subtle :)(adsbygoogle = window.adsbygoogle || []).push({});

Consider a system of two particles, distinguishable by some continuous parameter, which can each be in one of two places. Suppose each particle has an equal probability of being in either place, so the system is in the state

[tex]|\psi\rangle=\frac{1}{2}\left(|ab;0\rangle+|a;b\rangle + |b;a\rangle+|0;ab\rangle\right)[/tex]

where a and b represent the value of the parameter and the left- and right-hand sides of the semi-colon represent the two locations. This state is normalized for all distinct a and b, but has norm squared of 3/2 for b=a (since |a;b>+|b;a> becomes 2|a;a> when b=a). How does the norm of the state vector change discontinuously like that?

Next look at the probability of finding one particle in each location. Initially we have

[tex]p=|\langle a;b|\psi\rangle|^2+|\langle b;a|\psi\rangle|^2=\frac{1}{2}[/tex]

but when b=a we get instead

[tex]p=\frac{|\langle a;a|\psi\rangle|^2}{|\langle\psi|\psi\rangle|} =\frac{2}{3}.[/tex]

Again, how does it make sense for the probability to jump discontinuously like that? And why would it do that, it seems that if the particles are indistinguishable (bosons) the probability should still be 1/2 of finding one in each place.

I think I'm just overlooking something simple, but I've been hitting my head against this for a little while with no new thoughts, so hoped I'd find some help here.

Thanks,

Mike

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# Limit as distinguishable->indistinguishable

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