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Quantum Physics
What are the probabilities for different outcomes with Bose-Einstein statistics?
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[QUOTE="vanhees71, post: 5743698, member: 260864"] What's given is a basis consisting of three orthonormalized vectors, $$|b_1\rangle=|HH \rangle, \quad |b_2 \rangle=\frac{1}{2}(|HT \rangle+|TH \rangle), \quad |b_3 \rangle = |TT \rangle.$$ It's wrong to say that these are all allowable states, but these are given by all statistical operators in this 3D Hilbert space. What we can read from the above quoted text snippet is only that all states are equally probable. That's not complete information, and we need some objective criterion of how to associate a state with this information. The answer is provided by information theory: First one defines the Shannon-Jaynes-von-Neumann entropy as a measure for the missing information, given the state, $$S[\rho]=-\mathrm{Tr}(\hat{\rho} \ln \hat{\rho}).$$ Then we want a state of minimal prejudice, i.e., with maximal entropy, compatible with the information. Here, we have given the probabilities for the outcomes of measurements to be $$P_j=\langle b_j|\hat{\rho}|b_j \rangle.$$ This we have to minimize with these constraints and the constraint that ##\mathrm{Tr} \hat{\rho}=1##, i.e., we introduce Lagrange multipliers for these constraints $$\tilde{S}[\rho]=\sum_j \lambda_j \langle b_j|\hat{\rho}|b_j \rangle+\lambda \mathrm{Tr} \hat{\rho}-\mathrm{Tr}(\hat{\rho} \ln \hat{\rho}).$$ Variation of the statistical operator gives $$\delta \tilde{S}[\rho]=\sum_j [\lambda_j + \lambda-1] \delta \rho_{jj}-\mathrm{Tr} [\delta \hat{\rho} \ln \hat{\rho}] =0.$$ Since we can vary the 9 matrix elements of ##\hat{\rho}## independently now (thanks to the Lagrange parameters), the expression can only be 0 if ##[\ln \hat{\rho}]_{ij}=0## for ##i \neq j##, i.e., ##\hat{\rho}## is diagonal in the above given basis. After some algebra thus we get $$\hat{\rho}=\sum_{j=1}^{3} P_j |b_j \rangle \langle b_j|, \quad \sum_{j=1}^3 P_j=1.$$ if all the ##P_j## are equal, as said in the above text snippet, we have necessarily ##P_j=1/3##. [/QUOTE]
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What are the probabilities for different outcomes with Bose-Einstein statistics?
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