量子統計力学における密度行列の展開

抄録

A general theorem is proved on the density matrix of quantum statistics. Let the de sity matrix be ρ {M} =exp [-βΣMl=1H_l], in which the operators Hl's are not always commutable. ρ {M} can be expanded in series of the form
ρ {M} =ΣMΣmi=0 n≥0 Σ {ml} _M--- {m_n} Mρ* {mM₁} ---ρ* {m_n} ρ {M-m₁----m_n}
where. the sets {m₁}, --- {m_n} are groups of operators chosen from the set {M} and operators belonging to different groups are commutable. The summation is to be taken over all the possible choice of the sets {m₁} --- {m_n}.
ρ {k} is defined by
ρ {k} = [Π {k} exp (-βH_j)] s where suffix s means the symmetrization by changing the order of the products. And ρ* {m} is defined by
ρ* {m} =Σ _{k}l (-) ^m-k ρ {k} ρ {m-k}
which is proved to be 0 (β^m+1). the series expansion here proved is very useful in quantum statistics.