Example 5.7: Density matrix equation Show with the help of the Schrodinger equation i t (x,

Example 5.7: Density matrix equation Show with the help of the Schr¨odinger equation i

∂

∂t

ψ(x, t) = Hψ(x, t), (5.66)

that the density operator ρN, i.e.

ρN(β) = e−βH (5.67)

(without normalization), satisfies in the configuration space matrix representation the equation

∂ρN(x, x; β)

∂β

= −HxρN(x, x; β), (5.68)

where the subscript x indicates that H acts on x of ρN(x, x; β).‡‡ Note: Observe that the Schr¨odinger equation describes a single, so-called pure state. The thermal density equation, however, with temperature dependence (β = 1/kT) describes the overall state of a system with mixed states. The equation can also be found with the names of Liouville and/or von Neumann attached.

‡‡See e.g. H.J.W. M¨uller-Kirsten [47], Sec. 5.4.1.