Pressure in the Average-Atom Model

The (well-known) quantum mechanical expression for the stress tensor is derived and applied to obtain a formula for the pressure in the average-atom model. This average-atom pressure formula reduces to the (well-known) expression for the pressure in a classical free-electron gas when the average-atom continuum wave functions are replaced by free-electron wave functions. 1 Derivation We start with the time-dependent Schrödinger equation for an electron in a potential V (r), − h̄ i ∂ψ ∂t = − h̄ 2 2m ∇2ψ + V ψ (1) The expectation value of i-th component of the electron’s momentum inside a region R is 〈pi〉 = ∫ R dτ ψ†piψ. (2) The rate of increase of momentum in R is d dt 〈pi〉 = ∫ R dτ [ ∂ψ† ∂t piψ + ψ†pi ∂ψ ∂t ] = − ih̄ 2m ∫ R dτ [ ∇2ψ†piψ − ψ†pi∇2ψ ] − i h̄ ∫ R dτ ψ† [piV − V pi]ψ. (3)