Potentials with non-trivial scales and the quantum-mechanical Virial theorem

Summary

For general, central potentials with non-trivial intrinsic scales, it is shown that the quantum-mechanical Virial theorem$\left\langle {\phi _m \left| T \right|\phi _m } \right\rangle = \left( {1/2} \right)\left\langle {\phi _m \left| {r \cdot \nabla V} \right|\phi _m } \right\rangle$ is also valid for any multi-parameter trial wave function ψ_t = (1/√b)- φ_t(r/b,a ₀, (a₁, …, a_r) providedb is chosen to minimize 〈ψ_t|T + V|ψ_t〉,i.e. one requires$\left( {\partial /\partial b} \right)\left[ {\left\langle {\psi _t \left| {T + V} \right|\psi _t } \right\rangle } \right] = 0$.