A quantum mechanical curvature theorem

A general quantum mechanical curvature theorem of the form d²ϵ/dχ² ⩽ 〈ψ¦d²H/d벦ψ〉 is established by means of perturbational-variational theory; here, ψ and ϵ are the exact eigenfunctions and eigenvalues of the Schrödinger time-independent equation, H is the Hamilton operator, and λ is any real parameter occurring in H. The theorem is also established for arbitrary optimum variational solutions to the Schrödinger equation. Several applications of the curvature theorem are discussed in conjunction with perturbation theory and the stability of approximate wave functions.