Proof of non-extremal nature of excited-state energies in nonrelativistic quantum mechanics with the use of constrained derivatives

TL;DR

This paper proves that excited states in nonrelativistic quantum mechanics are saddle points of the energy functional using constrained second derivatives, and provides a method to determine their Morse index.

Contribution

It introduces a second derivative test based on constrained derivatives to rigorously classify excited states as saddle points and determine their Morse index.

Findings

Excited states are saddle points of the energy expectation value.

A method to determine the Morse index of excited states is provided.

The approach clarifies the stability properties of excited states.

Abstract

With the use of a second derivative test based on constrained second derivatives, a proof is given that excited states in nonrelativistic quantum mechanics are saddle points of the energy expectation value, and is shown, further, how to determine their Morse index.