An Improved Lower Bound to the Ground-State Energy.

Area of Science:

• Quantum mechanics
• Computational physics
• Linear algebra

Background:

• The Arnoldi iterative method is used for eigenvalue problems.
• A 90-year-old lower-bound estimate for ground-state energy by Temple exists.
• Hamiltonian operator effects can be decomposed into parallel and perpendicular contributions.

Purpose of the Study:

• To improve the lower-bound estimate of the ground-state energy.
• To apply the Arnoldi method to enhance Temple's bound.
• To demonstrate the improved bound's validity for Hermitian operators.

Main Methods:

• Utilizing the decomposition of Hamiltonian operator effects.
• Applying the Arnoldi iterative method.
• Calculating improved lower bounds for specific potentials.

Main Results:

• A significantly improved lower-bound estimate for the ground-state energy was derived.
• The method was exemplified using quartic and Morse potentials.
• The improved bound is applicable to any bounded-below Hermitian operator.

Conclusions:

• The enhanced lower bound provides a more accurate estimation of ground-state energy.
• The Arnoldi method offers a powerful tool for refining energy bounds in quantum mechanics.
• This approach has broad applicability for Hermitian operators with discrete spectra.