Is the variational estimate always too high?

For the ground state, yes; the principle guarantees the trial energy is an upper bound, so a lower estimate is always better. The bound is exact only when the trial wavefunction coincides with the true ground state.

Can the variational method find excited states?

Yes, with care; by restricting trial functions to be orthogonal to the ground state one can bound the first excited state, and the Rayleigh-Ritz method with a basis yields approximations to several low-lying states at once.

Variational Method in Quantum Mechanics

The variational method estimates the ground-state energy of a quantum system by guessing a trial wavefunction with adjustable parameters and minimizing the expected energy; the result is guaranteed never to fall below the true ground-state energy.

Definition

The variational method is an approximation technique in which the ground-state energy is estimated as the minimum, over a family of trial wavefunctions, of the expectation value of the Hamiltonian, which is rigorously an upper bound on the true ground-state energy.

Scope

The topic covers the variational principle that the expectation of the Hamiltonian in any normalized trial state is an upper bound on the ground-state energy, the use of parametrized trial wavefunctions and minimization over the parameters, the Rayleigh-Ritz method using a basis of trial functions, the extension to excited states by orthogonality, and applications such as the helium atom and molecular bonding.

Core questions

Why is the energy expectation in any trial state an upper bound on the ground-state energy?
How are trial wavefunctions chosen and their parameters optimized?
How does the Rayleigh-Ritz method extend the principle using a basis of functions?
How can the method be adapted to estimate excited states?

Key concepts

variational principle
trial wavefunction
upper bound on energy
Rayleigh-Ritz method
parameter optimization
excited-state estimates

Key theories

Variational principle: Because any trial state is a superposition of the true eigenstates, whose energies all exceed the ground-state energy, the expectation of the Hamiltonian is a weighted average bounded below by the lowest eigenvalue, so minimizing it over trial parameters approaches the ground-state energy from above.
Rayleigh-Ritz method: Choosing a finite basis of trial functions and minimizing the energy turns the problem into diagonalizing the Hamiltonian within that basis, providing systematically improvable upper bounds and forming the basis of practical electronic-structure calculations.

Clinical relevance

The variational method is the workhorse of quantum chemistry and condensed-matter theory: Hartree-Fock and configuration-interaction calculations rest on it, it gives accurate ground-state energies for helium and molecules, and it underlies modern variational and tensor-network methods for many-body systems.

History

The variational principle for energies originated with Rayleigh and was systematized by Ritz in 1909; in quantum mechanics it became central through Hartree's self-consistent field method and Fock's extension, which together founded computational quantum chemistry.

Key figures

Lord Rayleigh
Walther Ritz
Douglas Hartree
Vladimir Fock

Seminal works

griffiths2018
landau1977

Frequently asked questions

Is the variational estimate always too high?: For the ground state, yes; the principle guarantees the trial energy is an upper bound, so a lower estimate is always better. The bound is exact only when the trial wavefunction coincides with the true ground state.
Can the variational method find excited states?: Yes, with care; by restricting trial functions to be orthogonal to the ground state one can bound the first excited state, and the Rayleigh-Ritz method with a basis yields approximations to several low-lying states at once.