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The variational method is, in quantum mechanics, one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. The basis for this method is the variational principle. Wikipedia does not have an article with this exact name. ... Mathematical method to convert divergent power series in a small expansion parameter, say , into convergent series in powers , where is a critical exponent. ... Fig. ... In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... A variational principle is a principle in physics which is expressed in terms of the calculus of variations. ...

Introduction

Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the discrete spectrum of H and the corresponding eigenspaces of each eigenvalue λ (see spectral theorem for Hermitian operators for the mathematical background): In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Mathematical formulation In mathematics and physics, the spectrum of an operator, in particular self adjoint or normal operators, can be classified via its spectral measures. ... In mathematics and physics, discrete spectrum of an operator on Hilbert space is the part of the spectrum which corresponds to discrete spectral measures. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

with

and

.

Physical states are normalized, meaning that their norm is equal to 1. Once again ignoring complications involved with a continuous spectrum of H, suppose it is bounded from below and that its greatest lower bound is E₀. Suppose also that we know the corresponding state |ψ>. The expectation value of H is then In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ... In probability (and especially gambling), the expected value (or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are...

Ansatz

Obviously, if we were to vary over all possible states with norm 1 trying to minimize the expectation value of H, the lowest value would be E₀ and the corresponding state would be an eigenstate of E₀. Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters α_i (i=1,2..,N). The choice of the subspace is called the ansatz. Some choices of ansatzes lead to better approximations than others, therefore the choice of ansatz is important. Ansatz is a term (from German) often used by physicists. ...

Let's assume there is some overlap between the ansatz and the ground state (otherwise, it's a bad ansatz). We still wish to normalize the ansatz, so we have the constraints In physics, the ground state of a quantum mechanical system is its lowest-energy state. ...

and we wish to minimize

.

This, in general, is not an easy task, since we are looking for a global minimum and finding the zeroes of the partial derivatives of ε over α_i is not sufficient. If ψ (α_i) is expressed as a linear combination of other functions (α_i being the ceofficients), as in the Ritz method, there is only one minimum and the problem is straightforward. There are other, non-linear methods, however, such as the Hartree-Fock method, that are also not characterized by a multitude of minima and are therefore comfortable in calcualtions. A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ...

There is an additional complication in the calculations described. As ε tends toward E₀ in minimization calculations, there is no guarantee that the corresponding trial wavefunctions will tend to the actual wavefunction. This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method. A wavefunction different from the exact one is obtained by use of the method described above.

Although usually limited to calculations of the ground state energy, this method can be applied in certain cases to calculations of excited states as well. If the ground state wavefunction is known, either by the method of variation or by direct calculation, a subset of the Hilbert space can be chosen which is orthogonal to the ground state wavefunction.

| ψ > = | ψ_test > − < ψ_gr | ψ_test > | ψ_gr >

The resulting minimum is usually not as accurate as for the ground state, as any difference between the true ground state and ψ_gr results in a lower excited energy. This defect is worsened with each higher excited state.

See also

• Hartree-Fock method
• Ritz method