Variational methods

(quantum mechanics) A method of calculating an upper bound on the lowest energy level of a quantum-mechanical system and an approximation for the corresponding wave function; in the integral representing the expectation value of the Hamiltonian operator, one substitutes a trial function for the true wave function, and varies parameters in the trial function to minimize the integral.

Methods based on the principle that, among all possible configurations or histories of a physical system, the system realizes the one that minimizes some specified quantity. Variational methods are used in physics both for theory construction and for calculational purposes.

The earliest use of a variational principle for physics is Fermat's principle in optics, which states that when a light ray traverses a medium with nonuniform index of refraction its path is such as to minimize its travel time. An integral expresses the time that the light takes to travel from one point to another along a particular path, and an application of the calculus of variations to this integral makes it possible to determine the particular path for which the travel time is a minimum. This problem is mathematically identical to the variational principle that determines a geodesic, the path of shortest distance, in a given geometry. In that form, the same principle determines the world lines of all objects in the general theory of relativity. See also Relativity; Riemannian geometry.

Similarly, in mechanics, Hamilton's principle for the action is defined for any system of point particles by an integral (called the action) that extends over an arbitrarily prescribed path Γ in configuration space. Hamilton's principle asserts that the trajectories of all the particles are determined by the requirement that be such that, for given initial and final times, the action is a minimum; for this reason it is also called the principle of least action. If the calculus of variations is applied to implement this principle, the corresponding Euler-Lagrange equations are obtained. These are the lagrangian equations of motion, that is, Newton's equations of motion in lagrangian form. See also Action; Hamilton's principle; Lagrangian function; Least-action principle.

The principle of least action has been generalized to systems with infinitely many degrees of freedom, that is, fields. A Lagrange density function is then defined, which is a function of the fields and their time derivatives at any given point in space and time. For any field theory, only the Lagrange density needs to be given; the field equations are then derivable as the corresponding Euler-Lagrange equations. A similar technique makes it possible to derive the Schrödinger equation and the Dirac equation in quantum mechanics from specific Lagrange density functions. See also Quantum mechanics; Quantum theory of matter; Relativistic quantum theory.

This method has great procedural advantages. For example, it facilitates a check of whether the theory satisfies certain invariance principles (such as relativistic invariance or rotational invariance) by simply ascertaining whether the Lagrange density satisfies them. The corresponding conservation laws can also be derived directly from the lagrangian. See also Conservation laws (physics); Quantum field theory; Symmetry laws (physics).

The variational method also plays an important role in quantum-mechanical calculations. For the computation of needed quantities in terms of functions that result from the solution of differential equations, it is always of great advantage to use formulas that have the special form required to make them stationary with respect to small variations of the input functions in the vicinity of the unknown, exact solutions. See also Minimal principles.