Variational principle
Encyclopedia
A variational principle is a scientific principle used within the calculus of variations
, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions. For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy.
According to Cornelius Lanczos
, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint
. These expressions are also called Hermitian
. Such an expression describes an invariant
under a Hermitian transformation.
Felix Klein
's Erlangen program
attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem
, the Poincaré group
of transformations (what is now called a gauge group) for general relativity
defines symmetries under a group of transformations which depend on a variational principle, or action principle
.
H, is known, if the Schrödinger equation
cannot be solved explicitly for the ground state wavefunction
, then the ground state energy is bounded by the following: For any normalized wavefunction φ, the expectation value of the Hamiltonian for φ must be greater than or equal to the actual ground state energy. Or in other words:
This holds for any trial φ since, by definition, the ground state wavefunction has the lowest energy, and any trial wavefunction will have energy greater than or equal to it.
Proof:
φ can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal):
Then, to find the expectation value of the hamiltonian:
Now, the ground state energy is the lowest energy possible, i.e.
. Therefore, if the guessed wave function φ is normalized:
The variational principle states that
The variational principle formulated above is the basis of the variational method used in quantum mechanics
and quantum chemistry
to find approximations to the ground state
.
The Harris functional
method is anti-variational (it is a lower bound on the energy).
Another facet in variational principles in quantum mechanics is that since
and 
can be varied separately (a fact arising due to the complex nature of the wave function), the quantities can be varied in principle just one at a time. see Landau, Quantum Mechanics , pg. 58 for some elaboration.
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions. For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy.
According to Cornelius Lanczos
Cornelius Lanczos
Cornelius Lanczos Löwy Kornél was a Hungarian-Jewish mathematician and physicist, who was born on February 2, 1893, and died on June 25, 1974....
, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator 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...
. These expressions are also called Hermitian
Hermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...
. Such an expression describes an invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under a Hermitian transformation.
Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
's Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...
attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem
Noether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
, the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
of transformations (what is now called a gauge group) for general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
defines symmetries under a group of transformations which depend on a variational principle, or action principle
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
.
Examples
- Variational principleEkeland's variational principleIn mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems....
of Ivar EkelandIvar EkelandIvar Ekeland is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and... - Fermat's principleFermat's principleIn optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light...
in geometrical opticsGeometrical opticsGeometrical optics, or ray optics, describes light propagation in terms of "rays". The "ray" in geometric optics is an abstraction, or "instrument", which can be used to approximately model how light will propagate. Light rays are defined to propagate in a rectilinear path as far as they travel in...
. - The principle of least actionPrinciple of least actionIn physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
in mechanicsMechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
, electromagnetic theory, and quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. - Maupertuis principle in classical mechanicsClassical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. - The Einstein equation also involves a variational principle, the Einstein-Hilbert actionEinstein-Hilbert actionThe Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
. - Gauss' principle of least constraint.
- Hertz's principle of least curvature
Variational principle in quantum mechanics
In a system in which the HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
H, is known, if the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
cannot be solved explicitly for the ground state wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
, then the ground state energy is bounded by the following: For any normalized wavefunction φ, the expectation value of the Hamiltonian for φ must be greater than or equal to the actual ground state energy. Or in other words:
This holds for any trial φ since, by definition, the ground state wavefunction has the lowest energy, and any trial wavefunction will have energy greater than or equal to it.
Proof:
φ can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal):
Then, to find the expectation value of the hamiltonian:
Now, the ground state energy is the lowest energy possible, i.e.
. Therefore, if the guessed wave function φ is normalized:In general
For a hamiltonian H that describes the studied system and any normalizable function Ψ with arguments appropriate for the unknown wave function of the system, we define the functionalFunctional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
The variational principle states that
-
, where 
is the lowest energy eigenstate (ground state) of the hamiltonian -
if and only if 
is exactly equal to the wave function of the ground state of the studied system.
The variational principle formulated above is the basis of the variational method used in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
to find approximations to the ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
.
The Harris functional
Harris functional
In computational condensed-matter physics, the Harris energy functional is a non-self-consistent approximation to Kohn-Sham density functional theory...
method is anti-variational (it is a lower bound on the energy).
Another facet in variational principles in quantum mechanics is that since
and can be varied separately (a fact arising due to the complex nature of the wave function), the quantities can be varied in principle just one at a time. see Landau, Quantum Mechanics , pg. 58 for some elaboration.