Ritz method
Encyclopedia
The Ritz method is a direct method to find an approximate solution for boundary value problem
s. The method is named after Walter Ritz
.
In quantum mechanics
, a system of particles can be described in terms of an "energy functional" or Hamiltonian
, which will measure the energy of any proposed configuration of said particles. It turns out that certain privileged configurations are more likely than other configurations, and this has to do with the eigenanalysis ("analysis of characteristics") of this Hamiltonian system
. Because it is often impossible to analyze all of the infinite configurations of particles to find the one with the least amount of energy, it becomes essential to be able to approximate this Hamiltonian in some way for the purpose of numerical computations
.
The Ritz method can be used to achieve this goal. In the language of mathematics, it is exactly the finite element method
used to compute the eigenvectors and eigenvalues of a Hamiltonian system.
,
, is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration.
It can be shown that the ground state energy,
, satisfies an inequality:
That is, the ground-state energy is less than this value.
The trial wave-function will always give an expectation value larger than the ground-energy (or at least, equal to it).
If the trial wave function is known to be orthogonal
to the ground state, then it will provide a boundary for the energy of some excited state.
The Ritz ansatz function is a linear combination of N known basis functions
, parametrized by unknown coefficients:
With a known Hamiltonian, we can write its expected value as
The basis functions are usually not orthogonal, so that the overlap matrix
S has nonzero nondiagonal elements. Either
or 
(the conjugation of the first) can be used to minimize the expectation value. For instance, by making the partial derivatives of 
over 
zero, the following equality is obtained for every k = 1, 2, ..., N:
which leads to a set of N secular equations:
In the above equations, energy
and the coefficients 
are unknown. With respect to c, this is a homogeneous set of linear equations, which has a solution when the determinant
of the coefficients to these unknowns is zero:
which in turn is true only for N values of
. Furthermore, since the Hamiltonian is a hermitian operator, the H matrix is also hermitian and the values of 
will be real. The lowest value among 
(i=1,2,..,N), 
, will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the wave function of state i can be obtained by finding the coefficients 
from the corresponding secular equation.
is precisely the stiffness matrix of the Hamiltonian in the piecewise linear element space, and the matrix 
is the mass matrix. In the language of linear algebra, the value 
is an eigenvalue of the discretized Hamiltonian, and the vector 
is a discretized eigenvector.
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
s. The method is named after Walter Ritz
Walter Ritz
Walther Ritz was a Swiss theoretical physicist.His father, Raphael Ritz, a native of Valais, was a well-known landscape and interior scenes artist. His mother was the daughter of the engineer Noerdlinger of Tübingen. Ritz studied in Zurich and Göttingen...
.
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...
, a system of particles can be described in terms of an "energy functional" or Hamiltonian
Hamiltonian (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...
, which will measure the energy of any proposed configuration of said particles. It turns out that certain privileged configurations are more likely than other configurations, and this has to do with the eigenanalysis ("analysis of characteristics") of this Hamiltonian system
Hamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
. Because it is often impossible to analyze all of the infinite configurations of particles to find the one with the least amount of energy, it becomes essential to be able to approximate this Hamiltonian in some way for the purpose of numerical computations
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
.
The Ritz method can be used to achieve this goal. In the language of mathematics, it is exactly the finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
used to compute the eigenvectors and eigenvalues of a Hamiltonian system.
Discussion
As with other variational methods, a trial wave functionAnsatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
,
, is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration.It can be shown that the ground state energy,
, satisfies an inequality:That is, the ground-state energy is less than this value.
The trial wave-function will always give an expectation value larger than the ground-energy (or at least, equal to it).
If the trial wave function is known to be orthogonal
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...
to the ground state, then it will provide a boundary for the energy of some excited state.
The Ritz ansatz function is a linear combination of N known basis functions
, parametrized by unknown coefficients:With a known Hamiltonian, we can write its expected value as
The basis functions are usually not orthogonal, so that the overlap matrix
Overlap matrix
The overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basis vectors of a quantum system. In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basis vectors form an...
S has nonzero nondiagonal elements. Either
or (the conjugation of the first) can be used to minimize the expectation value. For instance, by making the partial derivatives of over zero, the following equality is obtained for every k = 1, 2, ..., N:which leads to a set of N secular equations:
In the above equations, energy
and the coefficients are unknown. With respect to c, this is a homogeneous set of linear equations, which has a solution when the determinantDeterminant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the coefficients to these unknowns is zero:
which in turn is true only for N values of
. Furthermore, since the Hamiltonian is a hermitian operator, the H matrix is also hermitian and the values of will be real. The lowest value among (i=1,2,..,N), , will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the wave function of state i can be obtained by finding the coefficients from the corresponding secular equation.The relationship with the finite element method
In the language of the finite element method, the matrix is precisely the stiffness matrix of the Hamiltonian in the piecewise linear element space, and the matrix is the mass matrix. In the language of linear algebra, the value is an eigenvalue of the discretized Hamiltonian, and the vector is a discretized eigenvector.Papers
- Walter Ritz (1909) "Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik" Journal für die Reine und Angewandte Mathematik, vol. 135, pages 1–61. Available on-line at: http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=261182 .
- J.K. MacDonald, "Successive Approximations by the Rayleigh–Ritz Variation Method", Phys. Rev. 43 (1933) 830