Synonym: Rayleigh-Ritz method
Schroedinger's differential equation can always be solved by the variation method -- even when analytical methods fail. Many-electron atoms and molecules, for which "exact" solutions are almost never possible, are commonly solved on modern computers using the variation method. In its commonest application, the variation method uses an expansion in a basis to transform the DE into a matrix equation or linear algebra problem.
Suppose the system hamiltonian operator is H and the problem is to solve the Schroedinger equation, Hyn = Enyn, for eigenvalues and eigenfunctions. None of the usual methods for analytical solutions may work if the system is complicated. Yet, we will see that the Rayleigh quotient provides the means to approximately solve the problem.
Let F be an arbitrary state
function of the system -- not necessarily an energy eigenfunction. Assume only that F belongs to the space spanned by the (unknown) eigenfunctions; for
instanceF is continuous, finite, differentiable,
and satisfies the boundary conditions. Define the Rayleigh quotient W(F) as
According to a postulate of quantum mechanics, W(F) is the average value of a great many energy measurements on the system prepared in the state F.
Theorem: If H is a self-adjoint (Hermitian) operator with a purely discrete spectrum and if E0 is its smallest eigenvalue, then the minimum value of W(F) is E0. This minimum is attained when F = y0, the eigenfunction corresponding to E0. The proof begins with the expansion for F in the basis of yn: where cn are the (unknown) expansion coefficients. Such an expansion is known to exist because the eigenfunctions, yn, are a complete basis (even though they are unknown). Next, the expansion is substituted into the Rayleigh quotient and the integrals are performed using Hyn = Enyn and the orthornormality of eigenfunctions. The result is . Finally, the order of eigenvalues is used, E0£En for all n to replace En by E0 and the equality by £ to obtain: W(F) £ E0. If we choose F=y0, then W becomes W(y0)=E0. QED.
The idea for using this result is to allow the function F to vary through some extended subspace of possible state functions and to select the state with lowest Rayleigh quotient. Then this lowest W is a more or less good approximation to E0, and the corresponding F is an approximation to y0. We use the name "trial function" for F and trial energy for W(F).
A powerful way to use the variation theorem is to expand the trial function in a basis of possible trial functions and minimize W with respect to the expansion coefficients. That is, let . Here there are M basis functions and M unknown linear coefficients, an. Trial basis functions are taken to be linearly independent but not necessarily orthonormal; instead their inner products form the elements of a M by M matrix called the "overlap matrix": . The numerator of the Rayleigh quotient requires the M by M "hamiltonian matrix" defined by . Now W is implicitly a function of the an coefficients, namely, . The minimum trial energy is found by differentiating with respect to each an (treat an* as independent of an) and setting the derivatives of W to zero. The end result is a matrix eigenvalue eigenvector problem called the "secular equation":
or, in matrix notation
where H and S are the Hamiltonian and overlap matrices and a is the column of expansion coefficients. This linear algebra problem is an approximation for the original Schroedinger differential equation.
The secular equation has M eigenvalues and eigenvectors. The lowest eigenvalue is an approximation to the ground state system energy and the corresponding eigenfunction provides an approximation to the ground state wave function (when substituted into the expansion for F in the basis of fn). When the secular equation is rewritten as (H-WS)a = 0 (0 is the null vector with zero components), it is seen that a non-trivial solution exists only if det(H-WS)=0, called the "secular determinant". Since the secular determinant is M by M, it has M roots: the eigenvalues of the the secular equation.
The variation method is not limited to the ground state. It can be shown that the M roots of the secular determinant are sequentially upper bounds for excited states. [J.K.L. MacDonald, Phys. Rev., 43, 830 (1933)] Denote the roots of the secular equation by Wk for k=0,1,2,...M-1. Then, not only do we have E0£W0, but also Ek£Wk.
Solve the particle in a box (PIB) problem using linear variation and for trial function use a linear combination of polynomials. Let the center of the box be placed at the origin: V=0 when -L/2 < x < L/2. Solution
Another way to vary the trial function is to scale the distances. Suppose F(x) is a trial function; then introduce the family of functions f(a,x)= F(ax). As a varies the trial function expands and contracts. The trial energy becomes a function of a: W(a) = <f(a,x)|H|f(a,x)>/<f(a,x)|f(a,x)>. The objective is to minimize the Rayleigh quotient with respect to a.
Created or up-dated 08/03/99
by R.D. Poshusta