Guide to the notebooks on variational problems - Chap. 10

The variational technique is founded on the principle

δ ((⟨ψ|H|ψ⟩)/(⟨ψ|ψ⟩)) = 0

The ground state of the Hamiltonian H is the state on the unit sphere of the Hilbert space (i.e. normalizable states) which minimizes the expectation value  of H.

The practical way in which most of the applications of this principle are done is based on the choice of an ensemble of base function, usually a subset of a complete set of functions, possibly depending on some parameters λ. The state is expanded in this basis as

ψ = (∑_(i=1)^N)(Subscript[c, i])(Subscript[f, i])

Once substituted in () these relations give rise to an eigenvalue equation of the form

(Subscript[H, ij]) (Subscript[c, j]) = ℰ[λ] (Subscript[N, ij]) (Subscript[c, j]);

with

(Subscript[H, ij])=⟨(Subscript[f, i])|H|(Subscript[f, j])⟩;            (Subscript[N, ij])= ⟨(Subscript[f, i])|(Subscript[f, j])⟩.

The eigenvalue ℰ[λ] is the (normalized) expectation value of H and its subsequent minimization with respect to λ yields an approximate solution of our problem, with an optimal set of parameters (Subscript[λ, m]) and corresponding eigenvalues ℰ[(Subscript[λ, m])] and eigenvectors. This procedure will converge to the exact solution in (𝕃^2)norm as N → ∞.

In the text it has been shown that higher eigenvalues of equation () will provide an estimate for excited states.

In real life N is always finite and the number of independent parameters λ is limited, in our examples only an overall scale parameter will be used. This has at least three practical implications.

In the simplest case of one dimensional problems, and in multidimensional cases in which an analogous strategy is pursued, the choice of the basis is usually dictated by the domain. As a general rule the expansion in Chebyshev polynomials is the most stable and fruitful (it is essentially a Fourier series in disguise). On infinite symmetric domains  [-∞,+∞] the Hermite polynomials times Exp[- (x^2)] are preferred, and the reader will recognize harmonic oscillator basis, while in semi-infinite domains [0,+∞] the recommended basis is the Laguerre polynomials times decreasing exponents, very similar to the hydrogen wave functions. Fourier series in some way interpolate between all these methods: it is the preferred method for periodic problems, but if we imagine our system enclosed in a large box of dimension L, then for L large enough one can approximate the system with the eigenfunctions of a free particle in the box, closely related to a Fourier series.

In studying the variational method we have thus two possible routes: either we can follow the method or focus our attention on a particular physical system. We have chosen an intermediate path:

One of the weak point of all these algorithms is the computation of matrix elements of the potential. In principle we have just to compute some numerical integrals. If we choose a basis of 100 element the symmetric matrix of the potential V has more than 5000 elements, in general. A computation with NIntegrate of all these elements is usually prohibitive, also because in the process of minimization for the parameter λ one has to make a function call for each step of the minimization procedure to the function which diagonalize the system (), and at each step the matrix change. An important exception is the case of homogenous potentials, in this case if λ is an overall scale the integrals must be computed only once. In physical interesting cases as Helium, anharmonic oscillators etc. the integrals can be computed analytically and this usually (but not always) solves the above mentioned problem. In other cases it can be of some interest to have a fast albeit approximate routine for computing integrals, after all we do not pretend to compute every integral, but only products of more or less smooth functions with orthogonal polynomials like Laguerre or Hermite polynomials. Up not too large dimension of the basis and for not too crazy potential functions a substantial help can be found in the use of Gauss weights, especially suited for each particular class of orthogonal polynomials. As this technique is not in the ordinary curriculum studiorum of physics students we added a small introductory notebook with several examples, and this will help us in some otherwise intractable (in a reasonable amount of time)  problems.

List

Here is the list of the notebooks of this chapter, with a short description of their content. The color sequence ●, ●, ● is a rough indication of the difficulty.