We describe a systematic procedure for the construction of variational principles for the variational estimation of an extremely wide class of functionals $F({\ensuremath{\varphi}}^{\ifmmode\dagger\else\textdagger\fi{}},\ensuremath{\varphi})$. The quantities $\ensuremath{\varphi}$ are well defined but not known exactly because the equations defining the set $\ensuremath{\varphi}$ are not solvable exactly. The procedure is illustrated for
F, the off-diagonal matrix element ${\ensuremath{\varphi}}_{1}^{\ifmmode\dagger\else\textdagger\fi{}}\mathrm{W}{\ensuremath{\varphi}}_{2}$ of an arbitrary Hermitian linear operator
W, for ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ nondegenerate normalized bound-state eigenfunctions of the same Hamiltonian with a specified relative phase.
Upper and lower bounds were obtained previously on matrix elements of the form ${W}_{\mathrm{nm}}=({\ensuremath{\psi}}_{n},W{\ensuremath{\psi}}_{m})$, where
W is a Hermitian operator and ${\ensuremath{\psi}}_{n}$ and ${\ensuremath{\psi}}_{m}$ are the wave functions of the
nth and
mth states of the system. The bounds are variational but nonstationary; they are expressed in terms of trial wave functions ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$ containing variational parameters, but the error in the bound is of first order in the errors in the ${\ensuremath{\Psi}}_{\mathrm{nt}}$ and ${\ensuremath{\Psi}}_{\mathrm{mt}}$. The results have been either subject to rather restrictive conditions (for example, only for certain specific choices for
W and only for real wave functions) or have been very conservative. We remove most of these restrictions (
W need not be positive or negative definite, the wave functions may be complex, the system may not even be invariant under time reversal) but maintain rigorous bounds of good quality. The method of using Gram-determinant inequalities, which has been employed previously, especially by Weinhold, and which we adopt, leads to variational but nonstationary bounds on
Wnm in terms of "simple" upper bounds (which may be poor) on
W2nn. Here again, only for a very few particular choices of
W have such simple bounds on
W2nn been given previously (for example,
n restricted to be the ground state, and
W the operator
zi, the coordinate of the
ith electron). The main result of this paper is to show that such simple upper bounds can be obtained for a very wide class of operators
W in terms of the energy eigenvalues of the Hamiltonian. (They can be improved if given additional experimental information on oscillator strengths, for example). These simple bounds on
W2nn do not involve any trial wave functions. The method of variational but nonstationary bounds is illustrated for diagonal matrix elements of
r1 and
r12---we, therefore, require simple bounds on
r12 and
r14---for the states $1{s}^{2} ^{1}S$ and
1s2s3S of the helium atom, with rather good results.
We develop variational principles and variational identities for bound state and continuum wavefunctions in a general context, paying particular attention to the proper choice of defining equations and boundary conditions which will lead to unique and unambiguous wavefunctions even when these functions are complex. Any functional, such as a matrix element, calculated with such a variationally determined wavefunction, will also be accurate to second order in the error of the starting choice. This provides, therefore, an alternative procedure for getting variational estimates of matrix elements to the one that already exists in the literature and we establish the equivalence of the two. Of even more interest is the possibility which now seems open of going beyond the variational principle and generating ’’supervariational’’ estimates of wavefunctions and matrix elements which are good to better than second order. We also give a simple prescription for the construction of variational identities for wavefunctions, that is, identities which lead readily to variational principles and, more significantly, might well serve as a starting point for the development of variational bounds.
The use of variational principles as a calculational tool is reviewed, with special emphasis on methods for constructing such principles. In particular, it is shown that for a very wide class of problems it is possible to construct a variational principle (VP) for just about any given quantity
Q of interest, by routine procedures which do not require the exercise of ingenuity; the resultant VP will yield an estimate of
Q correct to second order whenever the quantities appearing in the VP are known to first order. The only significant requirement for application of the routine procedures is that the entities which enter into the definition of
Q be uniquely specified by a given set of equations; the equations may involve difference or differential or integral operators, they may be homogeneous or inhomogeneous, linear or nonlinear, self-adjoint or not, and they may or may not represent time-reversible systems. No numerical calculations are presented, but procedures for the construction of VP's are illustrated for numerous quantities
Q of physical interest, particularly those
Q arising in quantum-mechanical scattering and transition probability calculations. For pedagogical purposes VP's are also derived for several problems in classical and (simple) mathematical physics which the authors hope will prove instructive and perhaps even amusing. The quantum-mechanical quantities
Q whose VP's are examined include various matrix elements and the quantum-mechanical eigenfunctions themselves. Topics examined include some points which have not always been appreciated in the literature, such as the necessity for properly specifying the phase when complex eigenfunctions are involved, and the importance of avoiding, wherever possible, formulations requiring the inversion of singular operators. The basic element of the technique is the recognition that the defining equations of a system can be incorporated into the VP as constraints through the use of (generalized) Lagrange multipliers; these can be constants, scalar, vector, or tensor functions of one or more variables, operators, etc. In the typical problem, these Lagrange multipliers
L serve as a new set of adjoint functions or entities, and the construction of the VP simultaneously provides well-defined equations for the
L. Moreover, these
L often have ready physical significance themselves; for example, they often may be regarded as generalized Green's functions. The construction of the VP also readily yields so-called variational identities for the quantity
Q of interest; these identities give explicit (if formal) expressions for the error in the variational estimate of
Q. In some cases this error can be shown to have a definite sign, so that the VP actually is an extremum principle, that is, that it yields an upper or lower variational bound for
Q; however, our routine procedures for constructing VP's do not routinely yield extremum principles.
A minimum principle for the calculation of the scattering length, applicable when the ground-state wave function of the target system is known precisely, has been available for some time. When, as is almost always the case, the target wave function is imprecisely known, a minimum principle is available but the simple minimum principle noted above is not applicable. Further, as recent calculations show, numerical instabilities usually arise which severely limit the utility of even an ordinary variational approach. The difficulty, which can be traced to the appearance of singularities in the variational construction, is here removed through the introduction of a minimum principle, not for the true scattering length, but for one associated with a closely connected problem. This guarantees that no instability difficulties can arise as the trial scattering wave function and the trial target wave function are improved. The calculations are little different from those required when the target ground-state wave function is known, and, in fact, the original version of the minimum principle is recovered as the trial target wave function becomes exact. A careful discussion is given of the types of problems to which the method can be applied. In particular, the effects of the Pauli principle, and the existence of a finite number of composite bound states, can be accounted for.