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.
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.
Recent work [Phys. Rev. A 9, 108 (1974)] on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Phys. Rev. A 9, 108 (1974) originally were based.
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.
For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.
The second order error δF(2), made by employing the Schwartz−Dalgarno−Delves variational principle for the diagonal matrix element φ†Wφ of an arbitrary Hermitian operator W, is examined in the case that φ is the bound ground state eigenfunction of some given Hamiltonian H. This variational principle characteristically involves not only a trial estimate φt of φ, but also a trial estimate Lt(φt) Lt(φt) of a well−defined but generally not exactly known auxiliary function L (φ). It has previously been shown that, for certain special choices of a trial Hamiltonian Ht, the trial Lt(φt) can be found from a minimum principle. The present work finds that for these same special Ht it is possible to express δF(2) in comparatively simple closed form, depending only on known quantities, so that δF(2) should be calculable when the system described by H is not too complicated. However, these results for δF(2) are obtained on the assumption that Lt(φt) is known essentially exactly for any given φt; the practical utility of the formulas derived still must be tested, therefore. If Lt(φt) can be determined to this necessary accuracy, one expects that combining the computed δF(2) with the usual variational estimate of φ†Wφ will be a significant improvement over the usual variational estimate alone. Under the same circumstances, when W is a positive definite operator, the expression for δF(2) can provide nonrigorous but nonetheless potentially useful second order (variational) upper and lower bounds on the exact φ†Wφ.