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.
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.
Upper bounds are given for the weight distribution of binary weakly self-dual codes. To get these new bounds, we introduce a novel method of utilizing unitary operations on Hilbert spaces. This method is motivated by recent progress on quantum computing. This new approach leads to much simpler proofs for such genre of bounds on the weight distributions of certain classes of codes. Moreover, in some cases, our bounds are improvements on the earlier bounds. These improvements are achieved, either by extending the range of the weights over which the bounds apply, or by extending the class of codes subjected to these bounds.