The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.
Components of a field function are functions of coordinates and time and satisfy some definite differential equations, which are called field equations. Field equations can be given a very general variational or Lagrangian form, if one accepts, that a field as the dynamic system must be characterized by a definite Lagrangian function. For a mechanical system this function is written down as the sum over all the material points of the system. For a continuous system like a wave field, Lagrangian function is expressed by the space integral from Lagrangian function density: L(t) = ∫ Ld3x. (4.1) However, the variational principle deals not with the Lagrangian function but with an action S obtained from it by means of integration over x0 = ct S = c ∫ L(t)dt. (4.2) Therefore, noncovariant expression (4.1) in Lagrangian formalism is in fact intermediate and it is sufficient to consider the Lagrangian function density L, which depends on the four space-time coordinates x in an implicit way. Hereinafter, we shall simply call L by the Lagrangian. Analogously, the Hamiltonian function density H will be named the Hamiltonian. The Lagrangian depends on components of a field wavefunction ψk and on their first derivatives with respect to the space-time coordinates ∂ψk/∂x ν . In so doing we assign the meaning of generalized field coordinates to ψk, and the meaning of generalized field velocities to ∂ψk/∂x ν . The demand of lacking derivatives with respect to field functions above the first order in the Lagrangian is caused by the following reason. Dynamics of all known quantum fields is described either by equations of the first or the second order. On the one hand, these equations follow from the principle of a least action: δS = 0, (4.3) and under action variation the order of derivatives, entering L, is raised by one. The Lagrangian must be a relativistically covariant quantity, namely, a relativistic scalar L′(x′) = L(x). Thanks to the invariance under the space-time translations, we conclude that the Lagrangian must not manifestly contain any space-time coordinates x. Since dynamical observables (four-dimensional vector of the energy-momentum, the moment of momentum tensor, and so on) are real quantities, then the Lagrangian must be also a real quantity. We shall also impose the locality condition on the Lagrangian, that is, we assume that the Lagrangian depends only on the state of fields in the infinitely small vicinity of the point Particles, Fields, and Quantum x. So: L(x) = L(ψk, ∂ψk ∂xν ) = L (ψk(x), ψk;ν (x)) . General variation of an action, related with varying both field wave functions and of integration domain boundaries, is equal to: δS = δ ∫ L(x)dx = ∫ L(x′)dx′ − ∫ L(x)dx, (4.4) where dx = dx0dr. For L′(x′) we have the expression: L′(x′) = L (ψ′k(x′), ψ′k;ν(x′)) = L(x) + δL(x) = L(x) + δL(x) + dLdxλ δxλ, (4.5) In Eq. (4.5) we denote a form variation of corresponding quantities by the symbol δ. This way δL(x) represents varying the Lagrangian only on the account of the variation of field functions and field function derivatives: δL(x) = ∂L ∂ψk δψk + ∂L ∂ψk;ν δψk;ν , (4.6) and δψk(x) does a variation of a field function form: δψk(x) = ψ ′ k(x) − ψk(x).
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 variational formalism for the determination of the spinless single-particle density matrix $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ for an interacting many-particle system, correct to second order, is developed. The method is based on variational principles for determining single-particle expectation values, correct to second order, without requiring highly accurate wave functions in order to represent the interacting system. The expectation value of the operator
X=Y+iZ, where $Y=(\frac{1}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})+W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $Z=\ensuremath{-}(\frac{i}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})\ensuremath{-}W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j})=\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, ${T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})$ is a translation operator, and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}={\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, is the reduced single-particle density matrix. For a given choice of a Slater-determinant-type trial wave function it is shown that the density matrix may be obtained from the charge density of the system where the latter too is determined via the same formalism to second order as the expectation value of the operator
W employing the same system trial wave function. The applicability of the technique is then demonstrated by showing that for the model hydrogen-atom problem it leads to highly accurate results for the momentum density and the Compton profile in the impulse approximation, and that the virial theorem is closely satisfied, even when the calculations of the various properties employing an approximate trial wave function alone are substantially in error.
The variation principle has not been well defined in quantum mechanics for a Lagrangian containing a velocity dependent potential. We investigate the method of variation to derive an Euler-Lagrange equation in connection with Noether's theorem when the Lagrangian is invariant under some point transformation. Obtained are the Euler-Lagrange equation retaining the above invariance by means of the variation method and the Hamiltonian with which the canonical equation of motion leads to the Euler-Lagrange equation.