The Lagrangian could be written in terms of kinetic and potential energy, functions of coordinates and velocity and the amplitude of the fields at that point. The essential condition is that the Lagrangian assumed for the field, when studied with Lagrange’s equation, gives Maxwell’s equations as solution and leads to the same wave equation. There are many other fields of physics and many mechanical systems to which the Lagrangian approach can be applied. The goal of the present text is to introduce the beginner to the least action principle and the Lagrangian approach in an elementary way. The chapter argues that the Lagrangian chosen for the fields involved must be an expression that leads to those fields and equations. In wave mechanics, particle properties are described by a so-called wave representing the classical matter field ψ used in Schrodinger’s approach of wave mechanics.
The Lagrangian could be written in terms of kinetic and potential energy, functions of coordinates and velocity and the amplitude of the fields at that point. The essential condition is that the Lagrangian assumed for the field, when studied with Lagrange’s equation, gives Maxwell’s equations as solution and leads to the same wave equation. There are many other fields of physics and many mechanical systems to which the Lagrangian approach can be applied. The goal of the present text is to introduce the beginner to the least action principle and the Lagrangian approach in an elementary way. The chapter argues that the Lagrangian chosen for the fields involved must be an expression that leads to those fields and equations. In wave mechanics, particle properties are described by a so-called wave representing the classical matter field ψ used in Schrodinger’s approach of wave mechanics.
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).
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
This chapter expresses the whole of quantum mechanics in terms of a small set of postulates, which embraces the behaviour of all known forms of matter. It introduces the postulates and illustrates how they are used, including how to apply them to problems of chemical interest, such as atomic and molecular structure and the properties of molecules. It also establishes the full significance of the Hψ=Eψequation and provides a foundation for its application. The chapter includes the solutions of the Schrödinger equation for a particle and the operator, which is a symbol for an instruction to carry out some action on a function. The principal mathematical difference between classical mechanics and quantum mechanics is that whereas in the former physical observables are represented by functions, in quantum mechanics they are represented by mathematical operators.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
A brief historical review of the Principle of Least Action is given, and it's methodological importance is emphasized: the basic equations of Classical Mechanics can be derived from the conjunction of the Principle Least Action with the Galilean Principle of Relativity and the hypothesis of homogeneity and isotropy of space and the homogeneity of time. In classical electrodinamics, the situation is similar: one can build the action keeping now in mind the Einstein's Principle of Relativity. By means of the solution of the Euler-Lagrange equations one obtain the equations of motion of charges in given fields (Lorentz force) and the first pair of Maxwell equations. Also, starting from the sources of the field (charges and currents), the second pair of Maxwell equations is obtained.\nIn the case of the General Relativity, the trajectories of a free particle are given by the geodesic lines in space-time. The Einstein equations are also obtained by finding the extreme of some action. A brief historical review of quantum theory is given, starting from the introduction of the quantum of action in physics, to the Schrödinger equation, which can be obtained from a variational principle, and the Heisenberg Uncertainty Principle. Interesting is the role of the classical action in the formulation of quantum mechanics by means of the Feynman path integrals. In it, the classical trajectory in the motion of a system between two configurations, it corresponds the limit in which the Planck constant tends to zero. In quantum electrodinamics, it is required also to start from a Lagrangian together with the properties of space-time simmetry, the gauge invariance and the CPT invariance. Finally, mention is made to the electroweak theory and to quantum cromodynamics