Express the velocity of each point of a rigid body in terms of the time derivative of the rotation matrix applied to the initial position of the point and hence integrate its norm square over the entire rigid body to obtain the kinetic energy of the rigid body as a quadratic function of the time derivative of the rotation matrix with the quadratic function being determined in terms of the moment of inertia matrix of the body. Express the gravitational potential of the rigid body as a linear function of the rotation matrix and hence determine the total Lagrangian of the body in terms of the instantaneous rotation matrix and its time derivative. Perform the Legendre transformation on this Lagrangian to obtain the Hamiltonian of the rigid body in terms of the Lie algebra coordinates and their corresponding canonical momenta.
Express the velocity of each point of a rigid body in terms of the time derivative of the rotation matrix applied to the initial position of the point and hence integrate its norm square over the entire rigid body to obtain the kinetic energy of the rigid body as a quadratic function of the time derivative of the rotation matrix with the quadratic function being determined in terms of the moment of inertia matrix of the body. Express the gravitational potential of the rigid body as a linear function of the rotation matrix and hence determine the total Lagrangian of the body in terms of the instantaneous rotation matrix and its time derivative. Perform the Legendre transformation on this Lagrangian to obtain the Hamiltonian of the rigid body in terms of the Lie algebra coordinates and their corresponding canonical momenta.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
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.
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).
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.
Abstract A system of 4 point masses (without spin) is considered. The potential energy is assumed to be a function of the distances between the particles only. It is possible to express the kinetic energy relative to the center of mass as the kinetic energy of 3 "reduced" masses. The different possibilities for that procedure are discussed for systems with an arbitrary number of particles. The positions of the 3 reduced masses are described by 9 coordinates. 6 of them, the "internal" coordinates, are needed to determine the internal structure of the system, and 3, the "external" coordinates, for its orientation in space. The classical Hamiltonian in such a set of coordinates is derived. In a first step the Hamiltonian is expressed in terms of linear and angular momenta. Then these quantities are determined as functions of the coordinates and the conjugate momenta by infinitesimal trans-formations. From the classical Hamiltonian we get the quantum mechanical one by the well-known translation rule. It can be expressed in terms of the linear and angular momentum operators quite analogous to the classical case. For the solution of the time-independent Schrödinger equation the rotational symmetry of the system is used. The wave function is expanded in terms of the eigenfunctions of the total angular momentum. This leads to a system of coupled differential equations for the expansion-coefficients which depend on the 6 internal coordinates only. The further symmetries of the system (reflection and particle exchange) appear in symmetry properties of these functions. Differences and similarities in the solution theory for the 4-body-problem with respect to the 3-body-problem are discussed.