Spinless particle

Now, consider the case of spinless particles not subject to external electronic and magnetic fields. We may now choose the unitai7 operator U as the unit operator, that is, T = K. For the coordinate and momentum operators, one then obtains... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

For a spinless particle the total angular momentum, J, is equal to L, the orbital angular momentum. It may, however, be the case that the total angular momentum is not equal to Z1 as given by (9-52), but is of the form... 

In non-relativistic Schrodinger theory every component of the orbital angular momentum L = r x p, as well as L2, commutes with the Hamiltonian H = p2/2m + V of a spinless particle in a central field. As a result, simultaneous eigenstates of the operators H, L2 and Lz exist in Schrodinger theory, with respective eigenvalues of E, l(l + l)h2 and mh. In Dirac s theory, however, neither the components of L, nor L2, commute with the Hamiltonian 10. 

A system of N non-relativistic spinless particles is described, using standard terminology, by the classical Hamiltonian... 

The functions h (x, p) are the classical relativistic Hamiltonians of spinless particles with positive and negative energies, respectively. 

The solution of (2.3.69) is a purely mathematical problem well known in the theory of diffusion-controlled processes of classical particles. However, a particular form of writing down (2.3.69) allows us to use a certain mathematical analogy of this equation with quantum mechanics. Say, many-dimensional diffusion equation (2.3.69) is an analog to the Schrodinger equation for a system of N spinless particles B, interacting with the central particle A placed... 

C. Dewdney, P. R. Holland, A. Kyprianidis, and J. P. Vigier, Causal action at a distance in a relativistic system of two bound charged spinless particles Hydrogen-like models, Phys. Rev. D (Special Issue Particles and Fields) 31(10), 2533—2538 (1985). 

Figure 2. Envelope of the absorption edge part of the differential cross section dajdo scaled in 0 3 a.u. as a function of energy transfer scaled in units of 1000 for an electrical field strength of F = 1 a.u or vector potential A = 3186 a.u.. The initial election energy is W = 100 a.u.. The solid line denotes the result for electrons, the short dashed one the differential cross section for spinless particles and the long dashed one the result for the nonrelativistic limit.

We begin our discussion with the simple case of a spinless particle of mass m and kinetic energy E = h2k2/2m in a spherical, time-independent potential V(r), so that the Schrodinger equation can be decomposed into uncoupled partial waves l. For a particular l, the scattering matrix or the S matrix is defined as S(k) = exp[2/5(A )] in terms of the phase shift 8(k). Here and in the following, the subscript l on the S matrix and the phase shift is suppressed. The asymptotic form of the time-dependent radial wavefunction is expressible as... 

One-body systems of particular interest are the free spinless particle and the spinless particle bound in a Coulomb potential Uc (the nonrelativistic hydrogen atom). Spin is introduced in section 3.3.2. The state of a free spinless particle is an eigenstate of momentum. It is completely specified by the momentum p. The corresponding Schrodinger equation is... 

It is useful to keep in mind a simple problem for illustration. This is potential scattering (section 4.4), i.e. scattering of a spinless particle by a short-range potential V. The coordinate representation of the scattering quantities is, in atomic units,... 

The first term of Eq. (6) is just the non-relativistic current density for a spinless particle, while the second term arises from the electron spin and has the form of the curl of the spin density. In the relativistic case, this decomposition is at best an approximation since spin and linear motion of a particle are coupled. The kinetic energy (including the rest energy) of the Kohn-Sham reference svstem is given... 

For a reaction between spinless particles in which only one resonance level of the compound state is important the total cross section may be written in the amiliar Breit-Wigner form... 

These expressions are complicated but simplify in many practical applications, particularly when spinless particles only are concerned. The interpretation of angular distribution patterns is also simplified by a number of general results for unpolarised beams and non-relativistic reactions. These are now listed and apply equally to particle-particle, particle-quantum or quantum-quantum correlations angular correlations in beta-decay processes are not considered here. 

An analysis of the transformation properties of the Fock-Klein-Gordon equation and of the Dirac equation leads to the conclusion that satisfaction of the first of these equations requires the usual (i.e., scalar) wave function, whereas the second equation requires a bispinor character of the wave function. Scalar functions describe spinless particles (because they cannot be associated with the Pauli matrices), while bispinors in the Dirac equation are associated with the Pauli matrices, and describe a particle of 1/2 spin. 

We are considering a single spinless particle in a fixed local potential. Let us suppose that the orbit U(t) ip) describes the evolution of some scattering experiment. This means that when followed back to a time well before the collision U (t) ip) represents a wave packet that is localized far away from the scattering center and, therefore, behaves like a free wave packet. Now, the motion of a free particle is given by the free evolution operator... 

We found a positive ADMR signal at 0.45 eV that is associated with the photogenerated 5". As mentioned in Section III.B.5, however, a situation where a spinless particle shows an ADMR signal is not uncommon in ODMR spectroscopies. This comes about via spin-dependent recombination processes of companion excitations having a cross-recombination channel with the spinless particle. The experimental correlation found between 6N(HE) (<0) and SN(LE) (>0) may show that an apparent conversion process from into 5 -5 ... 

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