Nonrelativistic particles

For a free nonrelativistic particle of mass m, H has the following form in the q representation... 

The basic stopping power formula of Bethe has a structure similar to that of Bohr s classical theory [cf. Eq. (2)]. The kinematic factor remains the same while the stopping number is given hy B = Zln(2mv /7) for incident heavy, nonrelativistic particles. The Bethe... 

A nonrelativistic particle is moving five times as fast as a proton. The ratio of their de Broglie wave lengths is 10. Calculate the mass of the particle. 

Figure 10. The various contributions to the present universal mass/energy density, as a fraction of the critical density (Q), as a function of the Hubble parameter (Ho). The curve labelled Luminous Baryons is an estimate of the upper bound to those baryons seen at present (z ( 1) either in emission or absorption (see the text). The band labelled BBN represents the D-predicted SBBN baryon density. The band labelled by M (Om = 0.3 0.1) is an estimate of the current mass density in nonrelativistic particles ( Dark Matter ).

In nonrelativistic quantum mechanics, the degeneracy due to spherical symmetry of the Schrodinger equation is 2i +1, where f is the orbital angular quantum number. The spin just doubles the number of states, and if there is no magnetic field, the two spin-states are energetically indistinguishable. Hence, a nonrelativistic particle with spin has a 2(2f -I- l)-fold degeneracy due to spherical symmetry. 

Answer Equation 4.20 will be used, but first the range of an alpha particle with speed equal to that of a 5-MeV deuteron will have to be calculated. The kinetic energy of an alpha particle with the same speed as that of the deuteron will be found using the corresponding equations for the kinetic energy. Since T — for these nonrelativistic particles. 

All the symbols in Eqs. 13.3 and 13.4a have been defined in Sec. 4.3, except Z, the charge of the incident particle, and Z2, the atomic number of the stopping material. For nonrelativistic particles (/3 - 1), which are much heavier than electrons, Eq. 13.4a takes the form... 

Equations 13.20 to 13.22 were written in terms of the mass M of the particle. For nonrelativistic particles, M has a nonintegral value very close to the value of A, which in turn, is given by an integer. This is fortunate because Z also assumes integral values only, and the product MZ assumes unique values for many particles. For example, for protons, deuterons, tritons, and alphas, the value of MZ is 1, 2, 3, and 16, respectively. 

This is analogous in form to the vector equation describing Newton s Second Law for the trajectory of a nonrelativistic particle if the mass is taken as being equivalent to (377/ ) and N plays the role of an imaginary time (imaginary, because of the negative sign). 

The conformation of a flexible polymer is closely analogous to the possible paths of a nonrelativistic particle, which is described by the time dependent Schrodinger equation. 

A third condition is required to explain Sommerfeld s success. Biedenharn shows that, surprisingly, the nonrelativistic problem solved by Sommerfeld is that of a nonrelativistic particle with (dynamically independent) spin, rather than a spinless nonrelativistic particle. If the spinless Schrodinger equation is used, the operator... 

Taking advantage of the canonical momentum we will now derive the Hamiltonian describing this nonrelativistic particle within electromagnetic fields. After expression of all velocities by canonical momenta p in the Legendre transformation of Eq. (2.71) and basic algebraic manipulations one... 

M. L6vy-Leblond. Nonrelativistic Particles and Wave Equations. Commun. Math. Phys., 6 (1967) 286-311. 

The topological (or Berry) phase [9,11,78] has been discussed in previous sections. The physical picture for it is that when a periodic force, slowly (adiabatically) varying in time, is applied to the system then, upon a full periodic evolution, the phase of the wave function may have a part that is independent of the amplitude of the force. This part exists in addition to that part of the phase that depends on the amplitude of the force and that contributes to the usual, dynamic phase. We shall now discuss whether a relativistic electron can have a Berry phase when this is absent in the framework of the Schrddinger equation, and vice versa. (We restrict the present discussion to the nearly nonrelativistic limit, when particle velocities are much smaller than c.)... 

The first consistent attempt to unify quantum theory and relativity came after Schrddinger s and Heisenberg s work in 1925 and 1926 produced the rules for the quantum mechanical description of nonrelativistic systems of point particles. Mention should be made of the fact that in these developments de Broglie s hypothesis attributing wave-corpuscular properties to all matter played an important role. Central to this hypothesis are the relations between particle and wave properties E — hv and p = Ilk, which de Broglie advanced on the basis of relativistic dynamics. 

Quantum Mechanical Generalities.—It will be recalled that in nonrelativistic quantum mechanics the state of a particle at a given instant t is represented by a vector in Hilbert space (f)>. The evolution of the system in time is governed by the Schrodinger equation... 

When is a one component scalar function, one can take the square root of Eq. (9-237) and one thus obtains the relativistic equation describing a spin 0 particle discussed in Section 9.4. This procedure, however, does not work for a spin particle since we know that in the present situation the amplitude must be a multicomponent object, because in the nonrelativistic limit the amplitude must go over into the 2-component nonrelativistic wave function describing a spin particle. Dirac, therefore, argued that the square root operator in the present case must involve something operating on these components. 

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

The problem of nonrelativistic limit description for fundamental particles and their interactions may be solved in different ways. Although in all methods of nonrelativistic expansion the first terms of the Hamiltonians coincide, however the difference begins to arise at transition to the higher orders of expansion. The method of Foldy-Wouthuysen... 

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. 

Even in the framework of nonrelativistic quantum mechanics one can achieve a much better description of the hydrogen spectrum by taking into account the finite mass of the Coulomb center. Due to the nonrelativistic nature of the bound system under consideration, finiteness of the nucleus mass leads to substitution of the reduced mass instead of the electron mass in the formulae above. The finiteness of the nucleus mass introduces the largest energy scale in the bound system problem - the heavy particle mass. 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

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