The ill-posedness of (non-)periodic travelling wave solution for deformed continuous Heisenberg spin equation

Based on an equivalent derivative nonlinear Schr\”{o}inger equation, some periodic and non-periodic two-parameter solutions of the deformed continuous Heisenberg spin equation are obtained. These solutions are all proved to be ill-posed by the estimates of Fourier integral in ${H}^{s}_{\mathrm{S}^{2}}$ (periodic solution in ${H}^{s}_{\mathrm{S}^{2}}(\mathbb{T})$ and non-periodic solution in ${H}^{s}_{\mathrm{S}^{2}}(\mathbb{R})$ respectively). If $\alpha \neq 0$, the range of the weak ill-posedness index is $1

On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections

A classical system of conservation laws descriptive of relativistic gasdynamics is examined. In the two-dimensional stationary case, the system is shown to be invariant under a novel multi-parameter class of reciprocal transformations. The class of invariant transformations originally obtained by Bateman in non-relativistic gasdynamics in connection with lift and drag phenomena is retrieved as a reduction in the classical limit. In the general 3+1-dimensional case, it is demonstrated that Synge’s geometric characterization of the pressure being constant along streamlines encapsulates a three-dimensional extension of an integrable Heisenberg spin equation.

Dirac Notation and Transformation Theory

Chapter 1 opens with a brief review of some basic features of quantum mechanics, including the Schrödinger equation, linear and angular momentum and the theory of the hydrogenic atom: It also includes complete orthonormal sets of eigenfunctions, the translation operator, current, spin, equation of continuity, gauge transformation, determinant & permanent multiparticle energy eigenfunctions for noninteracting particles and the Pauli exclusion principle. Attention is then focused on Dirac bra-ket notation and complete sets of commuting observables. In this connection, representations and transformation among representations are discussed in detail for the Schrödinger system state vector and the eigenstates, as well as bra-ket matrix elements of operators. Finally, Schwinger’s interpretation of ket-bra matrix operator structures (Schwinger “Measurement Symbols”) in terms of annihilation and creation of systems in eigenstates is introduced.

A Stronger Multi-observable Uncertainty Relation

Uncertainty relation lies at the heart of quantum mechanics, characterizing the incompatibility of non-commuting observables in the preparation of quantum states. An important question is how to improve the lower bound of uncertainty relation. Here we present a variance-based sum uncertainty relation for N incompatible observables stronger than the simple generalization of an existing uncertainty relation for two observables. Further comparisons of our uncertainty relation with other related ones for spin-"Equation missing" and spin-1 particles indicate that the obtained uncertainty relation gives a better lower bound.