Nonlinear Schrodinger-Pauli Equations

Traveling Wave Form Description for Dirac Field and Its Deduction To Pauli Equation Type Forms in Quantum Mechanics

We derive an equivalent traveling wave form description for Dirac field. In the non-relativistic limit, such form can reduce to inverse-Galilean transformed Schrodinger-type equation. We find that, the resulting two-component Schrodinger-type equation from the reduction of traveling wave form description of Dirac field is different to the naive Galilean transformed Schrodinger equation. Taking into account the interactions of the system to electromagnetic field by adding proper forms of covariant derivative, the traveling wave form description for Pauli equation can be similarly obtained in th…

Solutions of Pauli-Dirac Equation in terms of Laguerre Polynomials within Perturbative Scheme

We search for first- and second-order corrections to the energy levels of the Pauli-Dirac equation within the Rayleigh-Schrödinger theory. We use some identities satisfied by the associated Laguerre polynomials to reach this aim. We give a list presenting analytical forms of some integrals including two associated Laguerre polynomials, or their derivatives.

The Schrodinger equation and negative energies

We present a nonrelativistic wave equation for the electron in (3+1)-dimensions which includes negative-energy eigenstates. We solve this equation for three well-known instances, reobtaining the corresponding Pauli equation (but including negative-energy eigenstates) in each case.

New Perspectives on the Schr{ö}dinger-Pauli Theory of Electrons: Part II: Application to the Triplet State of a Quantum Dot in a Magnetic Field

The Schrödinger-Pauli theory of electrons in the presence of a static electromagnetic field can be described from the perspective of the individual electron via its equation of motion or `Quantal Newtonian' first law. The law is in terms of `classical' fields whose sources are quantum-mechanical expectation values of Hermitian operators taken with respect to the wave function. The law states that the sum of the external and internal fields experienced by each electron vanishes. The external field is the sum of the binding electrostatic and Lorentz fields. The internal field is the sum of field…

The Quantum Mechanical Current of the Pauli Equation

We argue that the process of constructing the quantum mechanical current of the Pauli equation by copying the line of arguments used in the spin-0 case, i.e. the Schrödinger equation, is ambiguous. We show that a non-relativistic reduction of the relativistic Dirac four-vector current is, however, capable of fully resolving the problem. This analysis reveals that the non-relativistic current of the Pauli equation should include an extra term of the form $\vec \nabla \vec \times (ψ^{\dagger} \vec σψ)$. We present an initial exploration of the potential consequences of this new 'spin-term' by so…

Weak Values: Approach through the Clifford and Moyal Algebras

In this paper we calculate various transition probability amplitudes, TPAs, known as `weak values' for the Schrodinger and Pauli particles. It is shown that these values are related to the Bohm momentum, the Bohm energy and the quantum potential in each case. The results for the Schrodinger particle are obtained in three ways, the standard approach, the Clifford algebra approach of Hiley and Callaghan, and the Moyal approach. To obtain the results for the Pauli particle, we combine the Clifford and Moyal algebras into one structure. The consequences of these results are discussed.

Spin-dependent Bohm trajectories for Pauli and Dirac eigenstates of hydrogen

The de Broglie-Bohm causal theory of quantum mechanics is applied to the hydrogen atom in the fully spin-dependent and relativistic framework of the Dirac equation, and in the nonrelativistic but spin-dependent framework of the Pauli equation. Eigenstates are chosen which are simultaneous eigenstates of the energy H, total angular momentum M, and z component of the total angular momentum M_z. We find the trajectories of the electron, and show that in these eigenstates, motion is circular about the z-axis, with constant angular velocity. We compute the rates of revolution for the ground (n=1) s…

The Pauli equation in scale relativity

In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and continuous spacetime. Since such a generalized geometry implies the occurence of new discrete symmetry brea…

Derivation of the Schrödinger equation from QED

The Schrödinger equation relates the electron wavefunction and the electric potential, which are emergent physical quantities. At that emergent level, the Schrödinger equation is either postulated as a principle of quantum physics or obtained heuristically. However, the Schrödinger equation is a low energy condition we can derive from the foundations of QED. Due to the small value of the electromagnetic coupling constant, we show that, in low energy interactions, the electric potential accurately represents the contributions of the intermediate photon exchanges. Then, we see that the dominant …

求解薛定谔方程中的扎森豪斯展开

Zassenhaus Expansion in Solving the Schrödinger Equation

该研究针对量子模拟中近似时间演化算符的挑战，提出改进的固定深度李理论模拟方法，纳入二阶扎森豪斯换位子校正，保持电路深度同时降低局部误差，减少门计数，为编译特定哈密顿量成适合量子硬件的电路提供途径。

Derivation of Nonlinear Schrödinger Equation

We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak.

Ehrenfest theorem, Galilean invariance and nonlinear Schrödinger equations

Galilean invariant Schrödinger equations possessing nonlinear terms coupling the amplitude and the phase of the wave function can violate the Ehrenfest theorem. An example of this kind is provided. The example leads to the proof of the theorem: A Galilean invariant Schrödinger equation derived from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for any linear or nonlinear lagrangian.

非线性薛定谔方程的微扰理论

Perturbation theory for nonlinear Schrodinger equations

本文提出将Gross-Pitaevskii非线性薛定谔方程的非线性项作为微扰，证明所得Rayleigh-Schrodinger级数在参数绝对值低于阈值时收敛，并给出定态解。