Abstract A classical field theory for a Schrödinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego–Monteiro (NR) [Phys. Rev. A 88 (2013) 032105 ]. This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary field ϕ(x,t). It is here shown that the relation between the dynamics of the auxiliary field ϕ(x,i) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach. Indeed, we formulate a variational principle for the aforementioned Schrödinger equation which is based solely on the wavefunction Ψ(x,t). A continuity equation for an appropriately defined probability density, and the concomitant preservation of the norm, follows from this variational principle via Noether's theorem. Moreover, the norm-conservation law obtained by NR is reinterpreted as the preservation of the inner product between pairs of solutions of the variable mass Schrödinger equation.
Abstract A classical field theory for a Schrödinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego–Monteiro (NR) [Phys. Rev. A 88 (2013) 032105 ]. This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary field ϕ(x,t). It is here shown that the relation between the dynamics of the auxiliary field ϕ(x,i) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach. Indeed, we formulate a variational principle for the aforementioned Schrödinger equation which is based solely on the wavefunction Ψ(x,t). A continuity equation for an appropriately defined probability density, and the concomitant preservation of the norm, follows from this variational principle via Noether's theorem. Moreover, the norm-conservation law obtained by NR is reinterpreted as the preservation of the inner product between pairs of solutions of the variable mass Schrödinger equation.
A linear one-dimensional Schr\"odinger equation, defined by means of a non-Hermitian Hamiltonian characterized by position-dependent masses, was proposed lately. Herein we present an exact classical field theory for this equation, showing the need for an extra field $\ensuremath{\Phi}(x,t)$, in addition to the usual one, $\ensuremath{\Psi}(x,t)$, similar to what was done recently in the analysis of a class of nonlinear quantum equations. These generalizations of the Schr\"odinger equation depend on an index q, in such a way that the standard case is recovered in the limit $q\ensuremath{\rightarrow}1$. Particularly, the field $\ensuremath{\Phi}(x,t)$ becomes ${\ensuremath{\Psi}}^{*}(x,t)$ only when $q\ensuremath{\rightarrow}1$ and satisfies a similar Schr\"odinger equation for the Hermitian conjugate of the Hamiltonian operator. In terms of these two fields one may define a probability density following a standard continuity equation, leading to the preservation of probability in Cartesian space. Simple applications are performed by solving the equations for the two fields.
We analyze the properties that manifest hamiltonian nature of the Schrödinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation).It is used to show that any solution of the Schrödinger equation with time independent potential can be presented in the form Ψ = (-h2 2m + V )φ + ih∂ t φ , where the real field φ (t, x i ) is some solution of nonsingular Lagrangian theory being specified below.Preservation of probability turns out to be the energy conservation law for the field φ .After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.
Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the ( t ) PT −inner product, where ( t ) is a obtained from the charge conjugation operator through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the ( t ) PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.
Abstract The Hamilton principle is a variation principle describing the isolated and conservative systems, its Lagrange function is the difference between kinetic energy and potential energy. By Feynman path integration, we can obtain the Hermitian quantum theory, i.e., the standard Schrodinger equation. In this paper, we have generalized the Hamilton principle to the generalized Hamilton principle, which can describe the open system (mass or energy exchange systems) and nonconservative force systems or dissipative systems, and given the generalized Lagrange function, it has to do with the kinetic energy, potential energy and the work of nonconservative forces to do. With the Feynman path integration, we have given the non-Hermitian quantum theory of the nonconservative force systems. Otherwise, with the generalized Hamilton principle, we have given the generalized Hamiltonian for the particle exchanging heat with the outside world, which is the sum of kinetic energy, potential energy and thermal energy, and further given the equation of quantum thermodynamics. PACS: 03.65.-w, 05.70.Ce, 05.30.Rt
Abstract We analyze a general family of position-dependent mass (PDM) quantum Hamiltonians which are not self-adjoint and include, as particular cases, some Hamiltonians obtained in phenomenological approaches to condensed matter physics. We build a general family of self-adjoint Hamiltonians which are quantum mechanically equivalent to the non-self-adjoint proposed ones. Inspired by the probability density of the problem, we construct an ansatz for the solutions of the family of self-adjoint Hamiltonians. We use this ansatz to map the solutions of the time independent Schrödinger equations generated by the non-self-adjoint Hamiltonians into the Hilbert space of the solutions of the respective dual self-adjoint Hamiltonians. This mapping depends on both the PDM and on a function of position satisfying a condition that assures the existence of a consistent continuity equation. We identify the non-self-adjoint Hamiltonians here studied with a very general family of Hamiltonians proposed in a seminal article of Harrison (1961 Phys. Rev. 123 85 ) to describe varying band structures in different types of metals. Therefore, we have self-adjoint Hamiltonians that correspond to the non-self-adjoint ones found in Harrison’s article.