This paper develops an energy-domain formulation of time-independent non-relativistic quantum processes, including scattering, wave packets, transition amplitudes, and tunneling. A fixed-energy spectral component is written as Ψ E ( x,z )= ψ E ( x ) χ z ( E ) , while the complete state Ψ( x,z ) is obtained by summing or integrating over the spectral components. The energy derivative is defined on the spectral phase χ z ( E ) , with the energy-dependent spatial eigenfunction ψ E ( x ) retained as an element of the spectral basis. For a specified reference energy E ref , the spectral phase is χ z ( E )=exp[ −i ( E− E ref ) 2 ( z− z 0 )/ℏ ] . Fixed-energy reflection and transmission amplitudes, tunneling probabilities, probability currents, transition matrix elements, and selection rules retain their standard forms because this factor is a common global phase. For superpositions, however, the z-ordering is generated by ( H ^ x − E ref I ^ ) 2 and is generally not identical to the standard time evolution generated by H ^ x . Continuum states are treated with generalized normalization, channel labels, and the appropriate spectral measure. A right-moving Gaussian free-particle packet is worked out explicitly, yielding analytic narrow-band expressions for its centre and spatial width as functions of z . Transition probabilities are defined as Born-rule probabilities conditioned on a specified interaction event; transition rates, lifetimes, delay times, and tunneling times are not claimed to be reproduced without an operational clock. The paper therefore establishes the preservation of fixed-energy stationary observables and a consistent unitary spectral ordering for time-independent Hamiltonians, while clearly identifying the remaining limitations of the proposed dynamics.
This paper develops an energy-domain formulation of time-independent non-relativistic quantum processes, including scattering, wave packets, transition amplitudes, and tunneling. A fixed-energy spectral component is written as Ψ E ( x,z )= ψ E ( x ) χ z ( E ) , while the complete state Ψ( x,z ) is obtained by summing or integrating over the spectral components. The energy derivative is defined on the spectral phase χ z ( E ) , with the energy-dependent spatial eigenfunction ψ E ( x ) retained as an element of the spectral basis. For a specified reference energy E ref , the spectral phase is χ z ( E )=exp[ −i ( E− E ref ) 2 ( z− z 0 )/ℏ ] . Fixed-energy reflection and transmission amplitudes, tunneling probabilities, probability currents, transition matrix elements, and selection rules retain their standard forms because this factor is a common global phase. For superpositions, however, the z-ordering is generated by ( H ^ x − E ref I ^ ) 2 and is generally not identical to the standard time evolution generated by H ^ x . Continuum states are treated with generalized normalization, channel labels, and the appropriate spectral measure. A right-moving Gaussian free-particle packet is worked out explicitly, yielding analytic narrow-band expressions for its centre and spatial width as functions of z . Transition probabilities are defined as Born-rule probabilities conditioned on a specified interaction event; transition rates, lifetimes, delay times, and tunneling times are not claimed to be reproduced without an operational clock. The paper therefore establishes the preservation of fixed-energy stationary observables and a consistent unitary spectral ordering for time-independent Hamiltonians, while clearly identifying the remaining limitations of the proposed dynamics.
We present a renewed wave-packet analysis based on the following ideas: if a quantum one-particle scattering process and the corresponding state are described by an indivisible wave packet to move as a whole at all stages of scattering, then they are elementary; otherwise, they are combined; each combined process consists from several alternative elementary ones to proceed simultaneously; the corresponding (normed) state can be uniquely presented as the sum of elementary ones whose (constant) norms give unit, in sum; Born's formula intended for calculating the {\it expectation} values of physical observables, as well as the standard timing procedure are valid only for elementary states and processes; only an elementary time-dependent state can be considered as the quantum counterpart to some classical one-particle rajectory. By our approach, tunneling a non-relativistic particle through a static one-dimensional potential barrier is a combined process consisting from two elementary ones, transmission and reflection. In the standard setting of the problem, we find an unique pair of solutions to the Schrödinger equation, which describe separately transmission and reflection. On this basis we introduce (exact and asymptotic) characteristic times for transmission and reflection.
A new variant of the packet analysis to solve the tunneling time problem for the so-called completed elastic scattering (a potential barrier of an arbitrary form is located in the finite spatial interval) is presented. In particular, we propose an explanation of the "paradoxes" of wave packet tunneling, and elaborate a rule enabling one to extract the tunneling times for a particle from the analysis of tunneling the corresponding wave packet. It is shown that the quantum ensembles of transmitted and reflected particles are statistically independent not only after the scattering but also before this event. We find the corresponding counterparts for the transmitted and reflected packets, which describe separately both scattering channels at early times, and define the expectation values of delay times for each channel. Besides, to describe jointly all particles at the stage when they cannot be divided into two independent subensembles, we find the scattering time taking into account broadening the packets. We derive the condition to be satisfied for the completed scattering. The times above are defined in terms of the well-known quantum-mechanical averages for the position and wave-number operators.
The wave-particle duality introduced by de Broglie and developed mathematically by Schrödinger remains one of the deepest conceptual achievements of modern physics. Yet Schrödinger's original physical intuition contained a problem that has never entirely disappeared: if a particle is represented by a wave, how can an extended and generally dispersive wave structure correspond to the localized events observed in experiments? Schrödinger initially hoped that a particle could be understood as a localized wave packet formed by the coherent superposition of many elementary waves. The program was attractive because localization could then arise from interference rather than being imposed as an independent particle property. However, generic wave packets disperse under free evolution, while stable bound states of quantum systems are not generally localized packets moving along classical trajectories. The subsequent Born interpretation provided an extraordinarily successful connection between the wave function and experimental statistics, identifying ∣Ψ∣2|\Psi|^2 with a probability density. Nevertheless, this interpretation leaves open a distinct physical question: how does a spatially extended complex amplitude structure give rise to localized, discrete detection events? This paper revisits that question from the historical development of wave mechanics to the operational structure of early scattering experiments. Particular attention is given to the experiments of Davisson and Kunsman, Ramsauer's measurements of slow-electron scattering, and Dymond's measurements of the velocity and angular distribution of electrons scattered in helium. Although these experiments differ in physical arrangement and measured quantities, they share a common operational structure: an extended incident quantum state undergoes propagation and interaction, after which the experiment registers discrete events within a finite spatial, angular, or energy acceptance region. The central proposal of this paper is therefore not a new derivation of Born's rule, but a reformulation of the problem that precedes such a derivation. Instead of asking only why a quantum state has a probability distribution, one may ask how a propagating wave-amplitude structure is dynamically converted into a localized event rate. In symbolic form, Ψ⟶D⟶Γ,\Psi \quad \longrightarrow \quad \mathcal{D} \quad \longrightarrow \quad \Gamma, where Ψ\Psi denotes the propagating amplitude structure, D\mathcal{D} represents the interaction and detection dynamics, and Γ\Gamma denotes the resulting localized event rate. Born's rule then appears as the empirical relation Γ(r)∝∣Ψ(r)∣2,\Gamma(\mathbf r)\propto |\Psi(\mathbf r)|^2, whose dynamical origin remains to be established. The paper concludes by reconsidering Schrödinger's original wave intuition rather than simply declaring it defeated. The failure of the persistent localized wave packet may indicate not that wave ontology was fundamentally misguided, but that propagation and localization are physically distinct stages of quantum dynamics. This distinction provides a conceptual starting point for a subsequent series of investigations into the dynamical origin of quantum probability.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The quantum mechanics is most easily started from the experimentally observed discrete energy states of bound particles. The free particle may take on any energy whatever, which makes the quantum description of a free particle more awkward conceptually and mathematically than the quantum description of a bound particle. The construction of moving quantum spatial probabilities will require a superposition of stationary state wave functions of different energies. Any coherent superposition of energy eigen states for a particle represents a possible quantum state of that particle. The most important property of the spatial probability function for superposed energy states is that it changes with time. A moving probability that is limited spatially is called a packet and the corresponding quantum-mechanical description is called a packet state. The free-particle packet states examples satisfy the position-momentum uncertainty relation. Probably the most important application of the energy time uncertainty relation concerns the lifetimes of the excited states of atoms and nuclei.
Abstract We investigate the limits of reconstructing relativistic particle-mass information from finite causal-source records, with particular emphasis on stochastic event histories, spectral identifiability, and classical–quantum jump dynamics. Izanagi Core Ver.3.0.0 distinguishes information determined by a finite Markov source from additional structures required to define quantum time evolution, local field interactions, stable particle excitations, and physical mass ratios. Three interconnected problems are examined: the nonuniqueness of source-compatible quantum spectral realizations, the preservation of classical event histories under quantum operations, and the existence of stationary states in source-driven bosonic field models. For finite irreducible reversible continuous-time Markov sources, we analyze stationary occupation measures, relative event clocks, typed transition histories, and spectral representations. A specified Markov generator determines its own relaxation spectrum, but that spectrum does not uniquely determine a physically admissible quantum Hamiltonian or relativistic particle masses. Explicit constructions demonstrate that compatible quantum representations can possess different internal spectral-gap ratios while reproducing the prescribed classical source information. The canonical Hilbert-space representation of a stationary Markov record further distinguishes stochastic time translation from positive-energy quantum dynamics: the nontrivial spectrum of the record-shift generator is symmetric about zero. This obstruction applies to the specified representation and does not exclude alternative quantum reconstructions. We formulate classical–quantum jump instruments in which transitions of a classical source induce completely positive operations on an independently specified quantum system. Exact preservation of the complete typed source-history law, including transition identities and waiting-time distributions, is characterized by the scalar-effect condition J_ji*(I) = q_ij I on every directed transition, when preservation is required independently of the quantum state. For source-conditioned transition intensities whose total discrepancies are uniformly bounded by δ, a coupling construction yields the finite-time total-variation estimate TV ≤ 1 − exp(−δT). The estimate is sharp in an explicit comparison model, while control over finite observation intervals does not imply uniform agreement over arbitrarily long times. To examine locality and quantum measurement structure, we construct conditional interactions with externally supplied bosonic free fields using future-supported Weyl displacement operators. The analysis distinguishes relativistic locality, overlap of localized operations, commutation of quantum state channels, and phase-sensitive coherent control. Weyl displacement channels can commute even when their implementing unitary operators differ under an exchange of order by a nontrivial central phase. Such phases can be detected only when an additional coherent ordering interface is available. These constructions clarify the distinction between preserving a classical event record and reconstructing the full operational content of an interacting quantum field. Two stationary-state results identify further limitations of one-way classical–quantum coupling. First, a product state consisting of the stationary source distribution and a designated pure free-field vacuum is stationary precisely when every positive-rate jump operation preserves the corresponding vacuum projector, J_ji(P) = q_ij P, under the stated autonomous-source dynamics. Nontrivial Weyl interactions can violate this condition without changing the classical source-history statistics. Second, for a stationary source driving a bosonic Fock field through equally weighted positive and negative Weyl displacements, we obtain an exact energy-balance relation, EH_f − EH_f = Γ_Q t. Here Γ_Q is the nonnegative transition-rate-weighted displacement energy. When Γ_Q > 0, the specified dynamics admit no normal joint stationary state with finite expected free-field energy. This heating obstruction depends on unbiased displacement kicks and nondissipative free-field evolution between transitions. It is not a universal consequence of classical-source autonomy: dissipative dynamics can balance energy injection and permit finite-energy stationary states. The results are placed in the context of reversible Markov processes, quantum spectral theory, GNS representations, canonical commutation relations, generalized free fields, completely positive quantum instruments, quantum dynamical semigroups, and classical–quantum hybrid dynamics. Particular attention is given to the distinction between established mathematical structures and their conditional application to source-to-matter reconstruction. The stationary-state and heating criteria describe a specified class of one-way quantum jump models; they do not constitute a general decoherence–diffusion theorem or a universal prohibition on stationary classical–quantum coupling. The principal unresolved issue is the physical generation of matter from causal-source dynamics. In the constructions studied here, the Markov generator, spacetime geometry, quantum carrier, interaction operators, and measurement interfaces are supplied as independent inputs rather than derived from the Minimal Realization Source Law (MRSL). Accordingly, the results do not establish a dynamically generated Lorentz-covariant matter sector, persistent physical identity between source events and particle excitations, stable isolated particle poles in the coupled relativistic theory, or a uniquely identifiable physical particle-mass ratio. Izanagi Core Ver.3.0.0 therefore provides a mathematically explicit account of the distinction between source-history preservation, quantum dynamical consistency, and physical mass identification, while leaving the source-generated origin and quantitative determination of relativistic particle masses open.