Feinberg-Horodecki Equation and Temporal Potentials

• The Feinberg-Horodecki equation is a space-like analogue of the Schrödinger equation that models quantum dynamics with time as the variable and quantized momentum due to time-translation symmetry.
• It employs techniques like the Nikiforov–Uvarov method to derive analytical solutions for temporally-dependent potentials, linking theory with applications in molecular and biological systems.
• The equation’s algebraic structure reveals su(1,1) coherent states and discrete momentum spectra, providing insights into time-dependent spectral phenomena and quantum evolution.

The Feinberg-Horodecki (FH) equation is a “space-like” or temporal analogue of the nonrelativistic Schrödinger equation, originating as the nonrelativistic limit of Feinberg’s tachyonic equation and formally introduced by Horodecki. The equation governs the dynamics of quantum systems where the traditional roles of space and time are exchanged: time acts as the variable, and the conserved quantity becomes a quantized momentum, corresponding to discrete translational invariance in time. This framework is particularly suited for investigating temporally varying potentials and “momentum” quantization in contexts ranging from diatomic molecular dynamics to biological growth phenomena.

1. Mathematical Formulation and Space-like Interpretation

In one spatial dimension, the space-like (temporal) counterpart of the Schrödinger equation, as proposed by Horodecki, reads in covariant notation: $$i\,\hbar\,\frac{\partial\Psi}{\partial x} \;+\;\frac{\hbar^2}{2m_0c}\frac{\partial^2\Psi}{\partial x_0^2} \;-\;\frac{V}{c}\,\Psi\; =\; 0,$$ where $x_0=ct$, $m_0$ is the mass, $c$ the speed of light, and $V$ a vector potential (Arda et al., 2017).

For stationary, plane-wave states in $x$ ($\Psi(x, t) = \psi(t) e^{-i\mathcal{P} x/\hbar}$), substitution yields an ODE: $$\left[\frac{d^2}{dt^2} + \frac{c}{M(c)}\,\mathcal{P} - \frac{1}{M(c)} V(t)\right]\,\psi(t) = 0,$$ with $M(c) \equiv \hbar^2/(2m_0c^2)$.

Physically, quantized values of $\mathcal{P}$ play the role analogous to energy eigenvalues in the standard time-independent Schrödinger equation. The space-like FH equation formalizes quantum evolution in time with momentum quantization due to time-translation symmetry (Farout et al., 2020, Farout et al., 2020).

2. Solvable Potentials and the Nikiforov–Uvarov Method

The FH equation has been solved for a variety of temporally-dependent potentials, including the trigonometric Pöschl–Teller, Kratzer, screened Coulomb, Hellmann, Manning–Rosen, improved deformed exponential-type, and others.

For a general time-dependent potential $x_0=ct$0, the canonical one-dimensional FH equation is: $x_0=ct$1 where $x_0=ct$2 is the quantized “space-like” momentum eigenvalue (Farout et al., 2020, Farout et al., 2020, Farout et al., 2020).

The Nikiforov–Uvarov (NU) method is systematically deployed to reduce the second-order ODE to a hypergeometric form. For most potentials investigated, change of variable and series expansion yields: $x_0=ct$3 with the parameters $x_0=ct$4 determined by the form and parameterization of $x_0=ct$5, and $x_0=ct$6 a function of $x_0=ct$7 (commonly exponential in nature) (Farout et al., 2020, Farout et al., 2020, Farout et al., 2020, Farout et al., 2020).

Polynomial solvability yields a discrete set of quantized momentum eigenvalues $x_0=ct$8, typically expressible in closed analytical form involving potential parameters, screening parameters, and system-specific constants.

3. Explicit Solutions: Momentum Spectra and Eigenfunctions

For the temporal trigonometric Pöschl–Teller potential, $x_0=ct$9, the quantized momentum solutions are obtained as

$m_0$0

with $m_0$1, and eigenfunctions in terms of Gauss hypergeometric or, equivalently, associated Legendre polynomials (Arda et al., 2017): $m_0$2 where $m_0$3, and $m_0$4.

For more complex potentials (e.g., screened Kratzer–Hellmann, improved deformed exponential), the momentum spectrum is given by higher-order algebraic expressions, always mapping to the zeros of an auxiliary quantization condition yielded by the NU method (Farout et al., 2020, Farout et al., 2020, Farout et al., 2020). The eigenfunctions are products of exponential or power-law prefactors and Jacobi polynomials: $m_0$5 with normalization constants determined by the orthogonality relations of Jacobi polynomials.

Negative values of $m_0$6 correspond to temporally-bound (attractive) states, mirroring the energy quantization in spatial Schrödinger analogues (Farout et al., 2020, Farout et al., 2020).

4. Algebraic Structures and Space-like Coherent States

A significant result for the class of exactly solvable FH equations, notably for the Pöschl–Teller and general molecular potentials, is the identification of su(1,1) algebraic structures governing the ladder operators of the momentum eigenstates (Arda et al., 2017). The raising ($m_0$7) and lowering ($m_0$8) operators act as: $m_0$9 and, together with $c$0, close the su(1,1) algebra: $c$1.

This algebraic foundation enables the construction of Barut–Girardello type space-like coherent states as eigenstates of the lowering operator: $c$2 Their normalized expansion in the momentum eigenbasis takes the form: $c$3 where $c$4 is the modified Bessel function. The resolution of the identity is ensured with an explicit integration measure involving Bessel functions (Arda et al., 2017).

Analogous coherent eigenstates are constructible for other solvable potentials using standard displacement-operator approaches, with annihilation and creation operators defined in the orthonormal momentum eigenbasis (Farout et al., 2020).

5. Special Cases and Parameter Dependence

Many special cases and limiting procedures recover familiar potential models:

• For the time-dependent Kratzer and screened Coulomb potentials, the FH equation admits closed-form quantization rules:

$c$5

with corresponding Jacobi-polynomial wavefunctions. In the Coulombic limit, $c$6 (Farout et al., 2020).

• For the general molecular potential, time-dependent Wei–Hua and Manning–Rosen potentials emerge by parameter specialization and yield spectrum and eigenfunctions in complete agreement with earlier models (Farout et al., 2020).
• Screening and deformation parameters (e.g., $c$7) control not only the depth and width of the potential “well” in the temporal variable but also the spacing and sign of the resulting momentum eigenvalues. For instance, as screening increases (large $c$8), momentum eigenvalues may cross zero and approach different asymptotics, altering physical interpretations (Farout et al., 2020, Farout et al., 2020).

The FH equation also admits modification to scenarios with time-dependent mass, leading to oscillator-like deformations and spectrum shifts governed by additional mass-distribution parameters, with solutions constructed via the Asymptotic Iteration Method (1609.00005).

6. Physical Interpretation and Applications

The primacy of the FH equation is its role as the mathematical medium for “space-like” quantum dynamics: temporal potentials induce discrete quantization of momentum due to translation invariance in time, inverting the standard Schrödinger prescription. This duality enables a trajectory-based interpretation of quantum phenomena, where quantized rates of temporal evolution manifest as distinct momentum levels (Farout et al., 2020, Farout et al., 2020).

The exact solutions for temporally-dependent molecular potentials enable:

• Construction of space-like coherent states, minimizing time–energy uncertainty (Arda et al., 2017, Farout et al., 2020).
• Analytical exploration of time-dependent spectral properties relevant to quantum diatomics, with direct applications to vibrational and growth models in chemical physics and biology (Farout et al., 2020).
• Investigation of extended relativity scenarios, time–dependent supersymmetric quantum mechanics, and the quantum mechanics of systems with time–dependent mass (1609.00005).

Tables, parameter plots, and explicit spectral formulas support the interpretation of FH quantized momenta as analogues of energy spectra in their respective spatial problems, with the potential for application in modeling crystallization dynamics, biological growth rates, and the time–dependent behavior of constrained quantum systems.

7. Summary Table of FH Equation Solutions (Representative Potentials)

Potential	Wavefunction Form	Quantized Momentum
Trig. Pöschl–Teller	$c$9	$V$0
Gen. Molecule	$V$1	$V$2 as in (Farout et al., 2020)
Screened Coulomb	$V$3	$V$4
TDM Harmonic Osc.	$V$5	$V$6

Forms and spectral quantization are direct quotes and reductions from the referenced works (Arda et al., 2017, Farout et al., 2020, Farout et al., 2020, Farout et al., 2020, Farout et al., 2020, 1609.00005).

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The FH equation and its solution hierarchy provide a mathematically rigorous and physically meaningful parallel to the established Schrödinger equation, expanding the toolkit for analyzing quantum systems dominated by time-dependent effects, and offering closed-form, model-agnostic methods for deriving space-like coherent and quantized states across a broad landscape of potentials.