TITLE:
Scattering States, Wave Packets, Transition Amplitudes, and Tunneling in the Energy Domain: A Time-Free Formulation of Quantum Processes
AUTHORS:
Michail Lazos, Neda Maria Kaizumi, John R. Woodward
KEYWORDS:
Quantum Mechanics, Quantum Foundations, Time-Free Quantum Mechanics, Energy-Domain Quantum Mechanics, Scattering Theory, Wave Packets, Transition Amplitudes, Quantum Tunneling, Quantum Processes, Energy-Based Evolution
JOURNAL NAME:
Journal of Modern Physics,
Vol.17 No.8,
August
31,
2026
ABSTRACT: 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.