Let H ^ x be a time-independent self-adjoint spatial Hamiltonian. Its discrete eigenstates satisfy

and its continuum generalized eigenstates satisfy

here E is a spectral label and α denotes degeneracy or channel data. For a one-dimensional free particle, for example, α = + and α = − distinguish the right-moving and left-moving momentum branches. In an energy-normalized representation,

with the understanding that continuum states are generalized, rather than square-normalizable, vectors. More generally, the continuum contribution is integrated with a spectral measure d μ ( E ) that contains the relevant density-of-states or Jacobian factor [16][17].

The corresponding completeness relation may be written schematically as

where σ c denotes the continuous spectrum.

Choose one reference energy E ref for the physical problem and define

The spectral phase is

The parameter z has dimensions [ z ] = [ ℏ ] / [ E ] 2 and is an auxiliary ordering parameter, not a laboratory clock reading.

The energy origin is part of the definition of the representation. Under an additive shift

the referenced energy ε ( E ) and all z-dependent phases are unchanged. For scattering calculations, E ref is chosen as the declared asymptotic threshold; for the free particle and the barrier examples below it is the zero of the asymptotic potential. For a bound problem, it may be chosen as a declared ground-state or threshold energy. Once this convention is fixed, the later fixed-energy examples write E for the referenced energy to avoid unnecessary notation.

For Δ z ≠ 0 , define the spectral differential operator

Then

This is a differential identity in the spectral coefficient. It is not an eigenvalue equation obtained by differentiating the full coordinate-space product ψ E ( x ) χ z ( E ) . Indeed,

The second term is generally non-zero. The operator D ^ E , z is therefore defined on the spectral coefficient χ z ( E ) , while the generalized eigenket | E , α 〉 is retained as the spectral basis element.

A fixed spectral component is denoted by

whereas the complete state is

In the position representation, Ψ E , α ( x , z ) = ψ E , α ( x ) χ z ( E ) and Ψ ( x , z ) = 〈 x | Ψ ( z ) 〉 . Thus E labels a component, while z orders the complete superposition.

If the initial spectral coefficients are normalized,

then normalization is preserved because | χ z ( E ) | = 1 . The complete state may be written as

with

For self-adjoint H ^ x , this operator is unitary.

For a single spectral component, χ z ( E ) is a global phase. Therefore,

and any expectation value or probability-current ratio formed entirely within the same fixed-energy component is unchanged. This observation explains the exact recovery of stationary bound-state quantities and fixed-energy scattering amplitudes.

For a superposition, relative phases are observable. The z-ordered state is generated by ( H ^ x − E ref I ^ ) 2 , whereas standard time evolution is generated by H ^ x . Consequently, the two evolutions are not identical for a general broadband state. The present paper treats this difference as a physical limitation and investigates the consequences that can be established without identifying z with laboratory time.

A single component of referenced energy E is written as

where the reference-energy convention of Equation (6) has already been applied. The complete state is denoted by Ψ ( x , z ) and is obtained by summing or integrating over the spectral labels.

For the discrete part of the spectrum,

For the continuum,

The channel label and measure are essential: an energy value alone may correspond to several propagation directions or scattering channels.

Figure 1.Spectral interpretation of a z-ordered superposition. A fixed-energy component carries only a global phase, whereas observable changes arise from relative phases among distinct spectral components.

The relative phase between two discrete components is

It is this relative phase, rather than the global phase of one fixed-energy state, that can change the probability density of a superposition.

The distinction between fixed-energy global phases and observable relative phases is summarized in Figure 1.

The probability density is

The diagonal spectral terms are independent of z , while the off-diagonal terms contain the relative phases. Scattering and tunneling coefficients at a fixed energy remain stationary quantities; packet displacement, packet deformation, and interference between distinct energies belong to the z-ordered superposition.

The contribution of the present framework is therefore limited but precise. It provides a spectral ordering of the complete state without assuming that z is a clock variable, and it separates three logically different objects: the stationary spatial eigenproblem, the global phase of one fixed-energy component, and the relative phases of a superposition. The framework does not imply that every quantity normally expressed per unit time has already been reconstructed.

For free-particle momentum eigenstates normalized by 〈 k | k ′ 〉 = δ ( k − k ′ ) , a right-moving packet is

where

The choice k > 0 fixes the right-moving branch. In an energy-normalized form, the same state contains the measure factor d k / d E = m / ( ℏ 2 k ) , or equivalently an energy-normalized coefficient that absorbs its square root. This is why a Gaussian A ( E ) by itself does not completely specify a localized directional packet.

Choose

where N normalizes the positive- k branch. When k 0 ≫ σ k , the negligible negative- k tail allows the lower limit to be extended to − ∞ and N ≃ ( 2 σ k 2 ) − 1 / 4 .

The branch-restricted Gaussian momentum distribution is shown in Figure 2.

Define the z-dispersion function

For a narrow packet, write q = k − k 0 and expand to second order,

where

and

Evaluation of the Gaussian integral gives a packet centred at

with spatial variance

The first term is the initial Fourier width. The second term is the leading narrow- k spreading produced by the curvature of the quadratic-energy phase. These formulas provide an explicit consequence of the proposed ordering rather than a purely qualitative statement.

For a packet centred at E 0 = ℏ 2 k 0 2 / ( 2 m ) , the local identification

reproduces the standard group displacement, x c − x 0 = ( p 0 / m ) t eff . The full width in Equation (31), however, retains the curvature of E 2 ( k ) and displays the difference between exact z-ordering and standard time evolution. Standard packet evolution is recovered only in the stronger regime in which the contribution of ( E − E 0 ) 2 Δ z / ℏ is negligible across the spectral width.

Because U ^ z is unitary,

whenever the initial packet is normalized. For broad spectra the quartic phase in Equation (23) produces non-Gaussian deformation, and the complete integral must be evaluated numerically.

Strictly speaking, a free particle does not scatter when the potential vanishes everywhere. The free-particle solutions are nevertheless essential because they describe the state sufficiently far from a localized interaction region, where the potential approaches a constant value. They therefore provide the incoming and outgoing asymptotic states used in scattering theory.

For a free particle,

The stationary Schrödinger equation becomes

For positive energy, the general solution is

where

The component e i k x has positive momentum and represents propagation towards increasing x , while e − i k x has negative momentum and represents propagation towards decreasing x .

Within the energy-domain formulation, the complete state of definite energy is

Both spatial components correspond to the same energy and therefore carry the same energy-domain phase. For a stationary scattering state, this factor is global and does not modify the relative amplitudes A and B .

The probability current associated with a one-dimensional state is

Figure 3.General one-dimensional scattering process. A state incident from the left interacts with a localized potential and produces reflected and transmitted outgoing components.

Since the energy-domain phase does not depend on position, it cancels from Equation (39). Consequently,

For the right-moving plane wave A e i k x , the current is

For the left-moving plane wave B e − i k x , the current is

The sign of the current gives the direction of propagation. The energy-domain phase does not change either current because it has unit magnitude and is common to both spatial components.

The incident, reflected, and transmitted asymptotic channels are illustrated in Figure 3.

We now consider a potential that approaches constant values far from the interaction region,

and

For an energy greater than both asymptotic potentials, the corresponding wavenumbers are

and

For a state incident from the left, the asymptotic spatial wavefunction is written as

and

The coefficient A is the incident amplitude, B is the reflected amplitude, and C is the transmitted amplitude. No left-moving component is included in the right asymptotic region because we assume that no wave is incident from the right.

The corresponding energy-domain states are

and

The same phase multiplies all three components because the scattering is elastic and the energy is conserved. Consequently, the matching conditions imposed on the complete states reduce directly to the usual matching conditions for the spatial wavefunctions.

For a general two-sided scattering problem, waves may be incident from both the left and the right. The asymptotic state can then be written as

and

The amplitudes A and D describe the incoming channels, while B and C describe the outgoing channels. When the asymptotic wavenumbers are different, the amplitudes must be normalized with respect to probability flux. We therefore define

To avoid unnecessary dependence on a matrix environment, we define the incoming and outgoing amplitude vectors as

The relation between the incoming and outgoing channels is then

For a real potential and elastic scattering, the flux-normalized scattering matrix is unitary,

This condition expresses probability-current conservation and remains unchanged in the energy-domain formulation.

For incidence from the left, the reflection amplitude is defined as

The transmission amplitude is

The incident probability current is

The reflected probability current is

The transmitted probability current is

The reflection coefficient is

and the transmission coefficient is

For a real potential with no absorption, current conservation requires

The result is unchanged in the energy-domain formulation because the common phase cancels from all three probability currents.

The scattering of a localized packet is obtained by combining the stationary amplitudes over a range of energies. Let D E denote the range of energies for which the relevant asymptotic channels are open, and let A 0 ( E ) represent the incident spectral distribution. The incident, reflected, and transmitted packets may be written as

and

These expressions show that the functions r ( E ) and t ( E ) change the amplitude and phase of each energy component, while the factor

determines the relative energy-domain phase between the different components.

The stationary Schrödinger equation, asymptotic plane waves, matching conditions, and probability currents retain their standard forms. For one state of definite referenced energy, the spectral phase is common to the incident, reflected, and transmitted components. It therefore cancels from the boundary conditions and from the ratios defining R ( E ) and T ( E ) .

In the energy-domain formulation, localized incident and outgoing packets are constructed from channel-resolved stationary states using the spectral z-phase. Each energy component is weighted by its stationary amplitudes r ( E ) and t ( E ) , while the complete reflected and transmitted packets are determined by interference across the spectral distribution. The stationary boundary-value problem therefore determines the energy-resolved amplitudes, and the spectral ordering determines the structure of their superposition.

We first consider the potential step

and

where V 0 > 0 . A particle is incident from the left with energy E .

Figure 4.Scattering at a potential step for E > V 0 . The incident state produces reflected and transmitted components, with different wavenumbers in the two spatial regions.

For E > V 0 , both regions admit oscillatory solutions. The wavenumbers are

and

The spatial wavefunction in the first region is

and in the second region it is

Only a right-moving component is included in the second region because no wave is assumed to be incident from positive infinity.

The potential profile and the three scattering components for E > V 0 are shown in Figure 4.

For a finite discontinuity in the potential and a constant particle mass, the wavefunction and its first derivative must be continuous at x = 0 . Therefore,

and

Substitution gives

and

For 0 < E < V 0 , the first region remains oscillatory, but the second region is classically forbidden. We define

The physically acceptable solution in the second region is

The single decaying exponential carries no probability current, and the particle is completely reflected by a semi-infinite step even though the probability density is non-zero for a finite distance inside the forbidden region. Applying the boundary conditions gives

so that

Consequently,

For E > V 0 , the reflection amplitude is

and the transmission amplitude is

Solving Equation (78) and Equation (79), we obtain

and

The reflection coefficient is

and the transmission coefficient is

Adding the two coefficients gives

The physical behaviour of the coefficients becomes clearer when the energy is written in units of the step height,

For ε > 1 , the coefficients become

and

The resulting energy dependence of the reflection and transmission coefficients is plotted in Figure 5.

Figure 5.Reflection and transmission coefficients for a potential step with E > V 0 . Close to the threshold, reflection dominates. At high incident energy, transmission approaches unity.

As E → V 0 + , the transmitted wavenumber approaches zero and the particle is almost completely reflected. In the high-energy limit, E ≫ V 0 , the step becomes a small perturbation and T ( E ) → 1 .

We next consider the finite rectangular barrier

and

The barrier has height V 0 and width a . Unlike the potential step, the forbidden region is finite. This allows the evanescent state inside the barrier to connect to a propagating state on the far side.

For E > V 0 , the spatial solutions are oscillatory in all three regions,

and

where

and

Figure 6.Rectangular potential barrier for E < V 0 . The state is oscillatory outside the barrier and evanescent inside it. Since the forbidden region has finite width, a transmitted component exists for x > a .

For 0 < E < V 0 , we define

The spatial solutions are then

and

Both exponential terms must be retained inside a finite barrier. Although one term increases with x , it remains finite over the interval 0 < x < a and is required to satisfy the boundary conditions at both interfaces.

The oscillatory and evanescent regions of the finite barrier are displayed in Figure 6.

The boundary conditions at x = 0 are

and

At x = a , the conditions are

and

For E < V 0 , substitution of the three regional solutions gives

and

Solving these equations gives the transmission coefficient

The reflection coefficient is

For a thick or high barrier, κ a ≫ 1 , the transmission coefficient becomes

For E > V 0 , the corresponding transmission coefficient is

Transmission resonances occur whenever

for which T ( E ) = 1 .

At fixed energy, the same Schrödinger equation is solved in every region, the same boundary conditions are imposed, and the same probability currents define reflection and transmission. The common spectral phase cancels at each interface. Consequently, r ( E ) , t ( E ) , R ( E ) , and T ( E ) are identical to the standard stationary results.

Probability-current conservation, total reflection from a semi-infinite forbidden region, finite-barrier transmission, exponential tunneling suppression, and above-barrier resonances are determined by the stationary spatial boundary-value problem. The z-dependence becomes non-trivial when the energy-resolved amplitudes are assembled into a packet, where r ( E ) and t ( E ) alter both the spectral weights and the relative phases.

Let

and attach the referenced spectral phases

Here and below the energies are measured from the reference declared in Section 2.2.

Write

where W ^ represents the specified interaction. The stationary coupling matrix element is

For the complete z -ordered states,

and therefore

The spectral ordering changes the phase of the amplitude but not the stationary coupling strength.

A matrix element alone is not a normalized probability. Suppose that the event under consideration is represented by the action of W ^ and that W ^ | Ψ i ( z ) 〉 ≠ 0 . The normalized post-event state is

The Born-rule probability of detecting the system in a normalized final channel | Ψ f ( z ) 〉 is

For a subspace of final outcomes ℱ with projector Π ^ ℱ , the corresponding probability is

If the final projectors resolve the identity, these conditional probabilities sum to unity. Continuum final channels are treated by replacing the sum with the appropriate channel integral and spectral measure.

This definition is the standard Born rule applied after a specified interaction event. It is appropriate here because the framework specifies the relative weights of possible final channels but does not specify when the event occurs or how long it lasts. It therefore must not be confused with Fermi’s golden-rule rate, a decay constant, or a lifetime.

A channel is forbidden when W f i = 0 . Consequently, parity and angular-momentum selection rules remain those of the stationary matrix element. For an electric-dipole interaction,

and the initial and final states have opposite parity.

The energy-domain description separates the stationary coupling, the conditional channel probability, and the phase assigned to the complete states. No temporal sequence of intermediate states is required to define the conditional distribution over final channels. Rates, lifetimes, linewidths, and detector response remain outside the present construction because they require an operational duration.

Consider a particle of mass m moving in one spatial dimension. The stationary Schrödinger equation is

In a region where V ( x ) > E , we define the local decay coefficient

For a rectangular barrier of constant height V 0 extending from x = 0 to x = a , the decay coefficient is

The spatial solution inside the barrier is

The complete energy-domain state in the forbidden region is

The probability density is therefore

Figure 7.Barrier penetration for E < V 0 . The upper panel shows the rectangular potential barrier and the incident, reflected, and transmitted components. The lower panel shows the corresponding wavefunction magnitude.

The energy-domain phase has disappeared because it has unit magnitude. The spatial density inside the barrier is therefore the same as in standard stationary quantum mechanics.

If the decreasing exponential is dominant, the wavefunction amplitude falls approximately as

while the corresponding probability density falls as

The characteristic penetration length is

The barrier profile and the corresponding attenuation of the wavefunction magnitude are shown in Figure 7.

The probability current inside the barrier is

A single real decaying exponential carries no current. The non-zero current through a finite barrier arises from the relative complex phase between the two exponential components.

For the rectangular barrier, the spatial solutions outside the barrier are

Figure 8.Exact tunneling probability for a rectangular barrier as a function of the normalized energy E / V 0 . The parameter α = a 2 m V 0 / ℏ measures the effective barrier width.

and

where k = 2 m E / ℏ . Since the asymptotic potentials are equal on both sides of the barrier, the incident and transmitted wavenumbers are the same. The tunneling probability is

The corresponding reflection probability is

For a high or wide barrier, κ a ≫ 1 , the transmission probability becomes

For a general smooth barrier, the leading WKB transmission probability is

The exact rectangular-barrier transmission probability is shown in Figure 8 for two effective barrier widths.

For an incident wave packet with normalized spectral distribution A ( E ) , the total transmitted probability is obtained by averaging the fixed-energy coefficient over the packet,

If the packet also contains components with E ≥ V 0 , the corresponding above-barrier transmission coefficient must be included in the same spectral average. The barrier therefore acts as an energy-dependent filter rather than as a simple constant attenuation factor.

For a stationary component of definite energy, the complete state in each spatial region has the form

The same energy-domain phase multiplies the incident, reflected, evanescent, and transmitted spatial components. It therefore cancels from the continuity conditions at the two boundaries and from the incident and transmitted probability currents. Consequently, the stationary tunneling coefficient is unchanged.

For a localized incident packet, the transmitted packet is

Writing t ( E ) = | t ( E ) | exp [ i δ t ( E ) ] , this becomes

The barrier modifies the packet in two ways. The magnitude | t ( E ) | changes the relative weight of each energy component, while the phase δ t ( E ) changes the interference between those components. The energy-domain phase supplies the additional relative phase associated with the parameter z .

The fixed-energy transmission coefficient answers whether a detector can register the particle beyond a finite classically forbidden region. It follows from the evanescent spatial solution, boundary matching, and probability-current ratios. No traversal duration is required for this calculation.

A different question asks how long the process takes. Wigner-Smith delay, dwell time, phase time, detector response, and attosecond tunneling delays are operational time observables or clock-dependent constructs [20][21]. They cannot be identified with the auxiliary parameter z without an additional physical relation between z and a clock. The present framework therefore reproduces the stationary tunneling probability but does not claim to determine a tunneling time.

The contribution of the energy-domain formulation is this explicit separation of the existence and probability of tunneling from the duration assigned to it. The first belongs to the stationary spatial boundary-value problem. The second requires an operational clock, a time-dependent measurement protocol, or a relational degree of freedom beyond the present model.

The complete state is ordered by

so that

The spectral differential identity and the operator evolution equation have distinct roles. Equation (10) acts on the spectral phase, whereas Equation (151) governs the complete state.

For self-adjoint H ^ x , the generator in Equation (151) is self-adjoint and the ordering is unitary. For an observable A ^ with no explicit z-dependence,

This differs from the standard Heisenberg equation generated by H ^ x and makes clear that the z-ordered theory is not a simple replacement t ↦ z .

The zero-of-energy issue is central because a bare H ^ x 2 generator is not invariant under H ^ x ↦ H ^ x + C I ^ . The referenced generator resolves this representational ambiguity by transforming the pair

Then

The physical prescription is therefore not an arbitrary shift of the Hamiltonian while keeping the reference fixed. A single declared threshold or baseline belongs to the definition of each problem and shifts covariantly with the Hamiltonian.

Let the referenced energies be concentrated near a non-zero value ε c ,

Then

If

the last term contributes a negligible relative phase over the occupied spectral width. The remaining relative phase has the standard form with

This is a local approximation tied to a selected spectral band. It is not a universal definition of time and does not apply automatically to broad distributions or to states centred on ε c = 0 .

The present construction assumes a time-independent Hamiltonian. If the physical Hamiltonian is explicitly time dependent, H ^ = H ^ ( t ) , its definition already contains an external clock parameter and the stationary spectral decomposition used above is generally unavailable. Replacing that problem by a z-ordered state would require an enlarged relational theory in which the driving field, clock, or measurement apparatus is included as a quantum subsystem. Such a construction is not supplied here.

This limitation is relevant to attosecond science, strong-field ionization, and tunneling-delay measurements. These experiments compare clock-resolved or phase-resolved signals and are commonly analysed with time-dependent wave packets, Wigner-Smith-type delays, or operational dwell-time constructions [20][21]. The present fixed-energy tunneling coefficient can be used as a stationary input, but the measured delay cannot be inferred from z alone.

Three limitations remain. First, no direct measurement protocol for z has been established. Second, the reference energy must be selected by a physical threshold, baseline, or enlarged closed-system description; different non-covariant choices would define different orderings. Third, broad superpositions evolve under a generator quadratic in the referenced Hamiltonian and generally differ from standard Schrödinger evolution. These limitations define the current boundary of the proposal rather than secondary qualifications.

The analysis establishes four principal results. First, fixed-energy spatial eigenfunctions, probability currents, reflection and transmission coefficients, barrier-penetration probabilities, matrix elements, and selection rules are unchanged because the spectral phase is global within one component. Second, a complete bound-plus-continuum state can be written with generalized normalization, explicit channel labels, and the appropriate spectral measure. Third, the z-ordering is unitary and is generated by the square of a referenced Hamiltonian. Fourth, the Gaussian example gives explicit mathematical consequences for packet displacement and spreading.

These results do not establish complete equivalence with standard time-dependent quantum mechanics. In particular, the exact broadband generator is quadratic in the referenced Hamiltonian. Transition rates, lifetimes, linewidths, time delays, tunneling times, and explicitly driven dynamics remain outside the present framework.

The fixed-energy coefficients derived in Sections 5-8 follow from the stationary spatial problem. The energy-domain structure enters through the separation of the spatial basis from the spectral phase, the channel-resolved continuum state, the reference-covariant generator, and the distinction between stationary observables and the ordering of a superposition.

This distinction also explains why the stationary results are recovered almost automatically while the broadband dynamics raises new questions. The fixed-energy calculations establish compatibility with the spatial boundary-value problem. The Gaussian packet and Equation (152) describe the distinct consequences of the proposed generator for superpositions.

Page-Wootters and related relational-clock approaches recover change through correlations between subsystems [1][6]. Quantum cosmology encounters a related problem because the Wheeler-DeWitt equation contains no external Schrödinger time [7]-[11]. Bohmian and wave-function-based cosmological models provide examples in which histories or semiclassical evolution are extracted from a timeless constraint [12][13].

The present proposal is not equivalent to these frameworks. It does not introduce a clock subsystem, and it is not derived from a Hamiltonian constraint of general relativity. Its narrower purpose is to examine whether a unitary ordering based on referenced energy can preserve stationary quantum predictions and generate a mathematically consistent ordering of superpositions. A future relational completion would need to explain how z is correlated with a physical clock and why the selected energy reference is operationally preferred.

The next steps are numerical comparison of exact z-ordered and standard time-ordered packets, extension to multi-channel and higher-dimensional scattering, and the construction of a relational clock model that can connect z with an experimental duration. Further work should also determine whether the referenced quadratic generator can be derived from a larger constrained system rather than postulated, and whether any experimentally distinguishable prediction survives once the clock and measurement interaction are included.