Quantum mechanics in configuration space in context

Abstract: To enhance the way in which wave-particle duality is implemented in the modelling of quantum mechanical systems, Bukhari et al. [New J. Phys. 27, 084501 (2025)] recently introduced an alternative approach to quantum mechanics, namely quantum mechanics in configuration space. This formalism is based on a physically motivated quantisation of Newtonian mechanics and promotes the classical position-velocity states (x,v) to pairwise distinguishable quantum states. The resulting |x,v> states form the basis of the Hilbert space of individual quantum mechanical particles and evolve along classical trajectories. In this paper, we consider the modelling of a mechanical particle in free space and put quantum mechanics in configuration space into context. It is shown that this formalism increases the continuity between quantum and classical mechanics by avoiding a conceptual inconsistency associated with the definition of momentum in canonical quantisation. In addition, we emphasise that standard quantum mechanics and quantum mechanics in configuration space are based on two distinct formulations of classical mechanics.

• The paper presents a novel QMCS framework that quantizes Newtonian mechanics by treating position and velocity as independent observables.
• It employs a reformulated Hilbert space based on (x, v) states to maintain classical trajectories and avoid the conceptual inconsistencies of standard quantum mechanics.
• The approach promises enhanced modeling in fields like molecular dynamics and deepens the understanding of quantum-classical continuity in foundational physics.

Quantum Mechanics in Configuration Space: A Comprehensive Analysis

Context and Motivation

The relationship between classical and quantum mechanical descriptions of physical systems has been a persistent area of conceptual and technical debate. The paper "Quantum mechanics in configuration space in context" (2606.17622) evaluates this relationship by re-examining the foundations on which standard quantum mechanics (SQM) is built, positing that a physically motivated quantization of Newtonian mechanics—not Hamiltonian mechanics—generates a more conceptually continuous framework between quantum and classical regimes. This formulation, termed quantum mechanics in configuration space (QMCS), redefines quantum kinematics and dynamics in terms of position-velocity $(x,v)$ states, directly reflecting the Newtonian perspective.

Classical Mechanics: Formulation Dependencies

The paper emphasizes that the three main formulations of classical mechanics—Newtonian, Lagrangian, and Hamiltonian—are not fundamentally equivalent, despite frequent pedagogical claims to the contrary. Newtonian mechanics provides a direct, causal representation of particles in terms of positions and velocities, while Hamiltonian mechanics introduces canonical conjugate variables through a Legendre transformation, opting for mathematical elegance at the expense of physical perspicuity.

Figure 1: The relations among Newtonian, Lagrangian, and Hamiltonian mechanics, highlighting derivations and non-equivalences in classical theory structures.

This choice of starting point for quantization has a profound impact: canonical quantization of Hamiltonian mechanics, the foundation for SQM, conflates momentum with the generator of spatial translations, leading to conceptual inconsistencies and the loss of an independent velocity observable. In contrast, Newtonian mechanics distinctly maintains position and velocity as foundational entities.

Quantum Mechanics in Configuration Space: Formal Structure

Physical Basis and Hilbert Space Construction

QMCS posits that the quantum description should preserve the physical degrees of freedom evident in Newtonian mechanics—namely, position $x$ and velocity $v$. Each $(x, v)$ classical state is directly promoted to an orthonormal quantum state $|x,v\rangle$. The collection of these states forms the complete Hilbert space for a single particle, with the quantum state given by

$$|\psi(t)\rangle = \iint dx \, dv \, \psi(x,v,t) |x,v\rangle,$$

where $\psi(x,v,t)$ is the wave function in the configuration space basis.

Figure 2: Four basis sets (e.g., $|x,v\rangle$, $|p,a\rangle$) interconnected by Fourier transforms, highlighting the distinction between classical and "least classical" quantum states.

This construction enables novel representations, including $|p,v\rangle$, $x$0, and especially $x$1, which is a basis with no classical analogue. Importantly, $x$2 are the most classical quantum basis states, while $x$3 exhibit maximal quantum delocalization.

Observables and Generators

In the QMCS framework, the position $x$4 and velocity $x$5 operators act diagonally on their basis:

$x$6

There also exist translation generators in both position and velocity—momentum $x$7 and acceleratum $x$8—with

$x$9

All nontrivial commutators stem from $v$0 and $v$1, enabling simultaneous sharp specification of position and velocity, a property not available in SQM due to the sole use of position and momentum as canonical conjugates.

Dynamics

QMCS dynamics are encoded in a Schrödinger-type equation with a dynamical Hamiltonian $v$2 chosen such that the basis states $v$3 evolve according to Newton's laws. For a free particle:

$v$4

which is Hermitian (since $v$5 and $v$6 commute), and ensures trajectories remain sharply localized if initially prepared as such. This is in marked contrast to SQM, where wave packets universally delocalize over time due to the quadratic kinetic energy operator.

Empirical and Theoretical Claims

Conceptual and Empirical Continuity

The central claim is that QMCS achieves strong conceptual continuity with Newtonian mechanics by:

• Representing both position and velocity as independent, simultaneously measurable observables.
• Ensuring localized quantum particles remain localized during free evolution, directly mirroring Newtonian trajectories.
• Avoiding the conflation of velocity with translation generator (momentum), which in SQM leads to differing treatments of massive particles and photons with respect to wave-particle duality.

These features allow for enhanced explanatory power in the classical-to-quantum transition, as exemplified by the faithful reproduction of classical motion and energy conservation in the quantum theory.

Relation to Standard Quantum Mechanics

A direct comparative analysis shows that SQM, based on Hamiltonian mechanics and canonical quantization, lacks several physical observables (such as an independent velocity operator) and thus cannot represent certain limits or features of classical systems without invoking approximations (e.g., Ehrenfest theorem, which is not universally valid in SQM).

Figure 3: Schematic relating QMCS and SQM to their respective classical theories, emphasizing conceptual continuity upwards from classical to quantum domains.

Implications and Future Directions

The QMCS framework suggests that the choice of classical theory formulation deeply influences quantum theory's physical content and modeling capacity. The potential to straightforwardly extend Newtonian concepts, including causal trajectory-based motion, into quantum theory has several implications:

• Practical Modeling: QMCS may provide more robust classical-quantum hybrid methods in molecular dynamics, quantum chemistry, and condensed matter, especially where decoherence and localization are central.
• Foundational Physics: The presence of independent velocity and acceleratum operators may provoke re-examination of quantum relativity, quantum gravity, and the quantum-field-theoretic formulation of matter, particularly given the closer analogy to quantum electrodynamics for photons.
• Measurement Theory: The improved treatment of wave-particle duality could unify the description of particles and fields in quantum theory without recourse to distinct quantization schemes or ambiguous identification of observables.

Further research is required to extend the formalism to interacting systems, relativistic regimes, and coupled fields, as well as to determine whether predictions of QMCS and SQM diverge in scenarios not trivially mappable between the two.

Conclusion

Quantum mechanics in configuration space offers a physically motivated alternative to standard quantum mechanics, predicated on the direct quantization of Newtonian mechanics and the preservation of position and velocity as independent degrees of freedom. The approach resolves several conceptual inconsistencies present in standard canonical quantization, leading to enhanced empirical and conceptual continuity between quantum and classical mechanics. The implications of this framework extend from improved practical modeling tools to foundational questions in quantum theory, warranting further investigation and development.