Classical Map for Quantum Electrons

• Classical mapping for quantum electrons is a framework that converts quantum properties, such as wave packet dynamics and many-body interactions, into classical variables for deeper insight.
• Various methodologies like hyperbolic quantization, classical fluid analogs, and momentum maps offer distinct computational tools for simulating complex quantum systems.
• These approaches enable precise predictions in regimes ranging from non-dispersive Gaussian states to chaotic quantum transport, bridging the quantum-classical divide.

A classical map for quantum electrons denotes any theoretical or computational framework in which the quantum mechanical properties of electrons—particularly many-body effects, wave packet dynamics, spin evolution, and correlation functions—are translated into an equivalent set of classical variables, classical equations, or classical statistical ensembles. The purpose of such mappings ranges from achieving computational efficiency to gaining deeper insight into quantum–classical correspondence, chaos, and the role of correlations in extended systems. Various methodologies realize these maps, including hyperbolic quantization procedures, classical fluid analogs, phase-space mappings, effective classical potentials, and structured reductions via Poisson geometry and block-encoded linear algebra. The following sections detail key approaches, foundational principles, explicit mapping schemes, applications, and open issues.

1. Alternative Quantization and Hyperbolic Quantum Maps

The alternative quantization procedure, as presented in "How to prepare quantum states that follow classical paths" (Pedram, 2010), diverges from the standard Schrödinger equation by quantizing a function defined from two classical solutions sharing identical energy and then replacing their momenta with differential operators according to

$$\mathcal{F} \Psi(u, v) = \left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial u^2} + \frac{\hbar^2}{2m}\frac{\partial^2}{\partial v^2} + V(u) - V(v)\right)\Psi(u, v) = 0,$$

yielding a hyperbolic partial differential equation for $\Psi(u,v)$.

This formulation allows both the initial wave function and its "slope" (i.e., first derivative) to be set independently, in analogy with classical initial-value problems. For free particles and particles in a box, the constructed wave packets remain sharply peaked along classical trajectories and do not disperse, as demonstrated by explicit formulas for non-spreading Gaussian states:

$$\Psi(u,v) = \frac{1-i}{2}\left[e^{-\alpha(u+v+d)^2} + e^{-\alpha(u+v-d)^2} + i e^{-\alpha(u-v+d)^2} + i e^{-\alpha(u-v-d)^2}\right].$$

Moreover, the minimum uncertainty product $\Delta u \Delta p_u \simeq \hbar/2$ holds for specially chosen initial conditions, preserving the essential quantum constraint while achieving full classical–quantum correspondence.

This alternative quantization offers an explicit classical mapping applicable to propagation in potential wells, enables the engineering of non-dispersive quantum states, and features analogies to Wheeler–DeWitt hyperbolic equations in quantum cosmology.

2. Classical–Map Hyper-Netted-Chain (CHNC) and Orbital-Free Methods

The CHNC method (Dharma-wardana, 2011, Dharma-wardana, 2019) replaces the quantum electron gas by an effective classical fluid at temperature

$T_{cf} = \sqrt{T_q^2 + T^2},$

where $T_q$ is a quantum temperature fitted to reproduce exact correlation energies. Quantum statistics are recast into classical interactions via a Pauli exclusion (Fermi-hole) potential and diffraction-corrected Coulomb interactions:

$$g_{ij}(r) = \exp\left[-\beta(P(r)\delta_{ij} + V_{cou}(r)) + h_{ij}(r) - c_{ij}(r)\right],$$

with $P(r)$ designed to yield the non-interacting Fermi pair distribution exactly, and $V_{cou}(r)$ including thermal diffraction corrections.

This mapping has been extensively validated against QMC and PIMC (Dharma-wardana, 2019). The resulting pair-distribution functions (PDFs) are $N$-representable, as the transformation from the Slater-determinant noninteracting PDF to the interacting case is well-behaved and invertible; the mapping gives accurate results for exchange-correlation energies and radial distribution functions in uniform and non-uniform systems, warm dense matter, and 2D electron-hole plasmas. The neutral pseudo-atom (NPA) model (Dharma-wardana, 2019) further shows rapid agreement with electronic density-functional theory in highly correlated regimes.

Compared to quantum approaches, CHNC is computationally efficient, independent of particle number and temperature, and readily extends to electron–ion mixtures. This makes it suitable for complex, high-density or low-temperature models where QMC or DFT-MD would be prohibitive.

3. Classical and Quantum Momentum Maps

The "momentum map" framework (Esposito, 2012) leverages symmetries in classical (Poisson) and quantum systems to reduce dimensionality and encode conserved quantities. In classical theory, the momentum map $\mu: M \to \mathfrak{g}^*$ associates conserved variables to group symmetries via

$\xi_M(m) = \pi^+(\mu^*(\theta_\xi))(m),$

with $\pi^+$ the Poisson structure and $\theta_\xi$ a left-invariant one-form. Upon reduction (e.g. Marsden–Weinstein), one constructs a lower-dimensional effective manifold with inherited symplectic structure.

Quantum mechanically, under deformation quantization and quantum groups, one defines a quantum momentum map $$\mu_\hbar: \mathcal{U}_\hbar(\mathfrak{g}) \to \Omega^1(C^\infty_\hbar(M))$$ to factorize Hopf-algebra actions on the star-product algebra:

$$\Phi_\hbar(\xi)(f) = \frac{1}{\hbar}\sum_i a^i_\xi [b^i_\xi, f]_\star.$$

The classical map is recovered in the $\hbar \to 0$ limit.

The use of momentum maps and Poisson reductions is central in modeling quantum electrons under symmetry constraints, notably enabling the derivation of effective models for Landau levels, band structures, and topologically nontrivial states by dimensionality reduction and quotienting the symmetry group.

4. Classical Mapping in Many-Body Quantum Dynamics and Transport

Quasi-classical mapping schemes (Li et al., 2012, Levy et al., 2019) translate fermionic operators into classical phase-space variables, maintaining anti-commutation via quaternionic or cross-product relations:

$a_n^\dagger a_n \to x_n p_{y,n} - y_n p_{x,n}.$

The time evolution of quadratic Hamiltonians is exactly mapped by imposing $i[\hat{H}_0, \hat{A}] \to \{\mathcal{A}, H_0\}$, preserving Heisenberg dynamics.

For two-body (electron–electron) interactions, occupation quantization is imposed in the classical equations using threshold (Heaviside) activation or high power modifications, ensuring strong repulsion for double occupancy as in the quantum Coulomb blockade regime. Quasi-classical ensemble sampling reproduces initial quantum statistics, and importance sampling strategies mitigate variance in observable averages, yielding correct steady-state and dynamic behaviors for complex many-body systems such as the Anderson impurity model.

This approach achieves linear scaling in computational cost with system size and accurately captures phenomena like the Coulomb blockade, with nontrivial extensions to current fluctuations and high-order correlation functions.

5. Quantum–Classical Mapping in Statistical and Chaotic Systems

Several works apply classical mapping principles to elucidate chaos, localization, and record statistics in quantum electron systems.

In standard (kicked rotor) maps (Srivastava et al., 2015), the record statistics for classical momentum square and quantum eigenfunction intensities reveal universal behavior corresponding to random walks (scaling as $\sqrt{t}$ in the chaotic regime) and Gumbel-type distributions for quantum states. Accelerator modes induce anomalous diffusion signaled by deviations in scaling exponents, providing a sensitive probe for nontrivial quantum transport dynamics.

In quantum linear algebra (Chen et al., 1 Nov 2024), block-encodings and quantum singular value transformations are used to represent exponentially large disordered electron Hamiltonians, enabling computation of observables such as the reduced density matrix, Green’s function, or conductivity. Quantum algorithms provide exponential and polynomial advantages over classical methods in extracting properties of the Anderson localization transition or spectral features in large disordered systems.

6. Mapping of Specific Wave Functions and Correlated Ground States

In the context of fractional quantum Hall states (Kramer, 2012), Laughlin’s wave function is recast into a classical statistical ensemble with a logarithmic "Pauli-potential," embodying antisymmetry and matching many-body quantum correlations. Newtonian dynamics and ensemble averaging, aided by modern hardware, reproduce equilibrium distributions and pair correlation functions, delineating how ground-state quantum statistics emerge from a classical mapping.

For few-body systems on curved geometries (Yang et al., 2018), quantum generalizations of the Thomson problem combine configuration interaction with semi-classical analysis. The quantum ground state, incorporating kinetic energy and symmetry constraints, converges to classical equilibrium as the confining radius grows, with quantum fluctuations captured quantitatively as vibrational modes and symmetry-dependent nodal patterns in reduced density matrices.

7. Classical–Quantum Correspondence in Spin Dynamics and Measurement Theory

A classical analog for spin-½ electron evolution (Wharton et al., 2011) is realized by coupled oscillators whose Lagrangian dynamics, encoded via quaternionic variables and structured coupling matrices, reproduce Zeeman splitting, geometric phase, and gyromagnetic ratio doubling. While exact for unmeasured states, classical analogs do not encapsulate quantum measurement outcomes or Born-rule probabilities, highlighting the distinction between unitary dynamics and measurement theory.

Furthermore, local electron correlations in the Hubbard model (Bellomia et al., 23 Jun 2025) are shown to be fully classical in the natural orbital basis, where the reduced density matrix is separable and mutual information measures ("nonfreeness") capture all intra-orbital correlation content. Nonlocal processes in the extended quantum environment renormalize these classical correlation values, forging a link between local information theory and many-body quantum physics.

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In summary, classical mapping frameworks for quantum electrons span rigorous quantization procedures, effective classical fluid analogs, phase-space mappings for many-body dynamics, transform-based reduction methods, statistical diagnostics in chaos, and analog models for spin and orbital correlations. These approaches yield practical computational tools, deep insight into quantum–classical transitions, and a systematic means for extracting essential physical quantities and structure from complex quantum electron systems.