Exact Quasiprobability Hierarchy of the Double-Morse Oscillator: From Potential Geometry to Operator Ordering

• URL: http://arxiv.org/abs/2608.23316v3
• Date: Sat, 29 Aug 2026 06:56:24 GMT
• Title: Exact Quasiprobability Hierarchy of the Double-Morse Oscillator: From Potential Geometry to Operator Ordering
• Abstract summary: We develop an exact, representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator.<n>In the double-Morse potential, the dimensionless parameter $A$ controls the separation of the minima and the central barrier.<n>We show that $A$ controls how the same non-Gaussian and nonclassical state is resolved across complementary representations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Phase-space and quasiprobability methods now play operational roles in quantum technologies, characterizing localization, non-Gaussianity, nonclassical resources, and coarse-graining. We develop an exact, representation-consistent analysis of the lowest quasi-exact ground state of the symmetric double-Morse oscillator. In the double-Morse potential, the dimensionless parameter $A$ controls the separation of the minima and the central barrier, thereby changing the physical ground state. At fixed $A$, the Cahill--Glauber parameter $s$ labels the quasiprobability $W_A^{(s)}(q,p)$: $s=0$, $-1$, and $1$ give the Wigner, Husimi $Q$, and Glauber--Sudarshan $P$ representations, respectively. Although the potential is double-welled for $0<A<1$, the exact ground-state amplitude is single-peaked at the origin and lies above the barrier; as $A$ approaches unity, the merged well remains locally quartic rather than harmonic. Closed analytical expressions are obtained for the Wigner function and Weyl characteristic function. The Wigner function displays the $A$-dependent exchange between position and momentum localization and retains negative regions, certifying nonclassicality and, for this pure state, non-Gaussianity. The Weyl function is its Fourier dual, generates symmetrically ordered moments and cumulants, and yields the full $s$-ordered hierarchy. For $s<0$, isotropic Gaussian smoothing suppresses fine sign-changing structure while preserving the large-scale localization envelope. The Husimi endpoint is nonnegative without implying classicality, whereas the $P$ representation remains distributional. Thus, $A$ controls the physical phase-space geometry, while $s$ controls how the same non-Gaussian and nonclassical state is resolved across complementary representations.

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