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
WA(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
We address the nonlinear properties of the double-Morse potential as a resource for single-mode quantum states due to its double-well structure and anharmonicity. We obtain analytical expressions for the ground-state wavefunction and the corresponding ground-state energy, using the asymmetry (width) parameter
α as the primary control parameter. We then assess non-Gaussianity and nonclassicality as quantitative signatures of nonlinearity and quantumness, and we find that both increase monotonically with
α. Furthermore, we analyze the metrological performance of the model for estimating the structural parameter
α. By evaluating the corresponding Fisher information, we show that position measurements are optimal and can saturate the Cramér-Rao bound. In particular, the estimation of
α is most precise in the shallow-well regime, where the quantum Fisher information is largest. For deep wells, enhanced sensitivity is instead obtained for the reparameterized control variable
A=2e−αx0, provided that
x0 is independently calibrated. These results establish the double-Morse potential as a controllable source of non-Gaussianity and nonclassicality, with a metrological behavior that depends on the chosen estimation parameter. We highlight possible applications of this model in quantum sensing, continuous-variable quantum information, and quantum simulation.
Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double‐well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation‐induced exact diagonalization procedure. The information entropy‐based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase‐space description provides a semi‐classical explanation for this feature. Total information entropy and phase‐space area show similar behavior with increasing barrier height. image
The bound state solutions of the Schrödinger equation (SE) under a deformed hyperbolic potential are used in this paper to investigate the correlation between the information content, such as Shannon entropies and Fisher information, and the variance of a quantum system in both momentum and position spaces. The variance was obtained from the expectation moments in conjugated spaces and used to get the uncertainty products, Fisher information products and Shannon entropic sums for different potential parameters. These information measures were observed to vary with the potential parameters and obey their lowest bound inequalities. An increase in the Fisher information leads to a lower uncertainty and less information while decreases in the Fisher information result in higher uncertainties. We proposed a relationship between the Shannon entropies, Fisher information and the variance where an increase in the Fisher information leads to a lower Shannon entropy or a lower spread of the probability distribution and vice versa. The numerical results obtained with the proposed relations agree with the most utilized method in the literature. This current work simplifies the calculation of the Shannon entropies notably for complex potential functions.
By employing the Nikiforov–Uvarov functional analysis (NUFA) method, we solved the radial Schrödinger equation with the shifted Morse potential model. The analytical expressions of the energy eigenvalues, eigenfunctions and numerical results were determined for selected values of the potential parameters. Variations of different thermodynamic functions with temperature were discussed extensively. Different quantum information-theoretic measures, including Shannon entropy, Fisher information and Fisher–Shannon product of the shifted Morse potential, were investigated numerically and graphically in position and momentum spaces for the ground and the first excited states. The quantum information theories considered satisfied their corresponding inequalities, including Bialynicki–Birula–Mycielski, Stam–Cramer–Rao inequalities and the Fisher–Shannon product relation.
We have studied the variation of the position space statistical complexity measure defined by López-Ruiz, Mancini, and Calbet such as the product of exponential of the Shannon information entropy and the disequilibrium by using the 1D-normalized probability densities derived from solutions of the Schrödinger equation corresponding to the q-deformed harmonic oscillator and q-deformed Morse potentials. An analysis of the numerical results in terms of Shannon information entropy, disequilibrium and complexity measure are presented. In q-deformed harmonic oscillator, q-dependence of the complexity shows a minimum point for all excited energy levels. In the case of q-deformed Morse Potential, complexity decreases with increasing
q for the investigated diatomic molecules.