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Register a new userExact Quantization Rule to the Kratzer-Type Potentials: An Application to the Diatomic Molecules
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For any arbitrary values of $n$ and $l$ quantum numbers, we present a simple exact analytical solution of the $D$-dimensional ($Dgeq 2$) hyperradial Schr% {o}dinger equation with the Kratzer and the modified Kratzer potentials within the framework of the exact quantization rule (EQR) method. The exact energy levels $(E_{nl})$ of all the bound-states are easily calculated from this EQR method. The corresponding normalized hyperradial wave functions $% (psi_{nl}(r))$ are also calculated. The exact energy eigenvalues for these Kratzer-type potentials are calculated numerically for the typical diatomic molecules $LiH,$ $CH,$ $HCl,$ $CO,$ $NO,$ $O_{2},$ $N_{2}$ and $I_{2}$ for various values of $n$ and $l$ quantum numbers. Numerical tests using the energy calculations for the interdimensional degeneracy ($D=2-4$) for $I_{2}, $ $LiH,$ $HCl,$ $O_{2},$ $NO$ and $CO$ are also given. Our results obtained by EQR are in exact agreement with those obtained by other methods.
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In the present study, the improved screened Kratzer potential (ISKP) is investigated in the presence of external magnetic and Aharanov-Bohm (AB) fields within the framework of non-relativistic quantum mechanics. The Schrodinger equation is solved via the Nikiforov-Uvarov Functional Analysis (NUFA) method and the energy spectra and the corresponding wave function for the ISKP in the presence of external magnetic fields are obtained in a closed form. The obtained energy spectra are used to study three selected diatomic molecules (H2, HCl and LiH). It is observed that the present of the magnetic and AB fields removes the degeneracy for different values of the control parameter. The thermodynamic and magnetic properties of the ISKP in the present of the magnetic and AB fields are also evaluated. The effects of the control potential parameter on the thermodynamic and magnetic properties of the selected diatomic molecules are discussed.
For one-dimensional power-like potentials $|x|^m, m > 0$ the Bohr-Sommerfeld Energies (BSE) extracted explicitly from the Bohr-Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones contrary to the negative parity states where BSE remain above the exact ones for $m>2$ but they are below them for $m < 2$. The ground state BSE as the function of $m$ are of the same order of magnitude as the exact energies for linear $(m=1)$, quartic $(m=4)$ and sextic $(m=6)$ oscillators but relative deviation grows with $m$ reaching the value 4 at $m=infty$. For physically important cases $m=1,4,6$ for the $100$th excited state BSE coincide with exact ones in 5-6 figures. It is demonstrated that modifying the right-hand-side of the Bohr-Sommerfeld quantization condition by introducing the so-called {it WKB correction} $gamma$ (coming from the sum of higher order WKB terms taken at the exact energies) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is small, bounded function $|gamma| < 1/2$ for all $m geq 1$, it is slow growing with increase in $m$ for fixed quantum number, while it decays with quantum number growth at fixed $m$. For the first time for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are written explicitly in closed analytic form with high relative accuracy $10^{-9 -11}$ (and $10^{-6}$).
Astrophysical molecular spectroscopy is an important means of searching for new physics through probing the variation of the proton-to-electron mass ratio, $mu$. New molecular probes could provide tighter constraints on the variation of $mu$ and better direction for theories of new physics. Here we summarise our previous paper citep{19SyMoCu.CN} for astronomers, highlighting the importance of accurate estimates of peak molecular abundance and temperature as well as spectral resolution and sensitivity of telescopes in different regions of the electromagnetic spectrum. Whilst none of the 11 astrophysical diatomic molecules we investigated showed enhanced sensitive rovibronic transitions at observable intensities for astrophysical environments, we have gained a better understanding of the factors that contribute to high sensitivities. From our results, CN, CP, SiN and SiC have shown the most promise of all astrophysical diatomic molecules for further investigation, with further work currently being done on CN.
We present a data-driven approach for the prediction of the electric dipole moment of diatomic molecules, which is one of the most relevant molecular properties. In particular, we apply Gaussian process regression to a novel dataset to show that dipole moments of diatomic molecules can be learned, and hence predicted, with a relative error <5%. The dataset contains the dipole moment of 162 diatomic molecules, the most exhaustive and unbiased dataset of dipole moments up to date. Our findings show that the dipole moment of diatomic molecules depends on atomic properties of the constituents atoms: electron affinity and ionization potential, as well as on (a feature related to) the first derivative of the electronic kinetic energy at the equilibrium distance.
We carry out an exact quantization of a PT symmetric (reversible) Li{e}nard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schr{o}dinger equation therein. We obtain explicitly the eigenvalues and eigenfunctions (in momentum space) and deduce the remarkable result that the spectrum agrees exactly with that of the linear harmonic oscillator, which is also confirmed by a semiclassical modified Bohr-Sommerfeld quantization rule, while the eigenfunctions are completely different.
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