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Register a new userParameter-free ansatz for inferring ground state wave functions of even potentials
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Schrodingers equation (SE) and the information-optimizing principle based on Fishers information measure (FIM) are intimately linked, which entails the existence of a Legendre transform structure underlying the SE. In this comunication we show that the existence of such an structure allows, via the virial theorem, for the formulation of a parameter-free ground states SE-ansatz for a rather large family of potentials. The parameter-free nature of the ansatz derives from the structural information it incorporates through its Legendre properties.
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Several ultra-compact accurate wave functions in the form of generalized Hylleraas-Kinoshita functions and Guevara-Harris-Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), or, equivalently, the Non-Relativistic QED (NRQED), for the ground state energies (4-5 significant digits (s.d.)) of He-like and Li-like iso-electronic sequences in the static approximation with point-like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in $Z$ by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He-like two-electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy $sim 10^{-3}$,a.u. and the same relative accuracies $sim 10^{-2}-10^{-3}$ for both the cusp parameters and the six expectation values, is found. For the Li-like three-electron sequence the most accurate ultra-compact function taken as the variational trial function provides absolute accuracy for energy $sim 10^{-3}$,a.u., 2-3 s.d. for the electron-nuclear cusp parameter for $Z leq 20$ and 3 s.d. for the two expectation values for $Z=3$.
The Schr{o}dinger equation $psi(x)+kappa^2 psi(x)=0$ where $kappa^2=k^2-V(x)$ is rewritten as a more popular form of a second order differential equation through taking a similarity transformation $psi(z)=phi(z)u(z)$ with $z=z(x)$. The Schr{o}dinger invariant $I_{S}(x)$ can be calculated directly by the Schwarzian derivative ${z, x}$ and the invariant $I(z)$ of the differential equation $u_{zz}+f(z)u_{z}+g(z)u=0$. We find an important relation for moving particle as $ abla^2=-I_{S}(x)$ and thus explain the reason why the Schr{o}dinger invariant $I_{S}(x)$ keeps constant. As an illustration, we take the typical Heun differential equation as an object to construct a class of soluble potentials and generalize the previous results through choosing different $rho=z(x)$ as before. We get a more general solution $z(x)$ through integrating $(z)^2=alpha_{1}z^2+beta_{1}z+gamma_{1}$ directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail.
We analytically evaluate the generating integral $K_{nl}(beta,beta) = int_{0}^{infty}int_{0}^{infty} e^{-beta r - beta r}G_{nl} r^{q} r^{q} dr dr$ and integral moments $J_{nl}(beta, r) = int_{0}^{infty} dr G_{nl}(r,r) r^{q} e^{-beta r}$ for the reduced Coulomb Greens function $G_{nl}(r,r)$ for all values of the parameters $q$ and $q$, when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions.
The discrete polymer model with random Boltzmann weights with homogeneous inverse gamma distribution, introduced by Seppalainen, is studied in the case of a polymer with one fixed and one free end. The model with two fixed ends has been integrated by Thiery and Le Doussal, using coordinate Bethe Ansatz techniques and an analytic-continuation prescription. The probability distribution of the free energy has been obtained through the replica method, even though the moments of the partition sum do not exist at all orders due to the fat tail in the distribution of Boltzmann weights. To extend this approach to the polymer with one free end, we argue that the contribution to the partition sums in the thermodynamic limit is localised on parity-invariant string states. This situation is analogous to the case of the continuum polymer with one free end, related to the Kardar--Parisi--Zhang equation with flat boundary conditions and solved by Le Doussal and Calabrese. The expansion of the generating function of the partition sum in terms of numbers of strings can also be transposed to the log-gamma polymer model, with the induced Fredholm determinant structure. We derive the large-time limit of the rescaled cumulative distribution function, and relate it to the GOE Tracy--Widom distribution. The derivation is conjectural in the sense that it assumes completeness of a family of string states (and expressions of their norms already used in the fixed-end problem) and extends heuristically the order of moments of the partition sum to the complex plane.
We develop a framework for characterizing and analyzing engineered likelihood functions (ELFs), which play an important role in the task of estimating the expectation values of quantum observables. These ELFs are obtained by choosing tunable parameters in a parametrized quantum circuit that minimize the expected posterior variance of an estimated parameter. We derive analytical expressions for the likelihood functions arising from certain classes of quantum circuits and use these expressions to pick optimal ELF tunable parameters. Finally, we show applications of ELFs in the Bayesian inference framework.
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