In this study, the approximate analytical bound state solutions with a newly proposed generalized Yukawa potential are solved using the extended Nikiforov–Uvarov method with the help of Greene–Aldrich approximation to the centrifugal term. The corresponding wave function is expressed in terms of the confluent Heun function. The numerical bound state solutions are reported for various values of the screening parameter and orbital angular quantum number. Furthermore, the bound state energy spectrum is used to study the thermodynamic properties of the system comprehensively, and we have discussed the influence of inverse temperature and maximum quantum number on the thermodynamic properties.
The state space of quantum light contains infinitely many photon-number levels, whereas numerical models usually retain only finitely many. Direct truncation can introduce Wigner negativity into an initially Wigner-positive state. We prove that every Wigner-positive state of finitely many modes admits physical approximations with finite photon-number support under the same mean-energy cap. For
m modes, mean total photon number at most
E, and trace-norm error at most
ε∈(0,1], a sufficient cutoff is
L=O((E+1)ε−2[m+lnεE+2]). The construction represents Wigner positivity by operator positivity in an enlarged system, where local photon-number projections preserve it. It yields finite semidefinite programmes whose optima
qL(A,E), for any bounded Hermitian observable
A, satisfy
0≤ΩE(A)−qL(A,E)≤εinfc∈R∥A−cI∥, where
ΩE(A) is the optimum over all Wigner-positive states under the same energy cap. At fixed mode number and energy, a continuity bound also provides finite approximations to Wigner entropy minima and finite witnesses for every strict entropy deficit. These bounds determine how much of the infinite state space a calculation must retain to certify its conclusions.
The static approximation, which neglects the distortion of the (complex) target, that is, virtual excitations, during the collision of a particle with the target, was shown previously to give an upper bound on the scattering length when the number of bound states supported by the static and true Hamiltonians are the same. It represents an underadjustment of the target to the field of the scattering particle. It is perhaps then to be expected that the adiabatic approximation, in which the target wave function is calculated for each value of the incident particle coordinate as if the incident particle were fixed at the given point and which represents therefore an overadjustment of the target to the field of the incident particle, might provide the opposite bound, the lower bound, on the scattering length. This possibility is explored for the determination of scattering lengths. The bound property is shown to hold in cases where both the true and adiabatic Hamiltonians support the same number of bound states; some (reasonable) subsidiary conditions must also be satisfied. Possible extensions of the formulation to scattering with nonzero energy and with exchange are briefly mentioned. The theory is then applied to
e+H and
e+He scattering at zero incident energy; a reasonably accurate bound for the latter system is obtained, but the adiabatic approximation gives only a very crude, though rigorous, bound on the
e+H scattering length. A connection between the generalized optical potential and the adiabatic potential is derived using projection-operator techniques, thus further clarifying the physical content of the adiabatic picture. Bounds are also obtained for atom-atom and ion-atom collisions; they are interesting only if the electron-proton mass ratio is not thought of as vanishingly small. Some consideration is given to
pH scattering. Lower bounds on electron-atom scattering lengths are more difficult to obtain.