The Energy-Levels and Transition Probabilities for a Bounded Linear Harmonic Oscillator

Exact diagonalization of the D-dimensional spatially confined quantum harmonic oscillator

“…We remark that an equivalent result to ours, for (2) r E , can also be found in reference BAIJAL and SINGH (1955). The sum was however left unevaluated in that paper.…”

“…Maple code to determine (2) r E ================================================= >summand:=q->(2*s+1+modp(q,2))^2/(2*r-2*s)^5/(q+2*s+2+modp(q,2))^5; >A2r:=sum(summand(2*r),s=0..r-1): # replace the last ":" with ";" to see the polygamma sums >B2r:=sum(summand(2*r),s=r+1..infinity): # replace ":" with ";" to see the polygamma sums >E2r:=expand(simplify(4*(2*r+1)^2*(A2r+B2r))): # we suppress the factor [lambda^4*epsilon] ># collect terms of the same order in Pi E2r:=collect(%,Pi): >E2r:=factor(coeff(E2r,Pi^4))*Pi^4+factor(coeff(E2r,Pi^2))*Pi^2+op(3,E2r);…”

Highly accurate solutions for the confined hydrogen atom

“…When Suryanarayana and Weil made their calculations by computing the zeros of the confluent hypergeomtric functions, efficient computational algorithms were not available and evaluation of hypergeometric functions produced energy eigenvalues were not accurate. This was the same problem faced by Baijal and Singh 39 in their calculations on the one‐dimensional confined oscillator.…”

Atoms and molecules in soft confinement potentials

“…for 0 ≤ M ≤ 2 . The expectation value r −1 follows directly from the virial theorem, and the r −2 follows immediately from (21). From these relations, the strict inequality…”

“…An example of such a system is the non-relativistic, artificially bounded harmonic oscillator, enclosed between potential walls. This model has been successfully applied to problems such as the fundamental mass-radius relation for white dwarf stars [17], the rate of escape of stars from galactic and globular clusters [18], the role of the symmetrically bounded linear harmonic oscillator in the theory of the specific heat of solids [19], second-order phase transitions [20], energy levels and transition probabilities for a bounded linear oscillator [21], anharmonic effects in solids [22], magnetic properties of metallic solids [23], and nuclear shell models [24]. Similar model systems have been employed in various research fields where the effects of pressure on energy levels [1,8,12] and properties such as polarizabilities [1,9], hyperfine splittings [7][8][9]11], nuclear magnetic shielding constants [9], hyperfine interaction energies [10] and electron (de)localization [25] have been of interest.…”