Hidden symmetries and thermodynamic properties for a harmonic oscillator plus an inverse square potential

Abstract The exact solutions of a one‐dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained. The ladder operators constructed directly from the normalized wavefunctions are found to satisfy a su(1, 1) algebra. Another hidden symmetry is used to explore the relations between the eigenvalues and eigenfunctions by substituting x → − ix . The vibrational partition function Z is calculated exactly to study thermodynamic functions such as the vibrational mean energy U , specific heat C , free energy F , and entropy S . It is both interesting and surprising to find that both vibrational specific heat C and entropy S are independent of the potential strength α. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

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