Elimination of angular dependency in quantum three-body problem made easy

Anjan Sadhukhan^1*, Henryk A. Witek¹

¹Department of Applied Chemistry, National Yang Ming Chiao Tung University, Hsinchu, Taiwan

* Presenter:Anjan Sadhukhan, email:anjanphy21@gmail.com

Separation of variables in the Schrödinger equation (SE) for non-relativistic quantum three-body systems (QTBS) has been a longstanding area of research. Leveraging the rotational and translational invariance, the nine-dimensional SE can, in principle, be reduced to a system of three-dimensional partial differential equations (PDEs), referred to here as the reduced radial SE (RRSE). Reducing the SE to a set of three-dimensional PDEs by separating angular variables has proven to be both challenging and intricate. Breit's [1] seminal work achieved this reduction for Pº states using internal coordinates proposed by Hylleraas [2]. The extension of this reduction to states of arbitrary angular momentum has been revisited by multiple researchers [3–9], yielding insights into the conceptual foundations of the problem.

In this work, we present a novel and straightforward technique to derive the RRSE for nonrelativistic QTBS using solid bipolar spherical harmonics (BSH) as the angular basis for arbitrary angular momentum and space parity states. The derived RRSE matches the previously reported equations in the literature [6]. We introduce a correspondence relation between bipolar spherical harmonics and Wigner-D functions, simplifying the evaluation of angular matrix elements compared to previous methods [5,10]. All the relations are validated by accurately estimating energy eigenvalues using an explicitly correlated Hylleraas-type basis for L = 0 to 7 natural parity states of the helium atom in octuple precision, with results demonstrating good agreement with the best reported values.

References:
[1] G. Breit, Phys. Rev. 35 (1930) 569–578.
[2] E. A. Hylleraas, Z. Phys. 54 (1929) 347–366.
[3] J. Hirschfelder and E. Wigner, Proc. Natl. Acad. Sci. 21 (1935) 113–119.
[4] A. K. Bhatia and A. Temkin, Rev. Mod. Phys. 137 (1965) 1050–1064.
[5] G. W. F. Drake, Phys. Rev. A 18 (1978) 820–826.
[6] V. Efros, Sov. Phys. JETP 63 (1986) 5–13.
[7] T. K. Mukherjee and P. K. Mukherjee, Phys. Rev. A 51 (1995) 4276–4278.
[8] M. Pont and R. Shakeshaft, Phys. Rev. A 51 (1995) 257–265.
[9] G. Pestka, J. Phys. A 41 (2008) 235202 1–13.
[10] J.-L. Calais and P.-O. Löwdin, J. Mol. Spectr. 8 (1962) 203–211.

Keywords: Atomic structure, Quantum Three-Body problem, Hylleraas type basis, Precision Measurements