[Paper Review] Algebraic derivation of Kramers-Pasternack relations based on the Schrodinger factorization method

Tomasz Szymanski, J. K. Freericks|arXiv (Cornell University)|Jul 22, 2020

Advanced Physical and Chemical Molecular Interactions6 references32 citations

TL;DR

The paper derives the Kramers-Pasternack relations algebraically via Schrödinger factorization, and computes the second inverse moment without Feynman-Hellman or brute-force integrals, enabling full recurrence for radial moments in hydrogenic atoms.

ABSTRACT

The Kramers-Pasternack relations are used to compute the moments of r (both positive and negative) for all radial energy eigenfunctions of hydrogenic atoms. They consist of two algebraic recurrence relations, one for positive powers and one for negative. Most derivations employ the Feynman-Hellman theorem or a brute-force integration to determine the second inverse moment, which is needed to complete the recurrence relations for negative moments. In this work, we show both how to derive the recurrence relations algebraically and how to determine the second inverse moment algebraically, which removes the pedagogical confusion associated with differentiating the Hamiltonian with respect to the angular momentum quantum number l in order to find the inverse second moment.

Motivation & Objective

• Motivate and clarify the calculation of all radial moments for hydrogenic atoms using algebraic methods.
• Derive both Pasternack relations (positive and inverse moments) algebraically.
• Provide a purely operator-based method to obtain the second inverse moment without differentiating with respect to l.

Proposed method

• Use Schrödinger factorization to construct ladder operators B_l and the intertwining relation among Hamiltonians H_l.
• Employ the hypervirial theorem in the form <[O,H_l]>=0 to derive the Kramers-Pasternack relation for positive moments (Eq. 2).
• Develop an algebraic procedure to compute the inverse second moment by relocating B_l and B_l^† through r^-2 and using the intertwining relation (Eqs. 23-28).
• Determine <n,l|1/r^2|n,l> algebraically via a chain of recursions in l and n, ending with a explicit expression (Eq. 43).
• Utilize the subsidiary condition B_{n-1}|n,n-1>=0 to fix right-hand side moments (Eq. 39-42).
• Show how to obtain higher inverse moments and connect positive and negative moments through the Pasternack inversion relation (Eq. 45-46).

Experimental results

Research questions

• RQ1 Can the Kramers-Pasternack relations be derived purely algebraically without recourse to the Feynman-Hellman theorem or direct Laguerre-polynomial integration?
• RQ2 How can the second inverse moment ⟨n,l|1/r^2|n,l⟩ be computed algebraically within the Schrödinger factorization framework?
• RQ3 Do the algebraic methods extend to obtain all inverse and positive radial moments for hydrogenic states via recurrence relations?
• RQ4 What is the role of the intertwining relation and the subsidiary condition in determining all degenerate hydrogenic states and their moments?

Key findings

• Successful algebraic derivation of both Pasternack relations using Schrödinger factorization and hypervirial theorem.
• An algebraic method yields ⟨n,l|1/r^2|n,l⟩ = 1/(a_0^2 n^3 (l+1/2)) (Eq. 43).
• The second inverse moment enables completing the negative-moment recurrence without brute-force integration or differentiating w.r.t. l.
• Derived a recursion that relates ⟨n,l|1/r^2|n,l⟩ to ⟨n,l+1|1/r^2|n,l+1⟩ (Eq. 34) and iterates to end states (Eq. 36).
• Equations provide a path to compute higher inverse moments via Eq. 22 and Eq. 33–46, linking positive and negative moments (Eq. 45–46).
• The approach highlights the pedagogical clarity of operator methods for hydrogenic radial moments.