[Paper Review] Normalization of Scattering States, Scattering Phase Shifts and Levinson's Theorem

Nathan Poliatzky|ArXiv.org|Apr 4, 1993

Quantum Mechanics and Applications9 references3 citations

TL;DR

This paper provides a rigorous derivation of the normalization of scattering states in quantum mechanics, showing that the normalization integral includes an additional term proportional to the derivative of the phase shift—though negligible in standard integrals, it encodes full phase shift information. The work proves Levinson's theorem in a general framework using state completeness, derives a new result for Dirac particles at positive and negative energies, and applies these insights to generalized quantum electrodynamics, proving that finite bound states imply zero total charge and that the coupling constant must be an eigenvalue with a fixed value.

ABSTRACT

We show that the normalization integral for the Schrödinger and Dirac scattering wave functions contains, besides the usual delta-function, a term proportional to the derivative of the phase shift. This term is of zero measure with respect to the integration over momentum variables and can be discarded in most cases. Yet it carries the full information on phase shifts and can be used for computation and manipulation of quantities which depend on phase shifts. In this paper we prove Levinson's theorem in a most general way which assumes only the completeness of states. In the case of a Dirac particle we obtain a new result valid for positive and negative energies separately. We also make a generalization of known results, for the phase shifts in the asymptotic limit of high energies, to the case of singular potentials. As an application we consider certain equations, which arise in a generalized interaction picture of quantum electrodynamics. Using the above mentioned results for the phase shifts we prove that any solution of these equations, which has a finite number of bound states, has a total charge zero. Furthermore, we show that in these equations the coupling constant is not a free parameter, but rather should be treated as an eigenvalue and hence must have a definite numerical value.

Motivation & Objective

• To establish a general normalization formalism for scattering states that includes phase shift derivative contributions.
• To re-derive Levinson's theorem without relying on specific potential forms, using only completeness of states.
• To extend high-energy phase shift behavior to singular potentials.
• To analyze a generalized interaction picture in quantum electrodynamics and derive constraints on charge and coupling constants.

Proposed method

• Derive the normalization integral for Schrödinger and Dirac scattering states, identifying a term proportional to the derivative of the phase shift.
• Use completeness of states to prove Levinson's theorem in a general setting, valid for both scalar and spinor particles.
• Apply the phase shift derivative term to analyze asymptotic high-energy limits, extending known results to singular potentials.
• Analyze a class of equations from a generalized interaction picture in QED, using phase shift information to constrain solutions.
• Demonstrate that finite bound states in these equations imply total charge zero, using the derived normalization structure.
• Show that the coupling constant in these equations is not free but must be an eigenvalue, hence fixed numerically.

Experimental results

Research questions

• RQ1 How does the normalization of scattering states depend on phase shifts beyond the standard delta-function term?
• RQ2 Can Levinson's theorem be proven in a general way without assuming specific potential forms?
• RQ3 What is the behavior of phase shifts in the high-energy limit for singular potentials?
• RQ4 What constraints do the derived normalization and phase shift relations impose on solutions of generalized QED equations?
• RQ5 Is the coupling constant in these generalized QED equations a free parameter or an eigenvalue?

Key findings

• The normalization integral for scattering states contains an additional term proportional to the derivative of the phase shift, which is of measure zero in momentum integrals but carries full phase shift information.
• Levinson's theorem is proven in a general form using only the completeness of states, valid for both Schrödinger and Dirac equations.
• A new result is derived for Dirac particles: Levinson's theorem holds separately for positive and negative energy states.
• High-energy phase shift asymptotics are generalized to include singular potentials, extending known results.
• In the generalized QED framework, any solution with a finite number of bound states must have zero total charge.
• The coupling constant in these equations is not a free parameter but must be an eigenvalue, implying a definite numerical value.