Constructing the Hamiltonian for a free 1D KFGM particle in an interval

TL;DR

The paper constructs a pseudo self-adjoint FV-free Hamiltonian for a truly free 1D Klein-Fock-Gordon-Majorana particle confined to an interval with penetrable walls. Starting from the FV momentum operator $\hat{p}=-i\hbar\,\hat{1}_2\,\partial_x$ on a $\hat{\tau}_3$-space under a Majorana constraint, the authors identify a general three-parameter BC family for the momentum domain via $[\phi_1(b,t)\phi_2(a,t)]=\hat{N}[\phi_2(b,t)\phi_1(a,t)]$ with $\hat{N}$ unitary (and extend to a four-parameter set before Majorana reductions). Requiring energy-current continuity $j_{\mathrm{en}}(b,t)=j_{\mathrm{en}}(a,t)$ and exploiting parity invariance reduce the FV-Hamiltonian domain to two quantum BCs: periodic and antiperiodic, which in the one-component KFG description correspond to $\phi(b,t)=\pm\phi(a,t)$ and $\phi'(b,t)=\pm\phi'(a,t)$; the Majorana constraint further implies $\phi$ is real or purely imaginary. The results connect two-component FV and one-component KFG formulations, clarify the boundary-value problem for strictly neutral relativistic particles in finite intervals, and extend to penetrable models on the real line.

Abstract

We analyze the problem of a free 1D Klein-Fock-Gordon-Majorana (KFGM) particle in an interval. By free, we mean that there is no potential within the interval and that its walls are penetrable; hence, the pertinent energy current density does not vanish at the walls. Certainly, quantization in an interval is not trivial because certain restrictions imposed by the domains of the operators involved arise. Here, our objective is to obtain the Hamiltonian for these particles. In practice, the Feshbach-Villars (FV)--free Hamiltonian is the proper operator for characterizing them and is a function of the momentum operator. Additionally, a Majorana condition must also be imposed on the wavefunctions on which these two operators can act. Thus, we start by calculating the pseudo self-adjoint momentum operator. A three-parameter set of boundary conditions (BCs) constitutes its domain. Up to this point, the domain of the Hamiltonian is induced by the domain of the momentum operator; however, we ensure that only the BCs for which the energy current density has the same value at each end of the interval are in its domain. All these BCs essentially belong to a one-parameter set of BCs. Moreover, because the FV equation is invariant under the operation of parity, the parity-transformed wavefunction is also a solution of this equation, which further restricts the domain of the free FV Hamiltonian. Finally, knowing the most general three-parameter set of BCs for the pseudo self-adjoint FV Hamiltonian for a 1D KFGM particle in an interval, we find that only two BCs can remain within the domain of the FV--free Hamiltonian: the periodic BC and the antiperiodic BC. These BCs are satisfied by both the two-component FV wavefunction, with these components being related, and the one-component KFG wavefunction, which can be real or imaginary.