Quantum Mechanics in a Relativistic Compton-Wavelength Representation

The Compton wavelength is conventionally introduced as the rest-mass scale $\lc=h/(mc)$, whereas moving particles are normally assigned the de Broglie wavelength $\ldb=h/p$. Haug recently introduced and subsequently refined a frame-dependent relativistic Compton wavelength, $\lcr=h/(\gamma mc)$, with reduced form $\lbarcr=\hbar/(\gamma mc)$. For a massive free particle with $p=\gamma mv$, the exact identity $\ldb=(c/v)\lcr=\lcr/\beta$ follows immediately. This article examines the consequences of taking $\lcr$, rather than $\ldb$, as the primary wavelength parameter for single-particle relativistic wave mechanics. The distinction is conceptually sharp at rest: $\lcr$ is finite and equals $\lc$ at $v=0$, while $h/p$ is not defined by ordinary division at $p=0$, although its limiting spatial wavelength is infinite and its wave number $k=p/\hbar$ is well defined and vanishes. We show that free-particle phase relations, Schr\"odinger eigenstates, box quantisation, diffraction conditions, uncertainty relations, and the plane-wave sectors of the Klein--Gordon and Dirac equations can all be written using $\lbarcr$ and $\beta$. In this parameterisation, $E=\hbar c/\lbarcr$ and $p=\hbar\beta/\lbarcr$, so the de Broglie wavelength is a derived spatial period generated by the velocity-dependent spatial gradient of a Compton-scale spacetime phase. The reformulation is algebraically equivalent to standard momentum-space quantum mechanics, but it exposes an unequal domain and an overlooked organisation of four-momentum. The quantity $E/c=\gamma mc$, already present as the temporal component of four-momentum, is precisely Haug's Compton momentum $p_{\mathrm C}=h/\lambda_{\mathrm C,r}$. Every finite de Broglie wavelength of a massive on-shell particle can be represented by the relativistic Compton wavelength and velocity through $\ldb=\lcr/\beta$.