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\begin{document}

\title{The relativistic feature of Hydrogen-like atoms in the Heisenberg
picture}



\author[1]{J Jahanpanah}%
\author[1]{A Vahedi}%
\author[1]{H . Khosrojerdi}%
\affil[1]{Kharazmi University}%


\vspace{-1em}



  
  \date{May 17, 2024}


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\selectlanguage{english}
\begin{abstract}
The relativistic properties of Hydrogen-like atoms (HLAs) are here
investigated in the Heisenberg picture for the first time. The
relativistic vibrational Hamiltonian (RVH) is first defined as a power
series of harmonic oscillator Hamiltonian by using the relativistic
energy eigenvalue . By applying the first-order RVH (proportional to )
to the Heisenberg equation, a pair of coupled equations is turned out
for the relativistic motion of the electron's position and linear
momentum. A simple comparison of the first-order relativistic and
nonrelativistic equations reveals this reality that the natural
(fundamental) frequency of HLA (like entropy) is slowly raised by
increasing the atomic number from . The second-order RVH (proportional
to ) has then been implemented to determine an exact expression for the
electron relativistic frequency in the different atomic energy levels.
In general, the physical role of RVH is fundamental because it not only
specifies the temporal relativistic variations of position, velocity,
and linear momentum of the oscillating electron, but also identifies the
corresponding relativistic potential, kinetic, and mechanical energies.
The results will finally be testified by demonstrating energy
conservation.%
\end{abstract}%



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\textbf{Hosted file}

\verb`Main document (clean version in word format).docx`  available at \url{https://authorea.com/users/436994/articles/543105-the-relativistic-feature-of-hydrogen-like-atoms-in-the-heisenberg-picture}

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