Linear Relativistic Hamiltonians

J. Jahanpanah, Ali Vahedi, H . Khosrojerdi

The relativistic behavior of Hydrogen-like atoms (HLAs) is investigated in 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 Heisenberg equation, a pair of coupled equations is turned out for the motion of electron position and its relativistic linear momentum. A simple comparison of the first-order relativistic and nonrelativistic equations reveals this reality that the natural (fundamental) frequency of electron oscillation (like entropy) is slowly raised by increasing the atomic number. The second-order RVH (proportional to ) have 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 oscillating electron, but also identifies the corresponding relativistic potential, kinetic, and mechanical energies. The results will finally be testified by demonstrating the energy conservation.

Fritz Rohrlich

J. Jahanpanah, Ali Vahedi, H . Khosrojerdi

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.

Richard F. Arens

J. Jahanpanah, Ali Vahedi, H . Khosrojerdi

Abstract The relativistic effects have been widely reported to affect the chemical and physical properties of the heavy elements such as lanthanides (), actinides (), and transactinides (). This effect is definitely weakened by reducing the atomic number in lighter elements such as hydrogen‐like atoms (HLAs). The aim of present paper is to investigate the relativistic effects of electron motion in Bohr orbits on the chemical and physical properties of HLAs. The theoretical model is based on the Heisenberg (rather than Schrodinger) picture where the relativistic vibrational Hamiltonian (RVH) is expanded as a power series of harmonic oscillator Hamiltonian for the first time. By applying the first‐order RVH (correct to ) to the Heisenberg equation, a pair of coupled equations is obtained for the relativistic position and linear momentum of electron. A simple comparison of the first‐order relativistic and nonrelativistic equations reveals that the relativistic natural frequency of an HLA (like entropy) is slowly raised by increasing beyond . In general, RVH plays a fundamental role because it specifies the temporal relativistic variations of position, velocity, and linear momentum of the oscillating electron. The results are finally verified by demonstrating energy conservation.