Influence of hydrogen-like nucleus mass on electronic state

Abstract

In an isolated two-body composite system, the discussion of how the change of one body affects the state of the other will help to know the relation of a single particle's mixed and pure states. Given 5 one-dimensional hydrogen-like atoms models, each Coulomb interaction potential keeps invariant, while the masses of the nuclei are different. These two-body composite systems stay in their respective entangled state, each electron stays in a mixed state. If we suppose a one-dimensional hydrogen atom model having infinite nuclear mass, the electron stays in a pure state. We select position representation. The wave function of the ground state of the atom has the form of the square root of a <i>δ</i> function. To avoid calculation trouble of <i>δ</i> function, we select the first excited state and the superposed state of the first and the second excited states. We compare the two pure states, the first excited state and the superposed state, with those corresponding mixed states by fidelity and <i>l</i>₁ norm coherence, and calculate the purities of the mixed states. The summations become integrations due to the position variable having a continuous eigenvalue spectrum. We investigate the changes in these quantities with the increase of the nuclear mass. The results show that the purities of the mixed states and the fidelities increase with the nuclear mass increasing. However, the coherences of the mixed states decrease with the nuclear mass increasing. This can be explained as that a mixed state with non-zero coherence may approach to a pure eigenstate, while the latter has zero coherence in the eigenspace. These mean that the greater a nuclear mass is, the closer the mixed state approaches to the corresponding pure state. Therefore, the two pure states are the approximations of the corresponding mixed states. The entangled state of the electron and the nucleus is related with the nuclear mass and the Coulomb interaction potential. So, each electron mixed state and its coherence are related with the nucleus and their Coulomb interaction potential. If the nuclear mass is great, the nucleus is approximately motionless or its state is approximately unchanged, and the Coulomb interaction potential approximates to the external Coulomb potential of the electron. The electron approximately stays in a pure state. The state and its coherence are related with the nucleus and the Coulomb interaction. From other point of view, the entangled state of the nucleus and the electron approximates to the product state of the unchanged nucleus state and the electron state. In this case, an electron mixed state approximates to its corresponding pure state, and then these states and their coherences are all related with the nucleus and the Coulomb interaction.