Gauge invariant hydrogen atom Hamiltonian

For quantum mechanics of a charged particle in a classical external electromagnetic field, there is an apparent puzzle that the matrix element of the canonical momentum and Hamiltonian operators is gauge dependent. A resolution to this puzzle is recently provided by us in [2]. Based on the separation of the electromagnetic potential into pure gauge and gauge invariant parts, we have proposed a new set of momentum and Hamiltonian operators which satisfy both the requirement of gauge invariance and the relevant commutation relations. In this paper we report a check for the case of the hydrogen atom problem: Starting from the Hamiltonian of the coupled electron, proton and electromagnetic field, under the infinite proton mass approximation, we derive the gauge invariant hydrogen atom Hamiltonian and verify explicitly that this Hamiltonian is different from the Dirac Hamiltonian, which is the time translation generator of the system. The gauge invariant Hamiltonian is the energy operator, whose eigenvalue is the energy of the hydrogen atom. It is generally time-dependent. In this case, one can solve the energy eigenvalue equation at any specific instant of time. It is shown that the energy eigenvalues are gauge independent, and by suitably choosing the phase factor of the time-dependent eigenfunction, one can ensure that the time-dependent eigenfunction satisfies the Dirac equation.Comment: 7 pages, revtex4, some further discussion on Dirac Hamiltonian and the gauge invariant Hamiltonian is added, one reference removed; new address of some of the authors added, final version to appear in Phys. Rev.

Mapping epigenetic modifications by sequencing technologies

The “epigenetics” concept was first described in 1942. Thus far, chemical modifications on histones, DNA, and RNA have emerged as three important building blocks of epigenetic modifications. Many epigenetic modifications have been intensively studied and found to be involved in most essential biological processes as well as human diseases, including cancer. Precisely and quantitatively mapping over 100 [1], 17 [2], and 160 [3] different known types of epigenetic modifications in histone, DNA, and RNA is the key to understanding the role of epigenetic modifications in gene regulation in diverse biological processes. With the rapid development of sequencing technologies, scientists are able to detect specific epigenetic modifications with various quantitative, high-resolution, whole-genome/transcriptome approaches. Here, we summarize recent advances in epigenetic modification sequencing technologies, focusing on major histone, DNA, and RNA modifications in mammalian cells

Note On Certain Inequalities for Neuman Means

In this paper, we give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$ and $N_{CA}$, and present the best possible upper and lower bounds for theses means in terms of the combinations of harmonic mean $H$, arithmetic mean $A$ and contraharmonic mean $C$.Comment: 9 page