Temporally stable coherent states in energy degenerate systems: The hydrogen atom

论电场存在下圆形盒中的二维氢原子

On the two-dimensional hydrogen atom in a circular box in the presence of an electric field

本文研究了电场中圆形盒内的二维氢原子，采用Rayleigh-Ritz方法和多项式基组计算能谱，探讨了大小盒半径极限与解的对称性，并揭示了电场引起的偶然简并及简并能级分裂现象。

Hydrogen Atom: Its Spectrum and Degeneracy Importance of the Laplace-Runge-Lenz Vector

Consider the problem: why does the bound state spectrum $E(n) < 0$, of hydrogen atom Hamiltonian $H$ have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, $\vec A$, of $H$, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of $H$ and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint o…

Population control of 2s-2p transitions in hydrogen

We consider the time evolution of the occupation probabilities for the 2s-2p transition in a hydrogen atom interacting with an external field, V(t). A two-state model and a dipole approximation are used. In the case of degenerate energy levels an analytical solution of the time-dependent Shroedinger equation for the probability amplitudes exists. The form of the solution allows one to choose the ratio of the field amplitude to its frequency that leads to temporal trapping of electrons in specific states. The analytic solution is valid when the separation of the energy levels is small compared …

The bound-state solutions of the one-dimensional hydrogen atom

The one-dimensional hydrogen atom is an intriguing quantum mechanics problem that exhibits several properties which have been continually debated. In particular, there has been variance as to whether or not even-parity solutions exist, and specifically whether or not the ground state is an even-parity state with infinite negative energy. We study a "regularized" version of this system, where the potential is a constant in the vicinity of the origin, and we discuss the even- and odd-parity solutions for this regularized one-dimensional hydrogen atom. We show how the even-parity states, with the…

Obtaining Hydrogen energy wave functions using the Runge-Lenz vector

The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate $|n,l,m\rangle$ with other degenerate states $|n, l\pm 1,m'\rangle$. Recursive relations can be obtained and the wave functions of the whole spectrum can be obtaine…

均匀静电场诱导的氢原子的重新探讨

A revisit on the hydrogen atom induced by a uniform static electric field

本文重新探讨了均匀静电场诱导的氢原子斯塔克效应，推导出连带拉盖尔多项式积分的一般公式，并结合不含时非简并微扰理论获得二次斯塔克效应，利用Siegert态方法进行数值基准比较，探讨了该公式的适用极限与精度。

The Stark Effect with Minimum Length

We will study the splitting in the energy spectrum of the hydrogen atom subjected to a uniform electric field (Stark effect) with the Heisenberg algebra deformed leading to the minimum length. We will use the perturbation theory for cases not degenerate ($n=1$) and degenerate ($n=2$), along with known results of corrections in these levels caused by the minimum length applied purely to the hydrogen atom, so that we may find and estimate the corrections of minimum length applied to the Stark effect.

Adjacency Graphs and Long-Range Interactions of Atoms in Quasi-Degenerate States: Applied Graph Theory

We analyze, in general terms, the evolution of energy levels in quantum mechanics, as a function of a coupling parameter, and demonstrate the possibility of level crossings in systems described by irreducible matrices. In long-range interactions, the coupling parameter is the interatomic distance. We demonstrate the utility of adjacency matrices and adjacency graphs in the analysis of "hidden" symmetries of a problem; these allow us to break reducible matrices into irreducible subcomponents. A possible breakdown of the no-crossing theorem for higher-dimensional irreducible matrices is indicate…

Hydrogen energy-levels shifts induced by the atom motion. Crossover from the Lamb shifts to the motion-induced shifts

When the hydrogen atom moves, the proton current generates a magnetic field which interacts with the hydrogen electron. A simple analyze shows that this interaction between the hydrogen momentum and the electron is of order of $α^3\frac{v}{c}m$, where $α$ is the fine structure constant, $v$ is the atom velocity, $c$ is the speed of light, and $m$ is the electron mass. Using the Bethe-Salpeter equation, the two velocity-dependent operators of this order are derived for the hydrogen velocity $\frac{v}{c}<<α$. As well known, the degeneracy of the energy levels with the same principal quantum numb…

Raman and non Raman EIT resonances in a degenerate four level system

We study coherence effects in an atom having three metastable levels and single life-time broadened upper state. Two laser fields couple ground states to an excited one and an radio-frequency (rf) field operates between ground state levels. We consider the case when the common level for the rf-field and one of the laser field (to which we refer as the probing field) transitions is twice degenerate. We predict a novel type of dark states (we call them non Raman dark states), which in contrary to Raman resonances, do not require the fulfillment of the two-photon resonance condition. We realize t…

狄拉克氢原子的非简并兰姆位移解

Nondegenerate, Lamb-shift Solution of the Dirac Hydrogen Atom

本文提出狄拉克方程的新解，利用兰姆位移消除氢原子所有简并性，以拉盖尔多项式给出径向本征函数并验证其可约化为薛定谔波函数。

The two-dimensional hydrogen atom revisited

The bound state energy eigenvalues for the two-dimensional Kepler problem are found to be degenerate. This "accidental" degeneracy is due to the existence of a two-dimensional analogue of the quantum-mechanical Runge-Lenz vector. Reformulating the problem in momentum space leads to an integral form of the Schroedinger equation. This equation is solved by projecting the two-dimensional momentum space onto the surface of a three-dimensional sphere. The eigenfunctions are then expanded in terms of spherical harmonics, and this leads to an integral relation in terms of special functions which has …

RGUP代数中由量子引力导致的斯塔克能量位移

Stark Energy Shifts due to Quantum Gravity in RGUP Algebra

本文将相对论性广义不确定原理（RGUP）引入氢原子斯塔克效应研究，推导能谱修正，得到含β正比项的能量位移，验证其极限可退化为经典结果，构建了相对论量子系统最小长度效应的通用分析框架。