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arXiv 1604.02478quant-phmath-phmath.MP
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相对论类氢原子束缚态到二分量态的变换

Transformation of bound states of relativistic hydrogen-l ike atom into two-component form

Tomasz M. Rusin

备注：
13 pages, 6 figures
发表信息：
Phys. Rev. A 94, 012117 (2016)
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摘要：

通过将相对论类氢原子的~$1S_{1/2}$、~$2P_{1/2}$和~$2P_{3/2}$态的变换函数表示为Dirac哈密顿量本征态的线性组合，精确地实现了它们的单步Eriksen变换.变换函数是四分量旋量，两个下分量为零，数值计算得到的变换函数具有与初态相同的对称性。对于所有核电荷~$Z \in [1\ldots 92]$，初态对传递函数的贡献超过总几率密度的86%.对TF的下一个大贡献来自具有接近~$-m_0c^2-E_b$的负能量的连续态，其中~$E_b$是初始态的结合能。其它态对传递函数的贡献小于总概率密度的~$0.1\%$。传递函数的其他分量几乎为零，这证实了埃里克森变换的有效性和数值计算的准确性。~$1S_{1/2}$和~$2P_{1/2}$态的传递函数分别接近于非相对论类氢原子的~$1s$和~$2p$态，但~$2P_{3/2}$态的传递函数与~$2p$态有质的不同.利用线性化的Eriksen变换计算的函数，等效于二阶Foldy-Wouthuysen变换，并与Eriksen变换得到的相应函数进行了比较。两个结果之间的一个非常好的协议。

英文摘要：

A single-step Eriksen transformation of~$1S_{1/2}$,~$2P_{1/2}$ and~$2P_{3/2}$ states of the relativistic hydrogen-like atom is performed exactly by expressing each transformed function (TF) as a linear combination of eigenstates of the Dirac Hamiltonian. The transformed functions, which are four-component spinors with vanishing two lower components, are calculated numerically and have the same symmetries as the initial states. For all nuclear charges~$Z \in [1\ldots 92]$ a contribution of the initial state to TFs exceeds 86\% of the total probability density. Next large contribution to TFs comes from continuum states with negative energies close to~$-m_0c^2-E_b$, where~$E_b$ is the binding energy of initial state. Contribution of other states to TFs is less than~$0.1\%$ of the total probability density. Other components of TFs are nearly zero which confirms both validity of the Eriksen transformation and accuracy of the numerical calculations. The TFs of~$1S_{1/2}$ and~$2P_{1/2}$ states are close to~$1s$ and~$2p$ states of the nonrelativistic hydrogen-like atom, respectively, but the TF of~$2P_{3/2}$ state differs qualitatively from the~$2p$ state. Functions calculated with use of a linearized Eriksen transformation, being equivalent to the second order Foldy-Wouthuysen transformation, are compared with corresponding functions obtained by Eriksen transformation. A very good agreement between both results is obtained.

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