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Theoretical Calculations of the Dynamic Polarizabilites and the Tune-Out Wave Length for H Atom
DOI: 10.12677/MP.2022.123008, PDF, HTML, XML, 下载: 145  浏览: 243  科研立项经费支持

Abstract: The theoretical method for calculating the dynamic polarizability of monovalent atoms by the variationally stable procedure is given. On this basis, the calculation program is independently developed by us. As an application, the dynamic polarizabilities at different frequencies are calculated with hydrogen atoms as an example, by this program. The calculation results are compared with the Green function data in the literature, with good consistency. The convergence of the variationally stable procedure is analyzed, and the corresponding tune-out wavelength is further determined by using the polarizability calculated in the present work.

1. 引言

2. 理论方法

2.1. 哈密顿和波函数

$\stackrel{^}{H}=-\frac{{\nabla }^{2}}{2}-\frac{Z}{r}=-\frac{1}{2r}\frac{{d}^{2}}{d{r}^{2}}r-\frac{Z}{r}+\frac{{l}^{2}}{2{r}^{2}}$ (1)

${E}_{n}=-\frac{{Z}^{2}}{2{n}^{2}}$ (2)

${\psi }_{nlm}\left(r,\theta ,\phi \right)={R}_{nl}\left(r\right){Y}_{lm}\left(\theta ,\phi \right)$ (3)

${R}_{nl}\left(r\right)={N}_{nl}{\left(\frac{2Zr}{n}\right)}^{l}{e}^{\frac{Zr}{n}}\underset{k=0}{\overset{n-l-1}{\sum }}\frac{\left(n+l\right)!}{\left(n-l-1-k\right)!\left(2l+1+k\right)!k!}{\left(-\frac{2Zr}{n}\right)}^{k}$ (4)

${N}_{nl}={\left(\frac{2Z}{n}\right)}^{3/2}\sqrt{\frac{\left(n-l-1\right)!}{\left(n+l\right)!2n}}$ (5)

2.2. 动态极化率

${\alpha }_{L}\left(\omega \right)=-\left[{T}_{i\to i}^{\left(N=2\right)}\left(\omega \right)＋{T}_{i\to i}^{\left(N=2\right)}\left(-\omega \right)\right]$ (6)

${T}_{i\to i}^{\left(2\right)}\left(\omega \right)=〈i|{\stackrel{^}{d}}_{L}\frac{1}{{E}_{i}+\omega -\stackrel{^}{H}}{\stackrel{^}{d}}_{L}|i〉$ (7)

${\stackrel{^}{d}}_{L}=\sqrt{\frac{4\pi }{2L+1}}{r}^{L}{Y}_{L0}\left(\theta ,\varphi \right)$ (8)

$\omega \to 0$ 时，即得到静态极化率。

$|\lambda 〉=\frac{1}{{E}_{i}+\omega -\stackrel{^}{H}}{\stackrel{^}{d}}_{L}|i〉$ (9)

${T}_{i\to i}^{\left(2\right)}\left(\omega \right)=2〈i|{\stackrel{^}{d}}_{L}|\lambda 〉-〈\lambda |{E}_{i}+\omega -\stackrel{^}{H}|\lambda 〉$ (10)

$|\lambda 〉={Y}_{l\text{'}m\text{'}}\left(\theta ,\phi \right)\underset{\mu =1}{\overset{M}{\sum }}{a}_{\mu }{\varphi }_{\mu }\left(r\right)$ (11)

${\varphi }_{\mu }\left(r\right)={N}_{\mu }{r}^{l\text{'}+\mu }{\text{e}}^{-r}$ (12)

${N}_{\mu }=\sqrt{\frac{{2}^{2\mu +2{l}^{\prime }+3}}{\left(2\mu +2{l}^{\prime }+2\right)!}}$ (13)

(11)式中 ${a}_{\mu }$ 是展开系数，M是基组的大小。将初态和中间态代入到(10)式，完成的积分，可得

${T}_{i\to i}^{\left(2\right)}\left(\omega \right)=2\underset{\mu }{\sum }{a}_{\mu }{N}_{\mu }{D}_{\mu }-\underset{\mu }{\sum }\underset{{\mu }^{\prime }}{\sum }{a}_{\mu }{a}_{{\mu }^{\prime }}{N}_{\mu }{N}_{{\mu }^{\prime }}\left({A}_{\mu {\mu }^{\prime }}+{B}_{\mu {\mu }^{\prime }}+{C}_{\mu {\mu }^{\prime }}\right)$ (14)

$\underset{\mu =1}{\overset{M}{\sum }}{a}_{\mu }{N}_{\mu }{N}_{\mu }\left({A}_{\mu {\mu }^{\prime }}+{B}_{\mu {\mu }^{\prime }}+{C}_{\mu {\mu }^{\prime }}\right)={N}_{{\mu }^{\prime }}{D}_{{\mu }^{\prime }}$ (15)

${D}_{\mu }={\epsilon }^{l}\left(n+l\right)!I\left(lmL0{l}^{\prime }{m}^{\prime }\right)\underset{k=0}{\overset{n-l-1}{\sum }}\frac{{\left(-\epsilon \right)}^{k}\left(k+\mu +\beta \right)!}{\left(n-l-1-k\right)!\left(2l+1+k\right)!k!{\left(1+\epsilon \right)}^{k+\mu +\beta +1}}$ (16)

$I\left(lmL0{l}^{\prime }{m}^{\prime }\right)={\left(-1\right)}^{m}\sqrt{\left(2l+1\right)\left(2{l}^{\prime }+1\right)}\left(\begin{array}{ccc}l& L& {l}^{\prime }\\ 0& 0& 0\end{array}\right)\left(\begin{array}{ccc}l& L& {l}^{\prime }\\ -m& 0& {m}^{\prime }\end{array}\right)$ (17)

$\begin{array}{l}{A}_{\mu {\mu }^{\prime }}=\left({E}_{i}+\omega +\frac{1}{2}\right)\frac{\stackrel{¯}{\mu }！}{{2}^{\stackrel{¯}{\mu }+1}}\\ {B}_{\mu {\mu }^{\prime }}=\left(\mu +l+1\right)\frac{\left(\stackrel{¯}{\mu }-1\right)！}{{2}^{\stackrel{¯}{\mu }}}\\ {C}_{\mu {\mu }^{\prime }}=\frac{\left(\mu -1\right)\left(\mu +2l\right)}{2}\frac{\left(\stackrel{¯}{\mu }-2\right)!}{{2}^{\stackrel{¯}{\mu }-1}}\end{array}$ (18)

3. 计算与讨论

Table 1. Theoretical calculation of the dynamic polarizabilities of H atom

Figure 1. Dynamic polarizabilities and tune-out wavelength of H atom in ground state

4. 总结

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