UDC 539.186.22

PHYSICS

M. I. CHIBISOV

DISPERSION FORMULA FOR THE HYDROGEN ATOM IN THE GROUND STATE

(Presented by Academician M. A. Leontovich on 23 II 1970)

1. According to quantum theory, the optical polarizability of the hydrogen atom in the ground state is equal to:

\[ \alpha(\omega)=2\sum_{n=2}^{\infty}\frac{\nu_n |z_{0n}|^2}{\nu_n^2-\omega^2} +2\int_0^\infty \frac{\nu |r_{0\nu}|^2}{\nu^2-\omega^2}\,d\nu, \tag{1} \]

where \(\nu_n\) are the transition frequencies from the ground state to the \(n\)-th excited state of the discrete spectrum; \(\nu\) is the same for a transition into the continuous spectrum of the hydrogen atom; \(z_{0n}\) is the matrix element of the \(z\)-component of the dipole moment. The atomic system of units is used: \(h=m=e^2=1\). With the aid of the apparatus of Green functions and the Laplace transform of the intermediate differential equation, the polarizability (1) can be represented in the form \((^{1,2})\)

\[ \alpha(\omega)=-\frac{2}{3\omega}[f(\omega)-f(-\omega)]; \tag{2} \]

\[ f(\omega)=\frac{96}{p_2^a p_1^{4-a}} \int_{p_1}^{2}\frac{(x-p_1)^{a-1}(x-p_2)^{3-a}}{x^5}\,dx; \tag{2′} \]

\[ p_1=1+\sqrt{1+2\omega};\qquad p_2=2-p_1;\qquad a=2-\frac{1}{p_1-1};\qquad -\infty<a\leq 2. \tag{2″} \]

As is seen from the definition of the parameters \(P_1\), \(P_2\), and \(a\), the integral (2) does not exist for \(\omega\leq 0\) or \(\omega\leq -\nu_1=-3/8\). To extend the domain of definition of (2′) to the region \(-\nu_n<\omega<\infty\), an \(n\)-fold integration of expression (2′) by parts is required \((^{1})\).

Analysis of expression (2′) in the plane of the complex \(x\) makes it possible to obtain another integral representation for \(\alpha(\omega)\), free of the indicated shortcoming, and in which the pole part is separated out as a simple analytic term.

2. The points \(p_1\) and \(p_2\) in the plane of the complex \(x\) are branch points of the integrand in formula (2′). The cut of the \(x\)-plane may be drawn along a segment of a curve connecting these points. For frequencies \(\omega\) for which \(a<-1\) and is not equal to an integer, \(p_1\) is an essentially singular point. At the origin \(x=0\) there is a fifth-order pole whose residue is

\[ \operatorname{Res} \left. \frac{(x-p_1)^{a-1}(x-p_2)^{3-a}}{x^5} \right|_{x=0} \equiv Q(\omega)= \frac{-2}{3p_1^4} \left(\frac{3-a}{1-a}\right)^a . \tag{3} \]

The contour of integration in (2′) must be such that, in the limit \(\omega\to 0\), expression (2′) gives the known value of the static polarizability \(\alpha(0)=4.5\). For this it is sufficient that, as \(\omega\to 0\) (\(p_1\to 2\)), the contour, if it became closed, would not contain the point \(x=0\) inside it. Otherwise \(f(\omega)\) would be proportional to the residue (3), which as \(\omega\to 0\) diverges as \(\omega^{-1}\). For \(a\geq 0\), as the contour of integra-

can be the segment of the real axis \((p_1,2)\), which also gives \(a(0)=4.5\) \({}^{(1)}\).

Restricting ourselves for the time being to frequencies for which \(a \geq 0\), we divide the integration contour in (2) into two parts. The first is the lower edge of the cut from \(p_1\) to \(p_2\), and the second is a certain curve satisfying the requirements given above and joining \(p_2\) and 2. We denote the integral over the first contour by \(J_3\), and over the second by \(J_2\). The first integral can be evaluated exactly by expressing it in terms of the residue \(Q(\omega)\), while the second is defined for any \(\omega\). Since on the upper edge of the cut the values of the integrand in (2) differ by the factor \(e^{2\pi i a}\) from the values of the function on the lower edge, we have

\[ (1-e^{2\pi i a})J_3=\int_C (x-p_1)^{a-1}(x-p_2)^{3-a}\frac{dx}{x^5}, \tag{4} \]

where the contour \(C\) goes around the cut clockwise. This contour, together with an infinitely large circle (the integral over which tends to zero), bounds a region inside which the point \(x=0\) lies, and therefore the integral (4) is equal to \(2\pi i Q(\omega)\). This gives

\[ f(\omega)=-\frac{96}{p_2^a p_1^{4-a}}\bigl[(\pi i-\pi\operatorname{ctg}\pi a)Q(\omega)+J_2(\omega)\bigr], \tag{5} \]

\[ J_2(\omega)=\int_{p_2}^{2}(x-p_1)^{a-1}(x-p_2)^{3-a}\frac{dx}{x^5}, \tag{6} \]

where the residue \(Q(\omega)\) is defined by formula (3).

We shall show that for real \(\omega\) formula (5) defines a real function \(f(\omega)\). To this end, consider the following contour of integration in (6): departure from \(p_2\) parallel to the imaginary axis to infinity and return from infinity along the real axis to the point \(x=2\). The second part of the contour gives a purely real contribution. When integrating along the first branch of the contour, note that the integrand in (6) satisfies the relation \(\varphi(x^*)=\varphi^*(x)\), which means that the required imaginary part of the integral (6) is equal to one half of the imaginary part of the same integral taken along the entire straight line \((p_2-i\infty;\,p_2+i\infty)\). The latter integral is equal to \(-2\pi i Q(\omega)\) (we are considering the frequencies \(-\nu_1<\omega<0\), for which \(p_2>0\) and \(a>0\)). Thus, instead of (5), for real \(\omega\) one may write

\[ f(\omega)=-\frac{96}{p_2^a p_1^{4-a}}\bigl[-\pi\operatorname{ctg}(\pi a)\cdot Q(\omega)+\operatorname{Re}J_2(\omega)\bigr]. \tag{7} \]

For integral \(a\), \(\operatorname{ctg}\pi a\) becomes infinite, and \(f(\omega)\) has poles. Indeed, \(a=2\) for \(\omega=+\infty\), \(a=1\) for \(\omega=0\), and \(a=-n\) for

\[ \omega=-\frac{1}{2}(1-1/n^2)=-\nu_n. \tag{8} \]

As is seen from formula (3), \(Q(\omega)\) does not vanish at these points. Consequently, the polarizability \(\alpha(\omega)\) has poles at \(\omega=\nu_n\), as it should. There are no other poles of \(\alpha(\omega)\).

Expressions (7) and (6) exist for all \(\omega\), which makes it possible to define \(f(\omega)\) in the entire \(\omega\)-plane as the analytic continuation of expression (7), since (7) has poles only at isolated points.

For positive values of \(\omega\), it is convenient to retain representation (2′) for \(f(\omega)\), and to use (7) for \(f(-\omega)\). Thus we obtain the polarizability of the hydrogen atom in the ground state:

\[ \alpha(\omega)=\frac{64}{\omega}\left\{\frac{1}{q_2^b q_1^{4-b}}\bigl[\pi Q(\omega)\operatorname{ctg}\pi a-\operatorname{Re}J_2(-\omega)\bigr]-\frac{J_1(\omega)}{p_2 p_1^{4-a}}\right\}, \tag{9} \]

\[ J_1=\int_{p_1}^{2}(x-p_1)^{a-1}(x-p_2)^{3-a}\frac{dx}{x^5}, \tag{10} \]

\[ J_2=\int_{p_2}^{2}(x-q_1)^{b-1}(x-q_2)^{3-b}\frac{dx}{x^5},\qquad b=a(-\omega),\qquad q_{1,2}=p_{1,2}(-\omega). \tag{11} \]

To study \(J_2\), we consider the contour defined above, consisting of two segments: the first \(X\) from \(p_2\) to \(p_2-i\infty\), and the second \(2<x<+\infty\). For the first segment we make the change of variable \(x=p_2-iy\), and for the second \(x=2+y\), and obtain

\[ \operatorname{Re}J_2=j_1+j_2, \]

\[ j_1=-\operatorname{Re}\int_0^\infty \frac{[i(p_2-p_1)+y]^{a-1}y^{3-a}}{(y+ip_2)^5}\,dy, \tag{12} \]

\[ j_2=-\int_0^\infty \frac{(y+p_2)^{a-1}(y+p_1)^{3-a}}{(y+2)^5}\,dy. \tag{13} \]

For \(\omega\to0\), from formulas (6) and (3) we obtain

\[ \operatorname{Re}J_2\to \frac{1}{12\omega^2},\qquad Q(\omega)\to-\frac{1}{12\omega^2}. \tag{14} \]

For \(\omega\to-\frac{1}{2}\), \(a\to-\infty\), and the integrals (12), (13) are transformed to the form

\[ \operatorname{Re}J_2\left(-\frac{1}{2}\right) =-\operatorname{Re}\int_0^\infty\frac{e^{2it}t\,dt}{(1+it)^5} -\int_0^1\frac{e^{2t}t\,dt}{(t+1)^5}\simeq -0.190, \]

\[ Q\left(-\frac{1}{2}\right)=\frac{2}{3}e^{-2}=0.0905. \tag{15} \]

For intermediate values of \(\omega\), the integrals \(J_1\) and \(J_2\) were obtained by numerical integration and are given in Table 1.

Since for \(\omega\to\frac{1}{2}\) the integrals \(J_1\) and \(J_2\) tend to constant values, near the ionization threshold (practically for frequencies \(\omega\) greater than the second eigenfrequency) the polarizability is close to the expression

\[ \alpha(\omega)=-4.96\,\operatorname{ctg}\left(\pi/\sqrt{1-2\omega}\right)+1.80, \tag{16} \]

which differs, in the last term, from the analogous expression obtained in work \({}^{(3)}\) using the asymptotic representation of the coordinate Green’s function near the ionization threshold.

Table 1

We now obtain a series for \(\alpha(\omega)\) in positive powers of \(\omega\) and determine the coefficients of this expansion. The function \(f(\omega)\) appearing in formulas (2) satisfies the differential equation \((^{1,2})\)

\[ [p(p-2)-2\omega] f_p'(p,\omega)+(4p-6)f=\frac{96}{p^5}. \tag{17} \]

Expanding \(f(p,\omega)\) in powers of \(\omega\) and substituting this expansion into (17), for the coefficients of the expansion we obtain:

\[ p(p-2)f_0' + (4p-6)f_0=\frac{96}{p^5}, \qquad f(p,\omega)=\sum_{n=0}^{\infty} f_n(p)\omega^n, \tag{18} \]

\[ \cdots\cdots\cdots\cdots\cdots \]

\[ p(p-2)f_n' + (4p-6)f_n=2f_{n-1}, \qquad f_0=12/p^4+24/p^5. \tag{18'} \]

From (18′) we obtain:

\[ f_n(p)=\frac{2}{p^3(p-2)}\int_{2}^{p} f_{n-1}'(z)z^2\,dz,\quad \to f_n(p)\bigg|_{p=2}=f_{n-1}'(2)= \frac{d^n}{dp^n}f_0(p)\bigg|_{p=2}. \tag{19} \]

Substituting (19) into (18), after differentiating \(f_0(p)\) \(n\) times, we obtain an expansion (in the form of an asymptotic series) for \(\alpha(\omega)\) and for the sums \(S(k)\) important in spectroscopy:

\[ \alpha(\omega)=\frac{1}{48}\sum_{n=0}^{\infty} \frac{(2n+9)(2n+4)!}{2^{2n}}\omega^{2n} \simeq \frac{9}{2}+\frac{165}{4}\omega^2+\frac{1365}{2}\omega^4+\cdots; \tag{20} \]

\[ S(k)\equiv 2\sum_n \frac{|z_{0n}|^2}{\nu_{0n}^k} =\frac{(k+8)(k+3)!}{3\cdot 2^{k+3}}, \qquad k=0,1,2,\ldots \tag{21} \]

The last equality (21) follows from comparing (20) with expansion (1) in powers of \(\omega^2\).

Received
5 I 1970

CITED LITERATURE

\({}^{1}\) M. N. Adamov, DAN, 133, 315 (1960).
\({}^{2}\) Ch. Schwartz, J. J. Tiemann, Ann. of Phys., 6, 178 (1959).
\({}^{3}\) S. V. Khristenko, S. I. Vetchinkin, Optics and Spectroscopy, 26, 310 (1969).