Why Do We Insist That the Following Expression is Satisfied for All Continuous Wave Functions

Asymptotic Form

The asymptotic form of the associated Legendre polynomials for the two exponentials in the simple expression of Eq. 75 then yields the asymptotic total wave amplitude solution Ψt, which readily allows comparison with the Schrödinger wave equation asymptotic solution Ψt.

From: Advances in Imaging and Electron Physics , 2012

Molecules

Yehuda B. Band , Yshai Avishai , in Quantum Mechanics with Applications to Nanotechnology and Information Science, 2013

11.2.1 Interatomic Potentials at Large Internuclear Distances

The asymptotic form of the potential between two atoms that are very far apart can be determined using perturbation theory, regarding the two isolated atoms as the unperturbed system and the potential energy of their electrical interaction as the perturbation. The potential of two systems of charges at large relative distance R can be expanded in powers of R. The multipole expansion (3.198) of the potential energy due to the charge distribution of a system contains terms from the system charge, dipole moment, quadrupole moment, etc. For a neutral atom, the total charge is zero, so the expansion then begins with the dipole term, etc.

For two neutral atoms, the potential energy in Eq. (3.201) can be expanded in multipoles to obtain (3.201). The lowest-order term for two neutral atoms is dipole–dipole (R ⁻³), followed by dipole–quadrupole (R ⁻⁴), quadrupole–quadrupole and dipole–octupole terms ( $R^{-5}$ ), and so on. The expectation value of the potential can now be used to get an expression for the first-order perturbation correction to the energy,

(11.1) $\Delta E^{\left(1\right)}\left(R\right)=\mathfrak{⟨}{\psi }_a{\psi }_b\mathfrak{|}U\left(R\right)\mathfrak{|}{\psi }_a{\psi }_b\mathfrak{⟩},$

where U (R ) is given by (3.202). For the two atoms in S states, no interaction between the atoms is possible in first-order perturbation theory. Therefore, the interaction energy of the atoms is determined by second-order perturbation theory. Let us restrict ourselves to the dipole–dipole interaction term in Eq. (3.202), because it decreases least rapidly as R increases, i.e., as R ⁻⁶. Because the nondiagonal matrix elements of the dipole moment, e.g., $\mathfrak{⟨}{\psi }_{npm}\mathfrak{|}p\mathfrak{|}{\psi }_{ns0}\mathfrak{⟩}$ , are in general different from zero, second-order perturbation theory yields a nonvanishing result which, being quadratic in U, is proportional to R ⁻⁶. The second-order perturbation energy for the lowest eigenvalue is always negative, and the atomic interaction energy is

(11.2) $\Delta E^{\left(2\right)}\left(R\right)=-\frac{C_6}{R^6},$

where the constant C ₆ is positive. The attractive forces between S-state atoms at large distances are called van der Waals forces. If only one of the atoms is in an S-state, Eq. (11.2) still gives the interaction energy, because the first-order perturbation vanishes when the multipole moments of one of the atoms is zero. Now, the constant C ₆ depends on the mutual orientation of the atoms.

However, if both atoms have nonzero orbital and nonzero total angular momenta, the situation is different. The expectation value of the dipole moment vanishes, because it has odd parity, but the expectation values of the quadrupole moment in states with $L\ne 0$ and $J\ne 0,1∕2$ are nonzero. The quadrupole–quadrupole term yields a finite first-order perturbation, and the interaction energy of the atoms goes as $\Delta E^{\left(1\right)}\left(R\right)=C_5∕R^5$ , where C ₅ may be either positive or negative.

The interaction of two "identical" atoms that are in different electronic states must be treated differently. The state of two atoms at large internuclear distance has an additional degeneracy due to atom interchange, and degenerate first-order perturbation theory must be used. If the states of the two atoms have different parities, and angular momenta L differing by ±1 or 0 (but not both zero), then the nondiagonal matrix elements of the transition dipole moment between these states do not vanish. First-order degenerate perturbation theory using the dipole–dipole interaction yields an interaction energy $U\left(R\right)=C_3∕R^3$ , where C ₃ can have either sign. However, when the interaction of the atoms is averaged over all possible orientations, the first-order interaction energy vanishes and the second-order perturbation, Eq. (11.2), becomes the lowest-order interaction energy.

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Further Topics in Analysis

George B. Arfken , ... Frank E. Harris , in Mathematical Methods for Physicists (Seventh Edition), 2013

Saddle Point Method

Now that we have identified z ₀ and the directions of steepest descent in |F (z, t)|, we complete the specification of the method of steepest descents, also called the saddle point method of asymptotic approximation, by assuming that the significant contributions to the integral are from a small range of 0 ≤ r ≤ a in each of the two directions along the path. Before obtaining a final result, we must make one more observation. Looking at the way in which the contour had to be deformed to pass through z ₀, we need to determine the sense of the path (i.e., we must decide whether the direction of travel is at $\theta =-\alpha /2+\frac{1}{2}\pi$ or at $\theta =-\alpha /2+\frac{3}{2}\pi$ ). Assuming that this has been decided, we can then identify, for the portion of the path in which we descend from F (z ₀), dz = e^iθ dr. The contribution in which we ascend to F (z ₀) will have the opposite sign for dz but we can handle it simply by multiplying the descending contribution by two. Then, noting that e ^{i(α + 2θ)} = −1, our approximation to f (t) is

(12.104) $f(t)\approx 2e^{w_0+i\theta }\underset{0}{\overset{a}{\int }}{e}^{-|w_0^{″}|r^2/2}dr,$

where the initial "2" causes inclusion of the ascent to z ₀. We now make the key assumption of the method, namely that $|w_0^{″}|$ , the measure of the rate of decrease in |F| as we leave z ₀, is large enough that the bulk of the value of the integral has already been attained for small a, and that the exponential decrease in the value of the integrand enables us to replace a by infinity without making significant error. In problems where the saddle point method is applicable, this condition is met when t is sufficiently large. We complete the present analysis by remembering that $e^{w_0}=F(z_0,t)$ and by evaluating the integral for a = ∞, where, cf. Eq. (1.148), it has the value $\sqrt{\pi /2|w_0^{″}|}$ . We get

(12.105) $f(t)\approx F(z_0,t)e^{i\theta }\sqrt{\frac{2\pi }{|w^{″}(z_0,t)|}}.$

We remind the reader that

(12.106) $\theta =-\frac{arg(w^{″}(z_0,t))}{2}+\left(\frac{\pi }{2}\text{\hspace{0.17em}or}\frac{3\pi }{2}\right),$

with the choice (which affects only the sign of the final result) determined from the sense in which the contour passes through the saddle point z ₀.

Sometimes it is sufficient to apply the method of steepest descents only to the rapidly varying part of an integral. This corresponds to assuming that we may make the approximation

(12.107) $f(t)=\underset{C}{\int }g(z,t)F(z,t)dz\approx g(z_0,t)\underset{C}{\int }F(z,t)dz,$

after which we proceed as before. Note that this causes g not to be considered when we define w or w″, and our final formula is replaced by

(12.108) $f(t)\approx g(z_0,t)F(z_0,t)e^{i\theta }\sqrt{\frac{2\pi }{|w^{″}(z_0,t)|}}.$

A final note of warning: We assumed that the only significant contribution to the integral came from the immediate vicinity of the saddle point z = z ₀. This condition must be checked for each new problem.

Example 12.7.1

Asymptotic Form of the Gamma Function

In many physical problems, particularly in the field of statistical mechanics, it is desirable to have an accurate approximation of the gamma or factorial function of very large numbers. As listed in Table 1.2, the factorial function may be defined by the Euler integral

(12.109) $t!=\mathrm{\Gamma }(t+1)=\underset{0}{\overset{\infty }{\int }}{\rho }^t{e}^{-\rho }d\rho =t^{t+1}\underset{0}{\overset{\infty }{\int }}{e}^{t(\ln z-z)}dz.$

Here we have made the substitution ρ = zt in order to convert the integral to the form given in Eq. (12.108). As before, we assume that t is real and positive, from which it follows that the integrand vanishes at the limits 0 and ∞. By differentiating the exponent, which we call w(z, t), we obtain

$\frac{dw}{dz}=t\frac{d}{dz}(\ln z-z)=\frac{t}{z}-t,w^{″}=-\frac{t}{z^2},$

which shows that the point z = 1 is a saddle point and arg w″ (1, t) = arg(−t) = π. Applying Eq. (12.106), we see that the direction of travel through the saddle point is

$\theta =-\frac{arg{w}^{″}}{2}+\left(\frac{\pi }{2}\text{\hspace{0.17em}or}\frac{3\pi }{2}\right)=0\text{\hspace{0.17em}or}\pi ;$

the choice θ = 0 is that consistent with deformation from a path that was originally along the real axis. In fact, what we have found is that the direction of steepest descent is along the real axis, a conclusion that we might have reached more or less intuitively.

Direct substitution into Eq. (12.108) with g = t ^t+1, F = e ^−t, θ = 0, and |w″| = −t yields

(12.110) $t!=\mathrm{\Gamma }(t+1)\approx \sqrt{\frac{2\pi }{t}}{t}^{t+1}{e}^{-t}=\sqrt{2\pi }{t}^{t+1/2}{e}^{-t}.$

This result is the leading term in Stirling's expansion of the gamma function. The method of steepest descents is probably the easiest way of obtaining this term. Further terms in the asymptotic expansion are developed in Section 13.4.

In this example the calculation was carried out assuming t to be real. This assumption is not necessary. We may show (Exercise 12.7.3) that Eq. (12.110) also holds when t is complex, provided only that its real part be required to be large and positive.

Sometimes the application of the saddle point method to a real integral results in a contour that goes through a saddle point that is not on the real axis. Here is a relatively simple example. A more complicated case of practical importance appears in the chapter on Bessel functions (see Section 14.6).

Example 12.7.2

Saddle Point Method Avoids Oscillations

As a second example of the method of steepest descents, consider the integral

(12.111) $H(t)=\underset{-\infty }{\overset{\infty }{\int }}\frac{e^{-t(z^2-1/4)}\cos tz}{1+z^2}dz,$

which we wish to evaluate for large positive t. When t is large, the integrand oscillates very rapidly, and ordinary quadrature methods become difficult. We proceed by bringing H(t) to a form appropriate for applying the saddle point method, replacing cos tz by cos tz + i sin tz = e^itz (a replacement that does not change the value of the integral because we added an odd term to the previously even integrand). We then have

(12.112) $H(t)=\underset{C}{\int }g(z)e^{-t(z^2-iz-1/4)}dz,$

with g(z) = 1/(1 + z ²). This form corresponds to $w(z)=-t(z^2-iz-\frac{1}{4})$ , so we have

(12.113) $w^{\prime }(z)=-t(2z-i),\text{\hspace{0.17em}which\hspace{0.17em}has\hspace{0.17em}a\hspace{0.17em}zero\hspace{0.17em}at}{z}_0=i/2\text{.}$

Then, at z ₀, which is a saddle point,

(12.114) $w_0=0,w^{″}(z_0)=-2t,g(z_0)=\frac{4}{3}.$

We also need the phase θ of the steepest-descent direction. Noting that arg(w″(z ₀)) = π and applying Eq. (12.106), we find θ = 0 (or π).

We are now ready to apply Eq. (12.108). The result is

(12.115) $H(t)\approx \frac{\sqrt{2\pi }(4/3)(e^0)}{|-2t|}=\frac{4}{3}\sqrt{\frac{\pi }{t}}.$

As a check, we compare this approximate formula for H(t) with the result of a tedious numerical integration: For t = 100, H _exact = 0.23284, and H _saddle = 0.23633.

Exercises

We present here a rather small number of exercises on the method of steepest descents. Several additional exercises appear elsewhere in this book, in particular in Section 14.6, where the technique is applied to the contour integral representations of Bessel functions.

12.7.1

Prove Jensen's theorem (that |F (z)|² can have no extremum in the interior of a region in which F is analytic) by showing that the mean value of |F|² on a circle about any point z ₀ is equal to |F (z ₀)|². Explain why you can then conclude that there cannot be an extremum of |F| at z ₀.

12.7.2

Find the steepest path and leading asymptotic expansion for the Fresnel integrals ${\int }_0^s\cos x^{2}dx,{\int }_0^s\sin x^{2}dx$ .

Hint. Use ${\int }_0^1{e}^{it{z}^2}dz$ .

12.7.3

Show that the formula

$\mathrm{\Gamma }(1+s)\approx \sqrt{2\pi s}{s}^s{e}^{-s}$

holds for complex values of s (with $\Re e(s)$ large and positive).

Hint. This involves assigning a phase to s and then demanding that $\Im m\left[sf(z)\right]$ be constant in the vicinity of the saddle point.

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Particles at Fluids Interfaces and Membranes

Peter A. Kralchevsky , Kuniaki Nagayama , in Studies in Interface Science, 2001

8.1.3. COMPARISON BETWEEN THE CAPILLARY FLOTATION AND IMMERSION FORCES

Equations (7.13) and (8.4) , and their asymptotic form for qL ≪ 1,

(8.19) $F=-2\pi \sigma \frac{Q_1{Q}_2}{L}{r}_k\ll L\ll q^{-1}$

(q ⁻¹  =   2.7   mm for water), show that the immersion and flotation forces exhibit the same functional dependence on the interparticle distance L. On the other hand, the "capillary charges", Q ₁ and Q ₂, can be very different for these two kinds of capillary force. To demonstrate that let us consider the case of two identical particles, for which R ₁  = R ₂  = R and α ₁  = α ₂  = α; then using Eqs. (8.5) and (8.16) one can derive [4,6,7]

(8.20) $\begin{array}{ll}F\propto \left(R^6/\sigma \right){\text{K}}_1\left(qL\right)\hfill & \text{for}\text{flotation}\text{force}\hfill \\ F\propto \sigma R^2{\text{K}}_1\left(qL\right)\hfill & \text{for}\text{immersion}\text{force}\hfill \end{array}$

Consequently, the flotation force decreases, while the immersion force increases, when the interfacial tension σ increases. Besides, the flotation force decreases much stronger with the decrease of particle radius R than the immersion force. Thus F _flotation negligible for R < 5–10 μm, whereas F _immersion can be significant even for R  =   2   nm [4,6]. This is illustrated in Fig. 8.3, where the two types of capillary interaction are compared, with respect to their energy $\Delta W\left(L\right)={\displaystyle {\int }_L^{\infty }F\left(L^{\prime }\right)d{L}^{\prime }}$ , for a wide range of particle sizes. The values of the parameters used are: particle mass density ρ _p   =   2   g/cm³, density difference between the two fluids Δρ  =   1   g/cm³, surface tension σ  =   40   mN/m, contact angle α  =   60°, interparticle distance L  =   2R, and thickness of the non-disturbed planar film l ₀  = R. Protein molecules of nanometer size can be considered as "particles" insofar as they are much larger than the solvent (water) molecules. For example, the radius of a water molecule is about 0.12   nm; then a protein of radius R ≥ 1.2   nm can be considered as being much larger.

Fig. 8.3. Energy of capillary attraction ΔW, in kT units, plotted vs. the radius R of two similar particles separated at a center-to-center distance L  =   2R. If ΔW &gt; kT, the capillary attraction is stronger than the Brownian force and can cause a two-dimensional aggregation of the particles [4,6].

The pronounced difference in the strength of the two types of capillary interactions, see Fig. 8.3, is due to the different magnitude of the interfacial deformation. The small floating particles are too light to create a substantial deformation of the liquid surface and then the lateral capillary force becomes negligible. In the case of immersion forces the particles are restricted in the vertical direction by the solid substrate (see Fig. 7.1b) or by the two surfaces of the liquid film (Fig. 7.1f). Therefore, as the film becomes thinner, the liquid surface deformation increases, thus giving rise to a strong interparticle attraction. For that reason, as already mentioned, the immersion forces may be one of the main factors causing the observed self assembly of μm-sized and sub-μm colloidal particles and protein macromolecules confined in thin liquid films or lipid bilayers.

In conclusion, the different physical origin of the flotation and immersion lateral capillary forces results in different magnitudes of the "capillary charges" Q_k , which depend in a different way on the interfacial tension σ and the particle radius R, see Eqs. (8.4) and (8.20). In this respect these two kinds of capillary force resemble the electrostatic and gravitational forces, which obey the same power law, but differ in the physical meaning and magnitude of the force constants (charges, masses) [7,8].

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RADIATION

V.B. BERESTETSKII , ... L.P. PITAEVSKII , in Quantum Electrodynamics (Second Edition), 1982

§ 56 The photoelectric effect: non-relativistic case

In §49–52we have discussed radiative transitions (with emission or absorption of a photon) between atomic levels of the discrete spectrum. The photoelectric effect differs from such a photon absorption process only in that the final state belongs to the continuous spectrum.

The cross-section for the photoelectric effect can be calculated in an exact analytical form for the hydrogen atom and for a hydrogen-like ion (with atomic numberZ≪ 137).

In the initial state, the electron is at a discrete level ɛ _i ≡ -I(whereIis the ionization potential of the atom) and the photon has a definite momentumk. In the final state, the electron has momentump(and energy ɛ _f ≡ ɛ). Sinceptakes a continuous series of values, cross-section for the photoelectric effect is

(56.1) $\text{d}\sigma =2\pi |V_{fi}{|}^2\delta \left(-I+\omega -\epsilon \right){\text{d}}^{3}p/{\left(2\pi \right)}^3$

(cf.(44.3)); the wave function of the final state of the electron is normalized to "one particle per unit volume". The wave function of the photon is, as before, normalized in the same way; in order to obtain the cross-sectiondσ, the probabilitydwthen has to be divided by the photon flux density (which isc/V=cwhenV= 1), but when relativistic units are used this does not affect the form of (56.1).

As in (45.2), we choose the three-dimensionally transverse gauge for the photon.

Then

$V_{fi}=-eA\cdot {\text{j}}_{fi}=-e\surd \left(4\pi \right)\frac{1}{\surd \left(2\omega \right)}{M}_{fi},$

where

(56.2) $M_{fi}={\displaystyle \int {\psi }^{\prime }*\left(\alpha \cdot e\right)e^{i\text{k}\cdot e}\psi {\text{d}}^{3}x,}$

with ψ ≡ ψ _i and ψ′ ≡ ψ _f the initial and final wave functions of the electron. Putting in (56.1) d ³ p→p ² d|p|do= ɛ|p|dɛdoand integrating to remove the delta function of ɛ, we can write this formula as

(56.3) $\text{d}\sigma =e^2\frac{\epsilon \left|p\right|}{2\pi \omega }|M_{fi}{|}^2{\text{d}}_0.$

The calculations will be given in two cases, which differ as regards the magnitude of the photon energy: Ω ≫Iand Ω ≪m. SinceI˜me ⁴ Z ²≪m, these two ranges partly overlap (whenI≪ Ω ≪m), and so an examination of the two cases gives an essentially complete description of the photoelectric effect.

We shall take first the case

(56.4) $\omega \ll m.$

The electron velocity is then small in both the initial and the final state, and the problem is therefore entirely non-relativistic as regards the electron. Accordingly, we replaceαin (56.2) by the non-relativistic velocity operator $\widehat{v}=-i\nabla /m$ (cf.§45). We can also use the dipole approximation, puttinge ^{i kτ}≈ 1, i.e. neglecting the momentum of the photon in comparison with that of the electron. Then

(56.5) $\begin{array}{ll}\hfill \text{d}\sigma & =e^2\frac{m\left|p\right|}{2\pi \omega }|e\cdot V_{fi}{|}^2{\text{d}}_0\hfill \\ \hfill V_{fi}& =-\frac{i}{m}{\displaystyle \int {\psi }^{\prime }*}\nabla \psi \cdot {\text{d}}^{3}x.\hfill \end{array}\}$

We shall consider the photoelectric effect from the ground level of the hydrogen atom (or of a hydrogen-like ion). Then

(56.6) $\psi =\frac{{\left(Z{e}^{2}m\right)}^{3/2}}{\surd \pi }{e}^{-Z{e}^{2}m\text{r}};$

in ordinary units,me ²becomes 1/a ₀, wherea ₀= ħ²/me ²is the Bohr radius.

The wave function ψ′ must be taken such that its asymptotic form comprises a plane wave ( e ^{i p˙r} ) together with an ingoing spherical wave; cf.QM,§136, where this function was denoted by ψ _p ^(-). On account of the selection rule forl, a transition from an s state can only be to apstate (in the dipole case). Thus, in the expansion ^†

(56.7) ${\psi }_p^{(-)}=\frac{1}{2p}{\displaystyle \sum _{\text{j}=b}^{\infty }{i}^l\left(2l+1\right)e^{-i{\delta }_l}{R}_{pl}\left(\text{r}\right)p_l\left(\text{n}\cdot {\text{n}}_1\right),}$

wheren=p/p,n ₁=r/r , it is sufficient to retain the term withl= 1. Omitting unimportant phase factors, we therefore have

(56.8) ${\psi }^{\prime }=\frac{3}{2p}\left(\text{n}\cdot {\text{n}}_1\right)R_{pl}\left(\text{r}\right).$

With the functions ψ and ψ′ given by (56.6) and (56.8), we have

$\begin{array}{cc}\hfill e\cdot V_{fi}& =\frac{2{\left(Z{e}^{2}m\right)}^{5/2}}{2\surd \pi mp}{\displaystyle \int {\displaystyle \int \left(\text{n}\cdot {\text{n}}_1\right)\left({\text{n}}_1\cdot e\right)}}{e}^{-Z{e}^{2}m\text{r}}{R}_{pl}\left(\text{r}\right){\text{d}}_{01}\cdot {\text{r}}^2\text{d}\text{r}\hfill \\ \hfill & =\frac{\surd \left(2\pi \right){\left(Z{e}^{2}m\right)}^{5/2}}{pm}\left(\text{n}\cdot e\right){\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{r}}^2{e}^{-Z{e}^{2}m\text{r}}{R}_{pl}\left(\text{r}\right)\text{d}\text{r}.}\hfill \end{array}$

According toQM(36.18),(36.24), the radial function is (in the units employed here)

$R_{p1}=\frac{\surd \left(8\pi \right)Z{e}^{2}m}{3}\surd \frac{1+{\nu }^2}{\nu \left(1-e^{-2\pi \nu }\right)}p\text{r}{e}^{-ip\text{r}}F\left(2+iv,4,2ip\text{r}\right),$

with

(56.9) $\nu =Z{e}^{2}m/p\left(=Z{e}^2/\hslash \nu \right).$

The integral can be calculated by means of the formula

${\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\lambda z}{Z}^{\gamma -1}F\left(\alpha ,\gamma ,\text{k}z\right)\text{d}z=\Gamma (\gamma ){\lambda }^{\alpha -\gamma }{\left(\lambda -\text{k}\right)}^{-\alpha };}$

cf.QM, (f.3). Noticing also that

${\left(\frac{\nu +i}{\nu -i}\right)}^{i\nu }=e^{-2\nu {\cot }^{-1}\nu },$

we obtain

$e\cdot v_{fi}=\frac{2^{7/2}\pi {\nu }^3\left(\text{n}\cdot f\right)}{\sqrt{p\cdot m{\left(1+{\nu }^2\right)}^{3/2}}}\frac{e^{-2\nu {\cot }^{-1}\nu }}{\sqrt{\left(1-e^{-2\pi \nu }\right)}}.$

The energy of ionization from the ground level of the hydrogen atom (or a hydrogen-like ion) isI=Z ² e ⁴ m/2. Hence

(56.10) $\omega =\frac{p^2}{2m}+I=\frac{p^2}{2m}\left(1+{\nu }^2\right).$

Using this relation, we obtain as the final expression for the cross-section for the photoelectric effect with emission of an electron into the solid-angle elementdo

(56.11) $\text{d}\sigma =2^7\pi \alpha a^2{\left(\frac{I}{\hslash \omega }\right)}^4\frac{e^{-4\nu {\cot }^{-1}\nu }}{1-e^{-2\pi \nu }}{\left(\text{n}\cdot e\right)}^2\text{d}o,$

wherea= ħ²/mZe ²=a ₀/Z(ordinary units are used here and below). The angular distribution of the emitted electrons is governed by the factor (n˙e)². This has maxima in the directions parallel to the direction of polarization of the incident photons, and is zero in directions perpendicular toe, including the direction of incidence. For unpolarized photons, formula (56.11) must be averaged over the directions ofe, which is equivalent to substituting

${\left(\text{n}\cdot e\right)}^2\to \frac{1}{2}{\left({\text{n}}_0\times \text{n}\right)}^2$

withn ₀=k/k; see (45.4b).

Integration of formula (56.11) over all angles gives the total cross-section for the photoelectric effect:

(56.12) $\sigma =\frac{2^9{\pi }^2}{3}\alpha a^2{\left(\frac{I}{\hslash \omega }\right)}^4\frac{e^{-4\nu {\cot }^{-1}\nu }}{1-e^{-2\pi \nu }}$

(M. Stobbe, 1930).

The limiting value of σ as ħΩ →I(i.e. asv→ ∞) is

(56.13) $\sigma =\frac{2^9{\pi }^2}{3e^4}\alpha a^2=\frac{2^9{\pi }^2}{3e^4}\frac{\alpha a_0^2}{Z^2}=0.23a_0^2/Z^2,$

whereein the denominator is the base of natural logarithms. The cross-section for the photoelectric effect tends to a constant limit near the threshold, as it must for a reaction forming charged particles (see QM,§147).

The case in which ħω ≫I(still with ħω ≫mc ²) corresponds to the Born approximation (v=Ze ²/ħv≪ 1). Formula (56.12) becomes

(56.14) $\sigma =\frac{2^8\pi }{3}\alpha a_0^2{Z}^5{\left(\frac{I_0}{\hslash \omega }\right)}^{7/2},$

whereI ₀=e ⁴ m/2ħ²is the ionization energy of the hydrogen atom.

The process inverse to the photoelectric effect is the radiative recombination of an electron with an ion at rest. The cross-section σ_recfor this process can be found from that for the photoelectric effect (σ_ph) by means of the principle of detailed balancing (QM,§144). This principle states that the cross-sections for the processesi→fandf→i(with two particles in each of the statesiandf) are related by

$g_i{p}_i^2{\sigma }_{i\to f}=g_f{p}_f^2{\sigma }_{f\to i},$

wherep_i ,p_f are the momenta of the relative motion of the particles, andg_i ,g_f the spin statistical weights of the statesiandf. Sinceg= 2 for the photon (which has two possible directions of polarization), we find for the hydrogen atom ground state

(56.15) ${\sigma }_{\text{r}ec}={\sigma }_{ph}\cdot 2{\text{k}}^2/p^2,$

wherep=m vis the momentum of the incident electron andkthat of the emitted photon.

PROBLEMS

PROBLEM 1

Derive formula (56.14) by direct use of the Born approximation in the non-relativistic case.

SOLUTION

In the Born approximation, ψ′ in (56.5) is simply the plane wave ψ′ =e ^{i p˙r} , and ψ is again the function (56.6). Then

$\begin{array}{cc}\hfill V_{fi}& =V_{if}=\frac{1}{m}{\displaystyle \int \psi \widehat{p}{\psi }^{\prime }{\text{d}}^{3}x}\hfill \\ \hfill & =\frac{p}{m}\frac{{\left(Z{e}^{2}m\right)}^{3/2}}{\sqrt{\pi }}{\left(e^{-Z{e}^{2}m\text{r}}\right)}_{p.}\hfill \end{array}$

The Fourier component is given by (57.6b), and so

$V_{fi}=8\surd \pi p^{-3}{m}^{3/2}{\left(Z{e}^2\right)}^{5/2}\text{n}.$

Substitution in (56.5) and integration overdoleads to (56.14)(here, with sufficient accuracy,p ²/2m˜ ω).

PROBLEM 2

Determine the total cross-section for radiative recombination of a fast but non-relativistic electron (I≪mv ¹≪mc ²) with a nucleus having chargeZ≪ 137.

SOLUTION

The cross-section for capture to theKshell (principal quantum numbern= 1) is obtained by substituting (56.14) in (56.15):

${\sigma }_1^{\mathrm{rec}}=\frac{2^7\pi }{3}{Z}^5{\alpha }^3{a}_0^2{\left(\frac{I_0}{\epsilon }\right)}^{5/2},$

where ɛ = 1/2mv ²is the energy of the incident electron, and ħω = ɛ. Among the other states of the resulting atom, onlysstates are important: in the calculation of the matrix element in the Born approximation, the important values are those of the wave function of the bound state whenris small (as will be seen in §57), and whenl> 0 these values are small compared with those forl= 0. It is sufficient to take the first two terms in the expansion of ψ in powers ofr. For states withl= 0 and anyn, these terms are

$\psi =\frac{1}{\sqrt{\left(\pi a^3{\text{n}}^3\right)}}\left(1-\frac{\text{r}}{a}\right),$

i.e. they containnonly as a common factorn ^−3/2this expression is obtained by expansion ofQM,(36.13). The total recombination cross-section is therefore

${\sigma }^{\mathrm{rec}}={\displaystyle \sum _{\text{n}=1}^{\infty }{\sigma }_{\text{n}}^{\mathrm{rec}}={\sigma }_1^{\mathrm{rec}}{\displaystyle \sum _{\text{n}=1}^{\infty }\frac{1}{{\text{n}}^3}=\zeta \left(3\right){\sigma }_1^{\mathrm{rec}}.}}$

The value of the zeta function is ζ(3) = 1.202.

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Advances in Atomic, Molecular, and Optical Physics

Vanja Dunjko , ... Maxim Olshanii , in Advances In Atomic, Molecular, and Optical Physics, 2011

3.2 General Features of 1D Scattering

In 1D scattering, there are just two kinds of partial waves: even and odd. The asymptotic form of the scattering wave function is thus

(14) $\psi (z)~e^{ikz}+f_{\text{even}}(k){\text{e}}^{ik\left|z\right|}+f_{\text{odd}}(k)\text{sign}(z){\text{e}}^{ik\left|z\right|}.$

Since in our case the 3D potential is zero-range, we expect that interactions in our effective 1D theory will also be zero-range, in which case the odd scattering amplitude f _odd must be zero: the odd component of the wave function is zero at the origin, and thus a zero-range scatterer at the origin cannot have any effect.

We may also consider a different sort of boundary conditions, namely one where ψ(z) is an even function. One reason why even-wave scattering wave functions are of interest is that the 1D even-wave scattering phase Δ(k) is defined using their asymptotic form, which is always

(15) $\psi (z)~\text{sin}\left[k\left|z\right|+Q(k))\right]\text{as}z\to \pm \infty ,$

where Q(k) is some function of k but not of z. By definition, the scattering phase in any particular case is just this function: Δ(k) = Q(k). The important concept of the 1D scattering length is then defined through

(16) $a_{1\text{D}}=\underset{k\to 0^{+}}{\text{lim}}\left[-\frac{d}{\text{d}k}\mathrm{\Delta }(k)\right],$

although it can also be read off from the scattering amplitude, see Equation (19).

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Vibrations of a Body of Revolution

J.D. Kaplunov , ... E.V. Nolde , in Dynamics of Thin Walled Elastic Bodies, 1998

5.1 Short-Wave Vibrations

In this chapter special features of the approximate methods developed in Chapters 2–4 2 3 4 are illustrated by the example of a thin walled body of revolution. Such a choice allows us to present all the results obtained above in a simpler form. First we consider short-wave approximations. Only these approximations are described by 3D equations. In the case of a body of revolution they can be considerably simplified.

We refer the midsurface of the body of revolution to the coordinate system (s, θ), where s(s ₁  ≤ s  ≤ s ₂) is the length of the meridian arc, θ (0   ≤ θ  <   2π) is the angular coordinate (see Fig. 5.1). Then the first quadratic form of the midsurface can be written as ds ²  + A ² R ² dθ ², where R is a typical radius of curvature of the midsurface, A  = A(s) is the distance from the axis of rotation, divided by R. We shall also use the dimensionless length of the meridian arc ξ   = s/R(ξ _[1]  ≤ ξ  ≤ ξ _[2]).

Figure 5.1.

Let us rewrite the equations of the leading short-wave approximation (4.1.5), (4.1.6) setting α ₁  = Rξ, α ₂  = Rθ, A ₁  =   1, A ₂  = A(ξ); in this case k ₁  =   0, k ₂  = k(ξ), where

(5.1.1) $k=\frac{1}{AR}\frac{dA}{d\xi }$

The result is

$\begin{array}{l}\frac{1}{R}\left(\frac{\partial {\sigma }_{11}}{\partial \xi }+\frac{1}{A}\frac{\partial {\sigma }_{12}}{\partial \theta }\right)+\frac{\partial {\sigma }_{31}}{\partial {\alpha }_3}+k\left({\sigma }_{11}-{\sigma }_{22}\right)-\rho \frac{{\partial }^2{\mathrm{\upsilon }}_1}{\partial t^2}=0,\\ \frac{1}{R}\left(\frac{1}{A}\frac{\partial {\sigma }_{22}}{\partial \theta }+\frac{\partial {\sigma }_{12}}{\partial \xi }\right)+\frac{\partial {\sigma }_{32}}{\partial {\alpha }_3}+2k{\sigma }_{12}-\rho \frac{{\partial }^2{\mathrm{\upsilon }}_2}{\partial t^2}=0\text{,}\end{array}$

(5.1.2) $\begin{array}{l}\frac{1}{R}\left(\frac{\partial {\sigma }_{31}}{\partial \xi }+\frac{1}{A}\frac{\partial {\sigma }_{32}}{\partial \theta }\right)+\frac{\partial {\sigma }_{33}}{\partial {\alpha }_3}+k{\sigma }_{31}-\rho \frac{{\partial }^2{\mathrm{\upsilon }}_3}{\partial t^2}=0,\\ {\sigma }_{11}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3}{\partial {\alpha }_3}+\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }+k{\mathrm{\upsilon }}_1\right)+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }\right],\\ {\sigma }_{22}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3}{\partial {\alpha }_3}+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }+k{\mathrm{\upsilon }}_1\right)+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_3}{\partial \theta }\right],\\ {\sigma }_{33}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }+\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }+k{\mathrm{\upsilon }}_1\right)+\frac{\partial {\mathrm{\upsilon }}_3}{\partial {\alpha }_3}\right],\\ {\sigma }_{31}=\frac{E}{2\left(1+v\right)}\left(\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_3}{\partial \xi }+\frac{\partial {\mathrm{\upsilon }}_1}{\partial {\alpha }_3}\right),\\ {\partial }_{32}=\frac{E}{2\left(1+v\right)}\left(\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_3}{\partial \theta }+\frac{\partial {\mathrm{\upsilon }}_2}{\partial {\alpha }_3}\right),\\ {\partial }_{12}=\frac{E}{2\left(1+v\right)}\left(\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \theta }+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \xi }-k{\mathrm{\upsilon }}_2\right)\text{.}\end{array}$

As before, we shall assume that the faces are free, i.e.

(5.1.3) ${\sigma }_{3\iota }=0\mathrm{at}{\alpha }_3=\pm h\left(l=1,2,3\right)\text{.}$

We focus our attention on vibrations initiated by edge loads. In doing so we impose the following boundary conditions on the edges

(5.1.4) ${\sigma }_{1l}={g_l}_n\mathrm{at}\xi ={\xi }_{\left[n\right]}\left(n=1,2;l=1,2,3\right)\text{,}$

Where g_ln   = g_ln (θ, α ₃, t) are given stresses.

Let us consider in detail the case in which vibrations of the body are short-wave ones only in the meridian direction. We assume that the loads Dynamics of Thin Walled Elastic Bodies satisfy the asymptotic relations

(5.1.5) $\begin{array}{l}\frac{{\partial }^j{g}_{\mathit{ln}}}{\partial {\theta }^j}~{\eta }^{-jp}{g}_{\mathit{ln}},\frac{{\partial }^j{g}_{\mathit{ln}}}{\partial t^j}~n^{-j}{\left(\frac{c_2}{R}\right)}^j{g}_{\mathit{ln}}\\ \left(n=1,2;l=1,2,3:j=1,2,\dots \right)\text{.}\end{array}$

Here p (0   ≤ p  <   1) is the variability index along along a parallel. As a simple of the loads considered, we mention those varying as exp $\left[i\left(m\theta -\omega t\right)\right]$ with $m~{\eta }^{-p},\omega ~{\eta }^{-1}{c}_2/R\text{.}$

Let us show that loads (5.1.5) can excite short-wave vibrations in the meridian direction (vibrations with the variability index q  =   1 along themeridian). To this end, we dilate the scale of the independent variables in equations(5.1.2) setting

(5.1.6) $\xi =\eta {\gamma }_1,\theta =n^p{\gamma }_2,{\alpha }_3=R\eta \zeta ,t=\eta \frac{R}{c_2}\tau \text{.}$

Two types of the SSS of the body correspond to the transformation performed. The asymptotics of these SSS have the form:

Asymptotics I

(5.1.7) $\begin{array}{cccc}\hfill {\sigma }_{ll}=E{\sigma }_{ll}^{\ast },\hfill & \hfill {\sigma }_{12}=E{\eta }^{1-p}{\sigma }_{12}^{\ast },\hfill & \hfill {\sigma }_{31}=E{\sigma }_{31}^{\ast },\hfill & \hfill {\sigma }_{32}=E{\eta }^{1-p}{\sigma }_{32}^{\ast },\hfill \\ \hfill {\mathrm{\upsilon }}_1=R\eta {\mathrm{\upsilon }}_1^{\ast },\hfill & \hfill {\mathrm{\upsilon }}_2=R{\eta }^{2-p}{\mathrm{\upsilon }}_2^{\ast },\hfill & \hfill {\mathrm{\upsilon }}_3=R\eta {\mathrm{\upsilon }}_3^{\ast }\hfill & \hfill \left(l=1,2,3\right)\text{;}\hfill \end{array}$

Asymptotics II

(5.1.8) $\begin{array}{cccc}\hfill {\sigma }_{ll}=E{\eta }^{1-p}{\sigma }_{ll}^{\ast },\hfill & \hfill {\sigma }_{12}=E{\sigma }_{12}^{\ast },\hfill & \hfill {\sigma }_{31}=E{\eta }^{1-p}{\sigma }_{31}^{\ast },\hfill & \hfill {\sigma }_{32}=E{\sigma }_{32}^{\ast },\hfill \\ \hfill {\mathrm{\upsilon }}_1=R{\eta }^{2-p}{\mathrm{\upsilon }}_1^{\ast },\hfill & \hfill {\mathrm{\upsilon }}_2=R\eta {\upsilon }_2^{\ast },\hfill & \hfill {\mathrm{\upsilon }}_3=R{\eta }^{2-p}{\mathrm{\upsilon }}_{3.}^{\ast }\hfill & \hfill \hfill \end{array}$

Here as usual the dimensionless stresses and displacements are of the same asymptotic order.

In the case of Asymptotics I the quantities σ_ll (l  =   1, 2, 3), σ ₃₁ and υ ₁, υ₃ are asymptotically principal among the stresses and displacements, respectively. Note that these stresses and displacements are characteristic of the plane problem of elasticity. In the case of Asymptotics II we have an opposite situation. The quantities characteristic of the antiplane problem of elasticity (the stresses σ ₁₂, σ ₃₂ and the displacement υ ₂) are asymptotically principal. Such an analogy with the plane and antiplane problems of elasticity will be employed below.

Substituting the asymptotic formulae(5.1.7), (5.1.8) into equations(5.1.2) and taking into account the change of variables (5.1.6) we obtain for Asymptotics I

(5.1.9) $\begin{array}{l}\frac{\partial {\sigma }_{11}^{\ast }}{\partial {\gamma }_1}+\frac{\partial {\sigma }_{31}^{\ast }}{\partial \zeta }+\eta k^{\ast }\left({\sigma }_{11}^{\ast }-{\sigma }_{22}^{\ast }\right)-\frac{1}{2\left(1+v\right)}\frac{{\partial }^2{\mathrm{\upsilon }}_1^{\ast }}{\partial {\tau }^2}+\frac{{\eta }^{2-2p}}{A}\frac{\partial {\sigma }_{12}^{\ast }}{\partial {\gamma }_2}=0,\\ \frac{\partial {\sigma }_{31}^{\ast }}{\partial {\gamma }_1}+\frac{\partial {\sigma }_{33}^{\ast }}{\partial \zeta }+\eta k^{\ast }{\sigma }_{31}^{*}-\frac{1}{2\left(1+v\right)}\frac{{\partial }^2{\mathrm{\upsilon }}_3^{*}}{\partial {\gamma }^2}+\frac{{\eta }^{2-2p}}{A}\frac{\partial {\sigma }_{32}^{\ast }}{\partial {\gamma }_2}=0,\\ {\sigma }_{11}^{*}=\frac{1}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial \zeta }+\eta k^{*}{\mathrm{\upsilon }}_1^{*}\right)+\frac{\partial {\upsilon }_1^{*}}{\partial {\gamma }_1}\right]+\frac{{\eta }^{2-2p}v}{2\left(1-v^2\right){\mathrm{\varkappa }}^{2}A}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_2},\\ {\sigma }_{22}^{\ast }=\frac{1}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial \zeta }+\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_1}\right)+\eta k^{\ast }{\mathrm{\upsilon }}_1^{*}\right]+\frac{{\eta }^{2-2p}}{2\left(1+v\right){\mathrm{\varkappa }}^{2}A}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_2},\\ {\partial }_{33}^{*}=\frac{1}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_1}+\eta k^{\ast }{\mathrm{\upsilon }}_1^{*}\right)+\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial \zeta }\right]+\frac{{\eta }^{2-2p}v}{2\left(1-v^2\right){\chi }^{2}A}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_2},\\ {\sigma }_{31}^{\ast }=\frac{1}{2\left(1+v\right)}\left(\frac{\partial {\mathrm{\upsilon }}_3^{\ast }}{\partial {\gamma }_1}+\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial \zeta }\right)\end{array}$

and

(5.1.10) $\begin{array}{l}\frac{\partial {\sigma }_{12}^{*}}{\partial {\gamma }_1}+\frac{\partial {\sigma }_{32}^{*}}{\partial \zeta }+\eta 2k^{*}{\sigma }_{12}^{*}-\frac{1}{2\left(1+v\right)}\frac{{\partial }^2{\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }^2}=-\frac{1}{A}\frac{\partial {\sigma }_{22}^{*}}{\partial {\gamma }_2},\\ {\sigma }_{32}^{*}=\frac{1}{2\left(1+v\right)}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial \zeta }+\frac{1}{2\left(1+v\right)A}\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial {\gamma }_2},\\ {\sigma }_{12}^{*}=\frac{1}{2\left(1+v\right)}\left(\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_1}-\eta k^{*}{\mathrm{\upsilon }}_2^{*}\right)+\frac{1}{2\left(1+v\right)A}\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_2}\text{;}\end{array}$

for Asymptotics II

(5.1.11) $\begin{array}{l}\frac{\partial {\sigma }_{12}^{*}}{\partial {\gamma }_1}+\frac{\partial {\sigma }_{32}^{*}}{\partial \zeta }+\eta 2k^{*}{\sigma }_{12}^{*}-\frac{1}{2\left(1+v\right)}\frac{{\partial }^2{\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }^2}+\frac{{\eta }^{2-2p}}{A}\frac{\partial {\sigma }_{22}^{*}}{\partial {\gamma }_2}=0,\\ {\sigma }_{32}^{*}=\frac{1}{2\left(1+v\right)}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial \zeta }+\frac{{\eta }^{2-2p}}{2\left(1+v\right)A}\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial {\gamma }_2},\\ {\sigma }_{12}^{*}=\frac{1}{2\left(1+v\right)}\left(\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_1}-\eta k^{\ast }{\mathrm{\upsilon }}_2^{*}\right)+\frac{{\eta }^{2-2p}}{2\left(1+v\right)A}\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_2}\text{,}\end{array}$

and

(5.1.12) $\begin{array}{l}\frac{\partial {\sigma }_{11}^{*}}{\partial {\gamma }_1}+\frac{\partial {\sigma }_{31}^{*}}{\partial \zeta }+\eta k^{\ast }\left({\sigma }_{11}^{*}-{\sigma }_{22}^{*}\right)-\frac{1}{2\left(1+v\right)}\frac{{\partial }^2{\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }^2}=-\underset{¯}{\frac{1}{A}\frac{\partial {\sigma }_{12}^{*}}{\partial {\gamma }_2}},\\ \frac{\partial {\sigma }_{31}^{*}}{\partial {\gamma }_1}+\frac{\partial {\sigma }_{33}^{*}}{\partial \zeta }+\eta k^{*}{\sigma }_{31}^{*}-\frac{1}{2\left(1+v\right)}\frac{{\partial }^2{\mathrm{\upsilon }}_3^{*}}{\partial {\gamma }^2}=-\underset{¯}{\frac{1}{A}\frac{\partial {\sigma }_{32}^{*}}{\partial {\gamma }_2}},\\ {\partial }_{11}^{*}=\frac{1}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial \zeta }+\eta k^{*}{\mathrm{\upsilon }}_1^{*}\right)+\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_1}\right]+\underset{¯}{\frac{v}{2\left(1-v^2\right){\mathrm{\varkappa }}^{2}A}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_2}},\\ {\partial }_{22}^{\ast }=\frac{1}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3^{\ast }}{\partial \zeta }+\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_1}\right)+\eta k^{*}{\mathrm{\upsilon }}_1^{*}\right]+\underset{¯}{\frac{1}{2\left(1+v\right){\mathrm{\varkappa }}^{2}A}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_2}},\\ {\partial }_{33}^{*}=\frac{1}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_1}+\eta k^{*}{\mathrm{\upsilon }}_1^{*}\right)+\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial \zeta }\right]+\underset{¯}{\frac{v}{2\left(1-v^2\right)\mathrm{\varkappa }A}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_2}},\\ {\partial }_{31}^{*}=\frac{1}{2\left(1+v\right)}\left(\frac{\partial {\mathrm{\upsilon }}_3^{*}}{\partial {\gamma }_1}+\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial \zeta }\right)\text{.}\end{array}$

In equations(5.1.9)–(5.1.12) k ^⁎  = A ^{−   1} ∂A/∂ξ  = kR.

Let us discuss the equations corresponding to Asymptotics I. The asymptotically principal parts of equations(5.1.9) [to within the error O(η  + η ^{2   −   2p} )] coincide with the equations of the plane problem of elasticity written in Cartesian coordinates. The asymptotically secondary stresses and displacements appear in these equations only with the factor η ^{2   −   2p} . Equations(5.1.10) without the underlined terms contain only asymptotically secondary stresses and displacements and are, to within the error O(η), identical to those of the antiplane problem of elasticity. The underlined terms in these equations are expressed in terms of the asymptotically principal quantities of the sought for SSS.

The opposite situation occurs in the case of Asymptotics II. The principal parts of equations(5.1.11) and equations(5.1.12) without the underlined terms coincide with the equations of the antiplane and plane problems of elasticity, respectively.

The equations considered can be simplified. First, the terms of order O(η) can be neglected in equations(5.1.10), (5.1.12). This leads to the additional truncation error δ_t   = O(η ^{3   −   2p} ) in equations(5.1.9), (5.1.11), which corresponds to the solution error δ_s   = O(η ^{2   −   2p} ). Second, equation(5.1.10) ₁ can be replaced by

(5.1.13) $\frac{1}{A}\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_2}=\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_1}=0\text{,}$

and the equation(5.1.12) ₁ and (5.1.12) ₂ can be replaced by

(5.1.14) $\begin{array}{l}\frac{\partial {\mathrm{\upsilon }}_1^{*}}{\partial {\gamma }_1}+\frac{1}{A}\frac{\partial {\mathrm{\upsilon }}_2^{*}}{\partial {\gamma }_2}=0,\\ {\mathrm{\upsilon }}_3^{*}=0\text{.}\end{array}$

It is possible to verify by means of direct calculations that to within the error O(η) equations(5.1.10) ₂, (5.1.10) ₃, (5.1.13) and (5.1.12) ₃–(5.1.12) ₆, (5.1.14) are equivalent to equations(5.1.10) and (5.1.12), respectively.

Conditions(5.1.13), (5.1.14) have an evident physical sense. They correspond to the absence of rotation around the normal to the midsurface, volume strains and transverse displacements. Equations(5.1.9), (5.1.10) with replacement (5.1.10) ₁ → (5.1.13) do not possess the modes characteristic of the antiplane problem of elasticity. At the same time, equations(5.1.11), (5.1.12) with replacements (5.1.12) ₁, (5.1.12) ₂ → (5.1.14) do not possess the modes characteristic of the plane problem of elasticity.

Taking into account these simplifications and introducing dimensional quantities, we have:

for Asymptotics I

(5.1.15) $\begin{array}{l}\frac{1}{R}\left(\frac{\partial {\sigma }_{11}}{\partial \xi }+\frac{1}{A}\frac{\partial {\sigma }_{12}}{\partial \theta }\right)+\frac{\partial {\sigma }_{31}}{\partial {\alpha }_3}+k\left({\sigma }_{11}-{\sigma }_{22}\right)-\rho \frac{{\partial }^2{\mathrm{\upsilon }}_{{}^1}}{\partial t^2}=0,\\ \frac{1}{A}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \theta }-\frac{\partial {\mathrm{\upsilon }}_2}{\partial \xi }=0,\\ \frac{1}{R}\left(\frac{\partial {\sigma }_{31}}{\partial \xi }+\frac{1}{A}\frac{\partial {\sigma }_{32}}{\partial \theta }\right)+\frac{\partial {\sigma }_{33}}{\partial {\alpha }_3}+k{\sigma }_{31}-\rho \frac{{\partial }^2{\mathrm{\upsilon }}_3}{\partial t^2}=0,\\ {\sigma }_{11}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3}{\partial {\alpha }_3}+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }+k{\mathrm{\upsilon }}_1\right)+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }\right],\\ {\sigma }_{22}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{\partial {\mathrm{\upsilon }}_3}{\partial {\alpha }_3}+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }\right)+\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }+k{\mathrm{\upsilon }}_1\right],\\ {\sigma }_{33}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^2}\left[\frac{v}{1-v}\left(\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }+\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }+k{\mathrm{\upsilon }}_1\right)+\frac{\partial {\mathrm{\upsilon }}_3}{\partial {\alpha }_3}\right],\\ {\sigma }_{31}=\frac{E}{2\left(1+v\right)}\left(\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_3}{\partial \xi }+\frac{\partial {\mathrm{\upsilon }}_1}{\partial {\alpha }_3}\right),\\ {\sigma }_{32}=\frac{E}{2\left(1+v\right)}\left(\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_3}{\partial \theta }+\frac{\partial {\mathrm{\upsilon }}_2}{\partial {\alpha }_3}\right),\\ {\sigma }_{12}=\frac{E}{2\left(1+v\right)}\left(\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \theta }+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \epsilon }\right)\text{;}\end{array}$

for Asymptotics II

(5.1.16) $\begin{array}{l}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }+\frac{1}{A}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }=0,\\ {\mathrm{\upsilon }}_3=0,\\ \frac{1}{R}\left(\frac{1}{A}\frac{\partial {\sigma }_{22}}{\partial \theta }+\frac{\partial {\sigma }_{12}}{\partial \xi }\right)+\frac{\partial {\sigma }_{32}}{\partial {\alpha }_3}+2k{\sigma }_{12}-\rho \frac{{\partial }^2{\mathrm{\upsilon }}_2}{\partial t^2}=0,\\ {\partial }_{11}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^{2}R}\left(\frac{v}{1-v}\frac{1}{A}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }+\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }\right),\\ {\partial }_{22}=\frac{E}{2\left(1+v\right){\mathrm{\varkappa }}^{2}R}\left(\frac{v}{1-v}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }+\frac{1}{A}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }\right),\\ {\partial }_{33}=\frac{Ev}{2\left(1-v^2\right){\mathrm{\varkappa }}^{2}R}\left(\frac{\partial {\mathrm{\upsilon }}_1}{\partial \xi }+\frac{1}{A}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \theta }\right),\\ {\sigma }_{31}=\frac{E}{2\left(1+v\right)}\frac{\partial {\mathrm{\upsilon }}_1}{\partial {\alpha }_3},\\ {\partial }_{32}=\frac{E}{2\left(1+v\right)}\frac{\partial {\mathrm{\upsilon }}_2}{\partial {\alpha }_3},\\ {\sigma }_{12}=\frac{E}{2\left(1+v\right)}\left(\frac{1}{AR}\frac{\partial {\mathrm{\upsilon }}_1}{\partial \theta }+\frac{1}{R}\frac{\partial {\mathrm{\upsilon }}_2}{\partial \xi }-k_{{\mathrm{\upsilon }}_2}\right)\text{.}\end{array}$

We shall refer to equations(5.1.15) as those of the quasi-plane and to equations(5.1.16) as those of the quasi-antiplane problem of elasticity. Similarly to the solutions to the equations of the plane and antiplane problems of elasticity the solutions to these equations have to satisfy two and one boundary conditions, respectively. As a result of ignoring asymptotically secondary stresses and displacements, the error of the order O(η ^{1   − p} ) appears in the boundary conditions on the edges and faces.

Equations(5.1.15), (5.1.16), as well as equations (4.1.5), (4.1.6), are applicable for describing not only short-wave vibrations but also long-wave ones provided that

(5.1.17) $q>max\left(q_0,p\right)\text{,}$

where

(5.1.18) $\begin{array}{ll}{q}_0=0\hfill & \begin{array}{l}(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{the}\mathrm{leading}\mathrm{tangential}\mathrm{low}‐\mathrm{frequency}\mathrm{and}\\ \mathrm{long}‐\mathrm{wave}\mathrm{high}‐\mathrm{frequency}\mathrm{approximations}),\end{array}\hfill \\ q_0=2/3\hfill & \begin{array}{l}(\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{the}\mathrm{leading}\mathrm{transverse}\mathrm{low}‐\mathrm{frequency}\\ \mathrm{approximation})\text{.}\end{array}\hfill \end{array}$

Comparing inequality (5.1.17) at q ₀  =   0 with inequality (4.3.3) we conclude that the range of applicability of equations(5.1.15) of the quasi-plane and (5.1.16) of the quasi-antiplane problems of elasticity, on the one hand, and that of equations (2.3.5) of the tangential low-frequency and (3.1.13), (3.2.7) of the long-wave high-frequency approximations, on the other hand, overlap in the region

(5.1.19) $p<q<2/3\text{.}$

The overlap region is non-empty only if the variability index along a parallel obeys the condition

(5.1.20) $p<2/3$

As for the general case considered in Section 4.3, the range of applicability of equations(5.1.15) does not overlap with that of equations (2.3.8) of the transverse low-frequency approximation. To provide overlapping, we insert the aggregate

$-\left({\beta }_{11}/R_1+{\beta }_{22}/R_2\right)$

into the left-hand side of equation(5.1.15) ₃ assuming that the quantities β ₁₁ and β ₂₂ obey equations (4.3.11). (Such a substitution has been suggested for a body of arbitrary shape in Section 4.3.) In the case of a body of revolution these equations can be considerably simplified. Let us introduce in equations (4.3.11) the dimensionless variables

(5.1.21) ${\alpha }_1=R{\eta }^q{\gamma }_1,{\alpha }_2=R{\eta }^p{\gamma }_2,\left(q>p\right)\text{.}$

Inspection of the equations obtained when q  > p leads to the following scaling for the unknowns of equations (4.3.11)

(5.1.22) $\begin{array}{l}{\beta }_{11}=E{\eta }^{q+d}{\gamma }_{11},{\beta }_{22}=E{\eta }^q{\gamma }_{22,}\\ {\beta }_{11}=E{\eta }^{2q-p}{\gamma }_{12},w_1=R{\eta }^{2q}{\psi }_1,\\ w_2=R{\eta }^{3q-p}{\psi }_2,{\upsilon }_3=R{\eta }^q{u}_3\end{array}$

where r ₁₁, r ₂₂, r ₁₂, ψ ₁, ψ ₂, u ₃ are of the same order. Further we need only equation (4.3.11) ₂. To within the error O(η^d ) it becomes

(5.1.23) $\begin{array}{l}\frac{\partial {\psi }_1}{\partial {\gamma }_1}+u_3\left(\frac{1}{R_1^{*}}+\frac{v}{R_2^{*}}\right)=0,\\ r_{22}=\frac{1}{1-v^2}\left[u_3\left(\frac{v}{R_1^{*}}+\frac{1}{R_2^{*}}\right)+v\frac{\partial {\psi }_1}{\partial {\gamma }_1}\right]\text{.}\end{array}$

Taking into account (5.1.22) and the estimate β ₁₁  ~ η^dβ ₂₂ we express the aggregate in terms of υ₃

(5.1.24) $-\left(\frac{{\beta }_{11}}{R_1}+\frac{{\beta }_{22}}{R_2}\right)=-\frac{E{\mathrm{\upsilon }}_3}{R_2^2}\left[1+O\left({\eta }^d+{\eta }^{2-2q}\right)\right]\text{.}$

Thus, introducing the above aggregate into equations(5.1.15) ₃ results in the formal substitution

(5.1.25) $\rho \frac{{\partial }^2{\mathrm{\upsilon }}_3}{\partial t^2}\to \left(\rho \frac{{\partial }^2}{\partial t^2}+\frac{E}{R_2^2}\right){\mathrm{\upsilon }}_3$

in this equation. The relative asymptotic order of aggregate (5.1.24) in the 3D equations of elasticity is equal to O(η ^{4q  −   2}). Therefore, the relative order of the O-term from formula(5.1.24) in these equations is O(η ^2q + η ^{5q  −   2} + η ^{6q  −   2   −   2p} ). Substituting the latter into (4.2.9) leads to the inequality

(5.1.26) $q>max\left(\frac{2+2p}{5},\frac{1}{2}\right)$

corresponding to the region of applicability of the refined equations of the quasi-plane problem of elasticity for describing transverse low-frequency vibrations.

The overlap region for the equations of the transverse low-frequency approximation (2.3.8) and equations(5.1.15) refined in accordance with (5.1.25) is of the form

(5.1.27) $max\left(\frac{2+2p}{5},\frac{1}{2}\right)<q<\frac{2}{3}\text{.}$

This interval is not empty when inequality (5.1.20) is valid.

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INTERACTION OF ELECTRONS WITH PHOTONS

V.B. BERESTETSKII , ... L.P. PITAEVSKII , in Quantum Electrodynamics (Second Edition), 1982

§ 95 Exact theory of pair production in the ultra-relativistic case

In §§93 and 94 we have discussed bremsstrahlung and pair production by a photon in the relativistic case, using the Born approximation, for which the condition Zα ≪ 1 must always be satisfied. In §§95 and 96 we shall describe a theory of these processes which is not subject to the limitation just mentioned, i.e. is valid even if Zα ˜˜ 1 (H. A. Bethe and L. C. Maximon, 1954). We shall assume that both the particles (the initial and final electrons, or the constituents of the pair) are ultra-relativistic, with energy ɛ m.

We have seen that in the ultra-relativistic case both particles move at small angles (θ, θ′ or θ₊, θ₋) to the direction of the photon: θ ≲ m/ɛ. This property is preserved in the exact (with respect to Zα) theory, and we shall therefore consider just this range of angles.

The momentum transfer to the nucleus in this range is q ˜ m. This means that in the wave functions the important values of the impact parameter are ρ ˜ 1/q ˜ 1/m, i.e. "large" distances. At such distances the wave function derived in §39 can be used. The calculations for pair production are as follows.

The pair production cross-section is similar in form to the photoelectric effect cross-section (cf. (56.1), (56.2)):

(95.1) $d\sigma =2\pi |e\surd \left(4\pi \right)\frac{1}{\surd \left(2\omega \right)}{M}_{fi}{|}^2\delta \left(\omega -{\epsilon }_{+}-{\epsilon }_{-}\right)\frac{d^3{p}_{+}{d}^3{p}_{-}}{{\left(2\pi \right)}^6},$

where

(95.2) $M_{fi}={\displaystyle \int {\psi }_{t_{-}.p_{-}}^{(-)*}\left(\alpha ·e\right)e^{ik·r}{\psi }_{-t_{+}.-p_{+}}^{(+)}}{d}^{3}x.$

Here ψ_{ɛ−,p −} ^(-) is the wave function of the electron, and ψ_{−ɛ+,-p +} ⁽⁺⁾ the wave function with negative energy -ɛ₊ and momentum -p ₊.

The function ψ_{ɛ−,p −} ^(-) which pertains to a particle in the final state, must have an asymptotic form which includes (besides the plane wave) an ingoing spherical wave; this is indicated by the superscript (-). According to (39.10), this wave function is ^†

(95.3) $\begin{array}{l}{\psi }_{{\epsilon }_{-},p_{-}}^{(-)}=\frac{C^{(-)}}{\sqrt{\left(2{\epsilon }_{-}\right)}}{e}^{i{p}_{-}·r}\left(1-\frac{i\alpha ·\nabla }{2{\epsilon }_{-}}\right)F\left(-iv,1,-i\left(p_{-}r+{\text{p}}_{-}·\text{r}\right)\right)u\left(p_{-}\right),\\ \hfill C^{(-)}=e^{\pi v/2}\Gamma \left(1+iv\right),v=Z\alpha .\hfill \end{array}\}$

The function ψ_{−ɛ+,-p +} ⁽⁺⁾ must have an asymptotic form which includes an outgoing spherical wave (indicated by the superscript (+)), since it denotes the wave function of an "initial state with negative energy". The asymptotic form of the wave function of the positron, obtained from ψ_{−ɛ+,-p +} ⁽⁺⁾, then has an ingoing wave, as is correct for a final particle. According to (39.11), this function is

(95.4) $\begin{array}{l}{\psi }_{-t_{+}.-p_{+}}^{(+)}=\frac{C^{(+)}}{\surd \left(2{\epsilon }_{+}\right)}{e}^{-i{\text{p}}_{+}.\text{r}}\left(1+\frac{i\alpha ·\nabla }{2{\epsilon }_{+}}\right)F\left(-iv,1,-i\left(p_{+}r+{\text{p}}_{+}·\text{r}\right)\right)u\left(-p_{+}\right),\\ \hfill C^{(+)}=e^{-}{}^{\pi v/2}\Gamma \left(1+iv\right).\hfill \end{array}\}$

The terms ˜1/ɛ in (95.3) and (95.4) have to be included because of the matrix structure of M _fi (95.2). The matrix element (α) _fi is a vector whose direction is close to that of k. The leading term in (α ˙ e) _fi is therefore small, and the correction terms are of the same order of magnitude as that term.

Substituting (95.3) and (95.4) in (95.2) and neglecting terms ˜1/ɛ₊ ɛ₋, we find

(95.5) $M_{fi}=\frac{N}{2\surd \left({\epsilon }_{+}{\epsilon }_{-}\right)}u*\left(p_{-}\right)\left\{\left(e·\alpha \right)\text{I}+\left(e·\alpha \right)\left(\alpha ·{\text{I}}_{+}\right)+\left(\alpha ·{\text{I}}_{-}\right)\left(e·\alpha \right)\right\}u\left(-p_{+}\right),$

where

(95.6) $N=C^{(+)}{C}^{(-)}=\pi v/\sinh \pi v,$

(95.7) $\begin{array}{l}\text{I}={\displaystyle \int e^{-\text{I}\text{q}·\tau }{F}_{-}^{*}{F}_{+}{d}^{3}x}\hfill \\ {\text{I}}_{+}=\frac{\text{I}}{2{\epsilon }_{+}}{\displaystyle \int e^{-\text{I}\text{q}·r}{F}_{-}^{*}\nabla F_{+}{d}^{3}x,}\\ {\text{I}}_{-}=\frac{\text{I}}{2{\epsilon }_{-}}{\displaystyle \int e^{-\text{I}\text{q}·r}\left(\nabla F_{-}\right)*F_{+}{d}^{3}x}\\ \text{q}={\text{p}}_{+}+{\text{p}}_{-}-\text{k}:\end{array}\}$

F ₋ and F ₊ are used for brevity to denote the hypergeometric functions which appear in (95.3) and (95.4). The integrals I, I ₊, I ₋ satisfy one identical relation: from

${\displaystyle \int \nabla \left(e^{-iq·r}{F}_{-}^{*}{F}_{+}\right)d^{3}x=0,}$

we have

(95.8) $q\text{I}+2{\epsilon }_{+}{\text{I}}_{+}+2{\epsilon }_{-}{\text{I}}_{-}=0.$

We average |M _fi |² over polarizations of the incident photon, and sum over directions of the electron and positron spins. ^† This is done by the tensor substitution

$e_i{e}_k^{*}\to \frac{1}{2}\left({\delta }_{ik}-n_i{n}_k\right),\text{n}=\text{k}/\omega ,$

and changing the bispinor products according to

$u_{\pm }{\overline{u}}_{\pm }\to 2{\rho }_{\pm }=\left({\epsilon }_{\pm }{\gamma }^0-{\text{p}}_{\pm }·{\gamma }^{\mp }m\right).$

Putting also α = γ⁰γ, we find

$\begin{array}{l}|M_{fi}{|}^2\to \left(N^2/2{\epsilon }_{+}{\epsilon }_{-}\right)\left\{\text{t}\text{r}{\rho }_{-}\text{Q}·{\rho }_{+}\overline{\text{Q}}-\text{tr}{\rho }_{-}\left(\text{n}·\text{Q}\right){\rho }_{+}\left(\text{n}·\overline{\text{Q}}\right)\right\},\\ \hfill \text{Q}=\gamma \text{I}-{\gamma }^0\gamma \left(\gamma ·{\text{I}}_{+}\right)-{\gamma }^0\left(\gamma ·{\text{I}}_{-}\right)\gamma ,\hfill \\ \hfill \overline{\text{Q}}=\gamma \text{I}*-{\gamma }^0\left(\gamma ·{\text{I}}_{+}^{*}\right)\gamma -{\gamma }^0\gamma \left(\gamma ·{\text{I}}_{-}^{*}\right).\hfill \end{array}$

The final result, obtained after making the appropriate approximations, for the ultra-relativistic case at small angles

(95.9) ${\theta }_{\pm }\sim m/\epsilon \ll 1,$

will be given here. We define the auxiliary vectors

(95.10) ${\delta }_{\pm }=\frac{1}{m}{\left(p_{\pm }\right)}_{\perp },{\delta }_{\pm }=\frac{{\epsilon }_{\pm }}{m}{\theta }_{\pm },$

where the suffix ⊥ denotes the component perpendicular to the direction of k. Then

(95.11) $|M_{fi}{|}^2\to \frac{1}{4}{N}^2\left\{\frac{m^2{\omega }^2}{2{\epsilon }_{+}^2{\epsilon }_{-}^2}|\text{I}{|}^2+2|I\frac{m{\delta }_{+}}{2{\epsilon }_{+}}+I_{+}{|}^2+2|I\frac{m{\delta }_{-}}{2{\epsilon }_{-}}+{\text{I}}_{-}{|}^2\right\},$

where we have used the fact that I ˜ ɛI _±/q ˜ ɛI/m (as is seen from (95.8)), and terms of higher order in m/ɛ are omitted.

The integrals I _± may be expressed as

(95.12) $\begin{array}{l}{I}_{\pm }=i\frac{p_{\pm }}{2{\epsilon }_{\pm }}\frac{\partial J}{\partial p_{\pm }},\\ J={\displaystyle \int \frac{e^{-iq·r}}{r}F\left(-iv,1,i\left(p_{+}r=p_{+}·r\right)\right)F\left(iv,1,i\left(p_{-}r=p_{-}·r\right)\right)d^{3}x}.\end{array}$

The integral J can be written in terms of the complete hypergeometric function: ^†

(95.13) $\begin{array}{l}J=\frac{4\pi }{q^2}{\left(\frac{q^2-2{\text{p}}_{+}·\text{q}}{{\text{q}}^2-2{\text{p}}_{-}·\text{q}}\right)}^{iv}F\left(-iv,iv,1,z\right),\\ z=2\frac{q^2\left({\text{p}}_{+}{\text{p}}_{-}-{\text{p}}_{+}·{\text{p}}_{-}\right)+2\left({\text{p}}_{+}·\text{q}\right)\left({\text{p}}_{-}·\text{q}\right)}{\left({\text{q}}^2-2{\text{p}}_{+}·\text{q}\right)\left({\text{q}}^2-2{\text{p}}_{-}·\text{q}\right)}.\end{array}\}$

The differentiation with respect to p _± must be carried out with q fixed, only thereafter putting q = p₊ + p₋ - k. The result, after making the approximations corresponding to the ultra-relativistic case and the conditions (95.9), is

(95.14) $I_{\pm }=\frac{4\pi }{q^2}\frac{{\epsilon }_{\mp }}{m^2\omega }{\left(\frac{{\epsilon }_{+}{\xi }_{+}}{{\epsilon }_{-}{\xi }_{-}}\right)}^{iv}\left\{\pm vq{\xi }_{\mp }F\left(z\right)+i\frac{q^2}{m^2}F\text{'}\left(z\right)\left(\text{q}{\xi }_{\mp }-m{\delta }_{\pm }\right)\right\},$

with, for brevity, the notation

(95.15) $\begin{array}{l}{\xi }_{\pm }=\frac{1}{1+{\delta }_{\pm }^2},z=1-\frac{q^2}{m^2}{\xi }_{+}{\xi }_{-},\\ F\left(z\right)=F\left(-iv,iv,1,z\right),\end{array}\}$

F(z) being a real function. The integral I is then found immediately from (95.8).

Substituting the values of the integrals in (95.11) and thence in (95.1), we find the required cross-section:

(95.16) $\begin{array}{ll}d\sigma & =\frac{4}{\pi }{\left(\frac{\pi v}{\sinh \pi v}\right)}^2{Z}^2\alpha r_e^2\frac{m^4}{q^4{\omega }^3}{\delta }_{+}d{\delta }_{+}·{\delta }_{-}d{\delta }_{-}.d\varphi d{\epsilon }_{+}\times \\ & \times \{F^2\left(z\right)[-2{\epsilon }_{+}{\epsilon }_{-}\left({\delta }_{+}^2{\xi }_{+}^2+{\delta }_{-}^2{\xi }_{-}^2\right)+{\omega }^2\left({\delta }_{+}^2+{\delta }_{-}^2\right){\xi }_{+}{\xi }_{-}+\\ & +2\left({\epsilon }_{+}^2+{\epsilon }_{-}^2\right){\delta }_{+}{\delta }_{-}{\xi }_{+}{\xi }_{-}\cos \varphi ]+\\ & +\frac{q^4}{m^4}\frac{{\xi }_{+}^2{\xi }_{-}^2}{v^2}F{\text{'}}^2\left(z\right)[-2{\epsilon }_{+}{\epsilon }_{-}\left({\delta }_{+}^2{\xi }_{+}^2+{\delta }_{-}^2{\xi }_{-}^2\right)+{\omega }^2\left(1+{\delta }_{+}^2{\xi }_{-}^2\right){\xi }_{+}{\xi }_{-}-\\ & -2\left({\epsilon }_{+}^2+{\epsilon }_{-}^2\right){\delta }_{+}{\delta }_{-}{\xi }_{+}{\xi }_{-}\cos \varphi ]\}.\end{array}$

When v → 0,

$\frac{\pi v}{\sinh \pi v}\to 1,F\left(z\right)\to 1,F\text{'}\left(z\right)\approx v^2\to 0.$

The expression (95.16) then reduces, as it should, to Bethe and Heitler's formula (94.3), which corresponds to the Born approximation. It also reduces to this formula for any v if the angles of emission of the pair satisfy the conditions

$|{\delta }_{+}-{\delta }_{-}|\ll 1,|\pi -\varphi |\ll 1.$

For then q ≪ m, so that the second term in the braces in (95.16) can be omitted because of the extra factor (q/m)⁴ as compared with the first term, and in the first term we have (since 1 - z ˜ q ²/m ² ≪ 1) ^†

(95.17) $\begin{array}{ll}F\left(z\right)\to F\left(1\right)& \equiv F\left(-iv,iv1,1\right)\\ & =\frac{1}{\Gamma \left(1-iv\right)\gamma \left(1+iv\right)}\\ & =\frac{\sinh \pi v}{\pi v},\end{array}$

so that the similar factor in front of the braces is cancelled.

Let us now consider the integration of the cross-section over the directions of emission of the pair. The integration over angles is divided into two regions I and II, in which we have respectively

$\left(I\right)1-z>1-z_1,\left(II\right)1-z<1-z_1,$

where z ₁ is a certain value such that 1 ≫ 1 - z ₁ ≫ (m/≫;)². Since in region II 1 - z ≪ 1, q ² ≪ m ², it follows from the above discussion that in this region dσ ˜ dσ _B ≡ dσ _v=0, where dσ _B is the cross-section in the Born approximation. The integral over angles is therefore

(95.18) $\begin{array}{ll}d{\sigma }_{\epsilon +}\equiv {\displaystyle \int d\sigma }& ={\displaystyle \underset{1}{\int }d\sigma +}{\displaystyle \underset{II}{\int }d{\sigma }_{v\to 0}}\\ & ={\left(d{\sigma }_{\epsilon +}\right)}_B+{\displaystyle \underset{1}{\int }\left(d\sigma -d{\sigma }_{v\to 0}\right),}\end{array}$

where (dσ_≪+) _B is the Born cross-section (94.5) integrated over angles.

In region I we have

$q^2/m^2\approx {\delta }_{+}^2+{\delta }_{-}^2+2{\delta }_{+}{\delta }_{-}\cos \varphi .$

We shall change from the variables δ₊, δ₋, ϕ to ξ₊, ξ₋, z. A direct calculation of the Jacobian for this transformation gives

$\delta +d{\delta }_{+}\cdot {\delta }_{-}d{\delta }_{-}\cdot d\varphi \to \frac{\varepsilon +\varepsilon }{8m^2}\frac{d\xi +d\xi -d\varphi }{{(\xi +\xi )}^{3}sin\varphi },$

where

$\begin{array}{ll}1-z& =\left(q^2/m^2\right){\xi }_{+}{\xi }_{-}\\ & ={\xi }_{+}+{\xi }_{-}-2{\xi }_{+}{\xi }_{-}+2\surd \left[{\xi }_{+}{\xi }_{-}\left(1-{\xi }_{+}\right)\left(1-{\xi }_{+}\right)\right]\cos \varphi .\end{array}$

Expressing sin ϕ and cos ϕ in terms of the other quantities by means of this equation and substituting in (95.16), we obtain after some simple algebra

$\begin{array}{l}d\sigma =Ad{\epsilon }_{+}\frac{2d{\xi }_{+}d{\xi }_{-}dz}{{\left[z\left(1-z\right)-\left(1-z\right){\left({\xi }_{+}+{\xi }_{-}-1\right)}^2-z{\left({\xi }_{+}-{\xi }_{-}\right)}^2\right]}^{1/2}}\times \\ \times \{\frac{F^2\left(z\right)}{{\left(1-z\right)}^2}\left[\left({\epsilon }_{+}^2{\epsilon }_{-}^2\right)\left(1-z\right)+2{\epsilon }_{+}{\epsilon }_{-}{\left({\xi }_{+}-{\xi }_{-}\right)}^2\right]+\\ +\frac{F{\text{'}}^2\left(z\right)}{v^2}\left[\left({\epsilon }_{+}^2+{\epsilon }_{-}^2\right)z+2{\epsilon }_{+}{\epsilon }_{-}{\left({\xi }_{+}+{\xi }_{-}-1\right)}^2\right]\},\\ A={\left(\frac{\pi v}{\sinh \pi v}\right)}^2\frac{Z^2\alpha r_e^2}{2\pi {\omega }^3}.\end{array}$

Finally, we replace ξ₊ and ξ₋ in terms of new "spherical" variables χ and ψ

$\begin{array}{l}{\xi }_{+}+{\xi }_{-}-1=\surd z\sin \chi \cos \psi ,\\ {\xi }_{+}-{\xi }_{-}=\surd \left(1-z\right)\sin \chi \sin \psi ,\hfill \\ \hfill 0\le \chi \le \frac{1}{2}\pi ,0\le \psi \le 2\pi ,\hfill \\ 2d{\xi }_{+}d{\xi }_{-}\to \surd \left[z\left(1-z\right)\right]\sin \chi \cos \chi d\chi d\psi .\hfill \end{array}$

These ranges of variation of χ and ψ correspond to the range 0 to 1 for ξ₊ and ξ₋, i.e. to the range 0 to ∞ for δ₊ and δ₋ (or, equivalently, θ₊ and θ₋); the rapid convergence of the integral allows the range of variation of the angles to be extended in this way. After the transformation, the root in the denominator becomes √[z(1 - z)] cos χ; the integration over χ and ψ is elementary, and the result is

$d\sigma =2\text{A}·2\pi dz\left({\epsilon }_{+}^2+{\epsilon }_{-}^2+\frac{2}{3}{\epsilon }_{+}{\epsilon }_{-}\right)\left[\frac{F^2\left(z\right)}{1-z}+\frac{z}{v^2}F{\text{'}}^2\left(z\right)\right]d{\epsilon }_{+}.$

An extra factor 2 has been included because the integration over z is to be taken from 0 to z ₁, whereas, when the azimuth ϕ varies from 0 to π and from π to π, each value of z occurs twice.

The integration over z is effected by means of formula (92.14), which, for v′ = - v (and F(z) accordingly real), becomes

$\frac{F^2}{1-z^2}+\frac{z}{v^2}F{\text{'}}^2=\frac{1}{v^2}\frac{d}{dz}\left(zFF\text{'}\right).$

The integral of this expression is z ₁ F(z ₁)F′(z ₁)/v ². The value of z ₁ F(z ₁) ˜ F(1) is taken from (95.17), and the limit of F′(z ₁ → 1) is given by ^†

$\begin{array}{ll}\frac{1}{v^2}F\text{'}\left(z\right)& =F\left(1-iv,1+iv,2,z\right)\\ & \approx -\left[\log \left(1-z\right)+2f\left(v\right)\right]\frac{\sinh \pi v}{\pi v},\end{array}$

where

(95.19) $\begin{array}{ll}f\left(v\right)& =\frac{1}{2}\left[\Psi \left(1+iv\right)+\Psi \left(1-iv\right)-2\Psi \left(1\right)\right]\\ & =v^2{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n\left(n^2+v^2\right)},}\\ \Psi \left(z\right)& =\Gamma \text{'}\left(z\right)/\Gamma \left(z\right).\end{array}$

Substituting the above expressions in (95.18), we obtain as the final formula

(95.20) $d{\sigma }_{r+}=4Z^2\alpha r_e^2\left({\epsilon }_{+}^2+{\epsilon }_{-}^2+\frac{2}{3}{\epsilon }_{+}{\epsilon }_{-}\right)\left[\log \frac{2{\epsilon }_{+}{\epsilon }_{-}}{m\omega }-\frac{1}{2}-f\left(\alpha Z\right)\right]\frac{d{\epsilon }_{+}}{{\omega }^3}.$

The total cross-section for pair production by a photon with energy ω is

(95.21) $\sigma =\frac{28}{9}{Z}^2\alpha r_e^2\left[\log \frac{2\omega }{m}-\frac{109}{42}-f\left(\alpha Z\right)\right].$

We see that the only change in these formulae is that a universal function f(αZ) of the atomic number is subtracted from the logarithm. Figure 18 shows a graph of this function. For v ≪ 1, f(v) ˜ 1.2v ¹.

Fig. 18..

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INSTABILITY THEORY

E.M. LIFSHITZ , L.P. , in Physical Kinetics, 1981

§61 Beam instability

According to the results of §34, the amplitude of a perturbation with wave vector k in a homogeneous unbounded medium has the asymptotic form (as t → ∞)

(61.1) $e^{-i\omega \left(\mathbf{k}\right)t},$

where ω(k) is the frequency of waves propagating in the medium. In particular, for longitudinal waves in a plasma the frequencies ω(k) are the roots of the equation†

(61.2) ${\in }_1(\omega ,\text{k})=0.$

The frequencies ω( k ) are in general complex. If the imaginary part im ω ≡ − γ < 0, the perturbation is damped in the course of time. If, however, γ < 0 in some range of k, such perturbations grow: the medium is unstable with respect to oscillations in that range of wavelengths, and |γ| is then called the instability growth rate. We should emphasize immediately that, in referring to an "unlimited" increase of the perturbation, as exp(|γ|t), we are considering here and subsequently only the behaviour in the linear approximation. In reality, of course, the increase is limited by non-linear effects.

In a collisionless plasma, the imaginary part of the frequency is due to Landau damping. The thermodynamic equilibrium state of the plasma, corresponding to the absolute maximum of the entropy, is stable with regard to any perturbation. However, it has already been noted in §30 that, for non-equilibrium distributions in plasmas, the absorption of energy of the oscillations may be replaced by amplification. This is shown by the appearance of a range of values of the independent variables k and ω (ω > 0) in which the imaginary part of the permittivity is negative: ∈″ _l (ω, k) < 0. It must be emphasized, however, that the existence of such ranges does not in itself necessarily signify that the plasma is unstable (at least in the linear approximation); some branch of the plasma oscillation spectrum must also actually fall in this range.

A typical instance of instability is afforded by a directed beam of electrons passing through a plasma at rest (A. I. Akhiezer and Ya. B. Fa

nberg 1949, D. Bohm and E. P. Gross 1949). The beam is assumed to be electrically compensated: the sum of the electron charge densities in the plasma and the beam is equal to the ion charge density in the plasma. The system is homogeneous and unbounded, i.e. both the beam and the plasma extend throughout space, and the directed velocity V of the beam is everywhere the same. We shall assume that V is non-relativistic.

Let us first suppose that both the beam and the plasma are cold, i.e. that the thermal motion of their particles is negligible. The necessary condition for this will be ascertained later.

In the electron oscillation frequency range, the longitudinal permittivity of the plasma-beam system has the form

(61.3) ${\in }_1(\omega ,\text{k})-1=-\frac{{\Omega }_e^2}{{\omega }^2}-\frac{{\Omega }_e^{,2}}{{(\omega -\text{k}\cdot \text{V)}}^2}.$

The first term on the right corresponds to the plasma at rest; Ω _e = (4πe ² N _e /m)^1/2 is the corresponding electron plasma frequency. The second term is due to the beam electrons. In a frame of reference K′ moving with the beam, the contribution of the beam electrons to ∈ _l − 1 is −(Ω _e ′/ω′)², where ω′ is the oscillation frequency in that frame, and Ω _e ′ = (4πe ² N _e ′/m)^1/2 (N _e ′ being the electron density in the beam). On return to the original frame K, the frequency ω′ is replaced by

(61.4) ${\omega }^{\prime }=\omega -\text{k}\cdot \text{V,}$

and we have (61.3).†

We shall assume the beam density to be small, in the sense that

(61.5) $N_e^{\prime }\ll N_e,$

and so Ω _e ′ ≪ Ω _e . Then the presence of the beam changes only slightly the principal branch of the spectrum of longitudinal oscillations of the plasma, i.e. the root of the dispersion relation ∈ _l = 0 for which ω ≈ Ω _e . As well as this branch, however, another branch appears, on account of the presence of the beam, which we have now to consider.

In order that the term with the small numerator Ω _e ′² should not disappear from the dispersion relation

(61.6) $\frac{{\Omega }_e^2}{{\omega }^2}+\frac{{\Omega }_e^{\prime 2}}{{(\omega -\text{k}\cdot \text{V})}^2}=1,$

this smallness must be compensated by that of the denominator. We therefore seek the solution in the form ω = k · V + δ, where δ is small. The equation then becomes

(61.7) $\frac{{\Omega }_e^2}{{(\text{k}\cdot \text{V})}^2}+\frac{{\Omega }_e^{\prime 2}}{{\delta }^2}=1,$

whence

(61.8) $\delta =\pm \frac{\Omega e^{\prime }}{{[1-{({\Omega }_e\text{/k}\cdot \text{V})}^2]}^{1/2}};$

the condition δ < k · V requires that |k · V| should not be too close to Ω _e . The assumption that the plasma is cold implies that kv _Te ≪ ω, and in the present case therefore that v _Te ≪ V: the speed of the beam is much greater than the thermal speed of the plasma electrons.

If (k · V)² > Ω _e ², then both roots (61.8) are real, and the oscillations do not grow. If, however,

(61.9) ${(\text{k}\cdot \text{V})}^2<{\Omega }_e^2,$

the two values of δ are imaginary, and the one for which im ω = im δ > 0 corresponds to growing oscillations. The system is thus unstable with respect to oscillations having sufficiently small values of k · V.

A different situation occurs when the thermal motion of the electrons is taken into account. In the general case, we have in place of (61.3)

(61.10) ${\in }_l(\omega ,\text{k})={\in }_l^{\left(pl\right)}(\omega ,\text{k})-{\Omega }_e^{\prime 2}/{(\omega -\text{k}\cdot \text{V})}^2,$

where ∈ _l ^(pl) pertains to the plasma in the absence of the beam. Solving the equation ∈ _l = 0 by the same method, we now find

(61.11) $\delta =\pm {\Omega }_e\prime /{\left[{\in }_l^{\text{(pl)}}\right(\text{k}\cdot \text{V,k}\left)\right]}^{1/2}.$

Because of the Landau damping, ∈ _l ^(pl) always has an imaginary part (for any k). Consequently, δ is always complex, and by virtue of the double sign in (61.11) im δ > 0 for one branch of the oscillations, i.e. these are unstable. For large V, corresponding to the cold-plasma case discussed above, the part of im ∈ _l due to the Landau damping becomes exponentially small, and we return to (61.8).

In the above analysis, the thermal spread of electron speeds in the beam has been neglected. This is justifiable if the amount of it

(61.12) $v_{Te}^{\prime }\ll \left|\delta \right|/\mathbf{k}.$

PROBLEMS

PROBLEM 1.

Determine the boundary of the beam instability region in a cold plasma for values of k · V close to Ω _e .

SOLUTION.

For small values of (k · V)² − Ω _e ², (61.7) is insufficiently accurate. Retaining the term of the next order in δ in the equation ∈ _l = 0, with ∈ _l from (61.3), we obtain

$\frac{{\Omega }_e^{\prime 2}}{{\delta }^2}\approx 1-\frac{{\Omega }_e^2}{{(\text{k}\cdot \text{V})}^2}+\frac{2{\Omega }_e^2\delta }{{(\text{k}\cdot \text{V})}^3}\approx \frac{2(\text{k}\cdot \text{V}-{\Omega }_e)}{{\Omega }_e}+\frac{2\delta }{{\Omega }_e}.$

With new variables ξ and τ defined by

$\begin{array}{cc}\delta =\xi {\left(\frac{1}{2}{\Omega }_e^{\prime 2}{\Omega }_e\right)}^{1/3},& \tau ={(2/{\Omega }_e^{\prime 2}{\Omega }_e)}^{1/3}(\text{k}\cdot \text{V}-{\Omega }_e),\end{array}$

we reduce this equation to

(1) ${\xi }^3+\tau {\xi }^2=1$

(taking the particular case where k · V is close to + Ω _e , not to − Ω _e ). All three roots of equation (1) are real if τ > 3.2^−2/3, and this determines the instability region. Two of them correspond to the two roots of (61.6), and the other corresponds to the oscillation frequency of the plasma at rest, which is close to them if Ω _e ≈ k · V.

PROBLEM 2.

Investigate the stability of ion-sound waves in a two-temperature plasma (T _e ≫ T _i ) in which the electron component moves relative to the ion component with a macroscopic velocity V, and V ≪ v _Te .

SOLUTION.

With the condition V ≪ v _Te , the directed motion of the electrons has little effect on the dispersion relation for the ion-sound waves, which is again given by (33.4) :

(2) $\frac{\omega }{\mathbf{k}}={\left(\frac{z{T}_e}{M}\right)}^{1/2}\frac{1}{{(1+{\mathbf{k}}^2{a}_e^2)}^{1/2}}.$

The electronic part of the damping rate is found from (33.6) by the change (61.4) :

(3) $\gamma =(\text{k}\cdot \text{V}-\omega ){\left(\pi zm\text{/}8M\right)}^{1/2}.$

The instability condition is k · V > ω; for this to be so, we must always have V > ω/k. Near the instability limit, the factor k · V − ω in (3) is small, and it may then be necessary to take account in γ of the ionic part of the damping, which in ordinary conditions is small.

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Advances in Atomic, Molecular, and Optical Physics

Robin Shakeshaft , in Advances In Atomic, Molecular, and Optical Physics, 2012

Abstract

The rate at which a system undergoes a transition to (or within) the continuum can be expressed in terms of the resolvent without knowing the detailed asymptotic form of the final-state wavefunction; only a crude approximation to this wavefunction, one which satisfies a simple constraint, is needed. For this and other reasons it is of considerable interest to have a method for efficiently calculating the resolvent, defined as $\mathrm{G}(E)\equiv 1/(E1-\mathrm{H})$ for a system whose Hamiltonian and energy are $\mathrm{H}$ and E, respectively. We develop an approach based on exploiting the underlying time scale, $t_0$ say, which is exposed by writing $\mathrm{G}(E)$ as the limit $\mathcal{T}\to \infty$ of ${\mathrm{G}}^{(\mathcal{T})}(E)\equiv -i{\int }_0^{\mathcal{T}{t}_0}\mathrm{d}t{\mathrm{e}}^{\mathit{iEt}}{\mathrm{e}}^{-i\mathrm{H}t}$ . In reality, only times $\mathcal{T}{t}_0$ up to some finite value, beyond which the probability of a transition is negligible, need be considered. Introducing a complex unit of time $t_{\varphi }\equiv t_0{\mathrm{e}}^{i\varphi }$ , and expanding the time-translation operator ${\mathrm{e}}^{-i\mathrm{H}t}$ in powers of $(t+{\mathit{it}}_{\varphi })/(t-{\mathit{it}}_{\varphi })$ , we show that ${\mathrm{G}}^{(\mathcal{T})}(E)$ can be reexpressed as

(1) $\begin{array}{c}\hfill {\mathrm{G}}^{(\mathcal{T})}(E)=-{\mathit{it}}_{\varphi }{\mathrm{e}}^{-\frac{1}{2}{t}_{\varphi }\mathrm{H}}\left({\mathcal{I}}_0(2{\mathit{Et}}_{\varphi }\text{,}{\mathcal{T}}_{-\varphi })-2t_{\varphi }\mathrm{H}\right.\\ \hfill \left.\times {\displaystyle \sum _{n=1}^{\infty }}\frac{1}{n}{\mathcal{I}}_n(2{\mathit{Et}}_{\varphi }\text{,}{\mathcal{T}}_{-\varphi })L_{n-1}^{(1)}(2t_{\varphi }\mathrm{H})\right){\mathrm{e}}^{-\frac{1}{2}{t}_{\varphi }\mathrm{H}}\text{,}\end{array}$

where ${\mathcal{T}}_{-\varphi }=\mathcal{T}{\mathrm{e}}^{-i\varphi }$ , and where the coefficients ${\mathcal{I}}_n(2{\mathit{Et}}_{\varphi }\text{,}{\mathcal{T}}_{-\varphi })$ satisfy a simple three-term recurrence formula, as do the Laguerre polynomials $L_{n-1}^{(1)}(2t_{\varphi }\mathrm{H})$ . The pre- and post-factors ${\mathrm{e}}^{-\frac{1}{2}{t}_{\varphi }\mathrm{H}}$ serve to cut off the high-energy components of the spectrum if $|\varphi |<\pi /2$ , so $\mathrm{H}$ can be treated as a bounded operator, and therefore can be represented by a finite matrix. The times $t_0$ and $\mathcal{T}{t}_0$ are associated with two branch points of (matrix elements of) ${\mathrm{e}}^{-\mathit{it}\mathrm{H}}$ in the complex-time-plane; $\mathcal{T}{t}_0$ is associated with a branch point at infinity which sets the "arrow of time," and $t_0$ is associated with a branch point at $\sim {\mathit{it}}_0$ which sets the time scale of evolution in the direction of this arrow. The branch structure of the resolvent is embedded in the coefficients ${\mathcal{I}}_n(2{\mathit{Et}}_{\varphi }\text{,}{\mathcal{T}}_{-\varphi })$ . The series can be terminated after roughly $({\mathit{Et}}_0){\mathcal{T}}^2$ terms, and convergence to the limit $\mathcal{T}\to \infty$ can be accelerated by averaging over the oscillations at large values of $\mathcal{T}$ . It is useful to break $\mathrm{H}$ into two parts, the Hamiltonian ${\mathrm{H}}_{\text{c}}$ for free-body motion distorted by a weak Coulomb tail, and a short-range interaction $\mathrm{H}-{\mathrm{H}}_{\text{c}}$ . It is only necessary to evaluate ${\mathrm{G}}_{\text{c}}(E)\equiv 1/(E1-{\mathrm{H}}_{\text{c}})$ since $\mathrm{G}(E)={\left[1-{\mathrm{G}}_{\text{c}}(E)(\mathrm{H}-{\mathrm{H}}_{\text{c}})\right]}^{-1}{\mathrm{G}}_{\text{c}}(E)$ . Using the series representation, and representing ${\mathrm{H}}_{\text{c}}$ on a basis of harmonic oscillator eigenfunctions with a dynamic length scale that is proportional to $\sqrt{\mathcal{T}}$ (to conform with unitarity), ${\mathrm{G}}_{\text{c}}(E)$ can be evaluated efficiently. However, ${\mathrm{G}}_{\text{c}}(E)$ should be re-represented on a set of basis functions that decrease exponentially in r (rather than in $r^2$ ) as r increases since the large-distance behavior of the response functions $〈\overrightarrow{r}|{\mathrm{G}}_{\text{c}}(E)|\psi 〉$ and $〈\overrightarrow{r}|\mathrm{G}(E)|\psi 〉$ is exponential in position space. Moreover, such a basis is more suitable for representing the core interaction $\mathrm{H}-{\mathrm{H}}_{\text{c}}$ , which in general has a Coulomb singularity at the origin. This singularity is not of great physicalimportance but it is numerically significant since it magnifies errors. However, if the basis functions are chosen to be the eigensolutions of a Sturm–Liouville eigenvalue problem tailored to the core interaction, the Coulomb singularity can be effectively removed. The singularity can also be removed if the analytically known Coulomb–Sturmian functions are employed, but additional matrix elements must be evaluated in this case. We illustrate the method using examples of a one-body system, for which exact results are known. In particular, we demonstrate that the rate for photoionization of hydrogen can be estimated to high accuracy using a final-state wavefunction whose asymptotic form is incorrect in all directions but one.

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ELASTIC COLLISIONS

L.D. LANDAU , E.M. LIFSHITZ , in Quantum Mechanics (Third Edition), 1977

§136. The system of wave functions of the continuous spectrum

In the analysis of motion in a centrally symmetric field (Chapter V) we have considered stationary states in which the particle has definite values of the energy, the orbital angular momentum l, and the component m of this angular momentum. The wave functions of such states of the discrete spectrum (ψ _nlm ) and the continuous spectrum (ψ _klm , energy ħ² k ²/2m) together form a complete set in terms of which the wave function of any state may be expanded. Such a set of functions is, however, not appropriate for problems in scattering theory. Here another set is convenient, in which the wave functions of the continuous spectrum are described by a particular asymptotic behaviour: at infinity there is a plane wave and an outgoing spherical wave. In these states the particle has a definite energy, but no definite angular momentum magnitude or component.

According to (123.6) and (123.7), such wave functions, here denoted by ψ _k ⁽⁺⁾, are given by

(136.1) ${\psi }_{\text{k}}^{(+)}=\frac{1}{2k}\sum _{l=0}^{\infty }{i}^l\left(2l+1\right)e^{i\delta }{R}_{kl}\left(r\right)P_l\left(\text{k}·\text{r}/k\text{r}\right).$

The argument of the Legendre polynomials is written as cos θ = k.r/kr, and the expression therefore does not involve any particular choice of the coordinate axes as it did in (123.6) (where the z-axis was the direction of propagation of the plane wave). By giving the vector k all possible values, we obtain a set of wave functions, which, as we shall now show, are orthogonal and normalized by the usual rule for the continuous spectrum:

(136.2) $\int {\psi }_{{\text{k}}^{\prime }}^{+\ast }{\psi }_{\text{k}}^{(+)}\text{d}V={\left(2\pi \right)}^3\delta \left({\text{k}}^{\prime }-\text{k}\right).$

To prove this,† we note that the product ${\psi }_{{\text{k}}^{\text{'}}}^{(+)}\ast {\psi }_{\text{k}}^{(+)}$ is expressed by a double sum over l and l′ of terms containing the products

$P_l\left(\text{k}·r/kr\right)P_{l^{\prime }}\left({\text{k}}^{\prime }·\text{r}/k^{\prime }r\right).$

The integration over the directions of r is effected by means of the formula

(136.3) $\int P_l\left(\text{k}·\text{r}/kr\right)P_l″\left({\text{k}}^{\prime }·r/{\text{k}}^{\prime }r\right)\text{d}o={\delta }_{l{l}^{\prime }}\frac{4\pi }{2l+1}{P}_l\left(\text{k}·{\text{k}}^{\prime }/k{k}^{\prime }\right);$

cf. (c.12) in the Mathematical Appendices. This leaves

$\int {\psi }_{{\text{k}}^{\prime }}^{(+)\ast }{\psi }_{\text{k}}^{(+)}\text{d}V=\frac{\pi }{k{k}^{\prime }}\sum _{l=0}^{\infty }\left(2l+1\right)e^{i\left[{\delta }_l\left(k\right)-{\delta }_l\left(k^{\prime }\right)\right]}{P}_l\left(cos\gamma \right)\underset{0}{\overset{\infty }{\int }}{R}_{k^{\prime }l}\left(r\right)R_{kl}\left(r\right)r^2\text{d}r,$

where γ is the angle between k and k′. The radial functions R_kl are orthogonal, however, and are normalized by

$\underset{0}{\overset{\infty }{\int }}{R}_{k^{\prime }l}{R}_{kl}{r}^2\text{d}r=2\pi \delta \left(k^{\prime }-k\right).$

Hence we can put k = k′ in the coefficients in front of the integrals; using also the relation (124.3), we have

$\begin{array}{l}\int {\psi }_{{\text{k}}^{\prime }}^{(+)\ast }{\psi }_{\text{k}}^{(+)}\text{d}V=\frac{2{\pi }^2}{k^2}\delta \left(k^{\prime }-k\right)\sum _{l=0}^{\infty }\left(2l+1\right)P_l\left(cos\gamma \right)\\ =\frac{8{\pi }^2}{k^2}\delta \left(k^{\prime }-k\right)\delta \left(1-cos\gamma \right)·\end{array}$

The expression on the right is zero for k ≠ k′; on being multiplied by 2πk ², sin γ dk dγ/(2π)³ and integrated over all k-space it gives 1, and this proves formula (136.2).

Together with the system of functions ψ _k ⁽⁺⁾, we can also introduce a system corresponding to states in which there are at infinity a plane wave and an ingoing spherical wave. These functions, which we denote by ψ _k ⁽⁻⁾, are obtained directly from the ψ _k ⁽⁺⁾:

(136.4) ${\psi }_{\text{k}}^{(-)}={\psi }_{-\text{k}}^{(+)\ast },$

since the complex conjugate of e^ikr /r (outgoing wave) is e^−ikr /r (ingoing wave), and the plane wave becomes e ^{−i k.r} , so that, in order to retain the previous definition of k (plane wave e ^{i k.r} ), we must replace k by −k, as in (136.4). Noticing that P_l (− cos θ) = (− 1) ^lP_l (cos θ), we obtain from (136.1)

(136.5) ${\psi }_{\text{k}}^{(-)}=\frac{1}{2k}\sum _{l=0}^{\infty }{i}^l\left(2l+1\right)e^{-i{\delta }_l}{R}_{kl}\left(r\right)P_l\left(\text{k}·\text{r}/kr\right).$

The case of a Coulomb field is of great importance. Here the functions ψ _k ⁽⁺⁾ (and ψ _k ⁽⁻⁾) can be written in a closed form, which is obtained directly from formula (135.7). We express the parabolic coordinates by

$\begin{array}{cc}\frac{1}{2}k\left(\xi -\eta \right)=kz=\text{k}·\text{r}\text{,}& \text{k}\eta =\text{k}\left(r-z\right)\end{array}=\text{k}r-\text{k}·\text{r}.$

Thus we obtain for a repulsive Coulomb field†

(136.6) ${\psi }_{\text{k}}^{(+)}=e^{-\pi /2k}\Gamma \left(1+i/k\right)e^{i\text{k}·r}F\left(-i/k,1,ikr-i\text{k}·\text{r}\right),$

(136.7) ${\psi }_{\text{k}}^{(-)}=e^{-\pi /2k}\Gamma \left(1-i/k\right)e^{i\text{k}·r}F\left(i/k,1,-ikr-i\text{k}·r\right).$

The wave functions for an attractive Coulomb field are found by simultaneously changing the signs of k and r:

(136.8) ${\psi }_{\text{k}}^{(+)}=e^{\pi /2k}\Gamma \left(1-i/k\right)e^{i\text{k}·r}F\left(i/k,1,ikr-i\text{k}·\text{r}\right),$

(136.9) ${\psi }_{\text{k}}^{(-)}=e^{\pi /2k}\Gamma \left(1+i/k\right)e^{i\text{k}·\text{r}}F\left(i/k,1,-ikr-i\text{k}·\text{r}\right).$

The action of the Coulomb field on the motion of the particle near the origin may be characterized by the ratio of the squared modulus of ψ _k ⁽⁺⁾ or ψ _k ⁽⁻⁾ at the point r = 0 to the squared modulus of the wave function ${\psi }_{\text{k}}=e^{ik.\text{r}}$ for free motion. Using the formula

$\begin{array}{l}\Gamma \left(1+i/k\right)\Gamma \left(1-i/k\right)=\left(i/k\right)\Gamma \left(i/k\right)\Gamma \left(1-i/k\right)\\ =\pi /ksinh\left(\pi /k\right),\end{array}$

we easily find, for a repulsive field,

(136.10) $\frac{{\left|{\psi }_{\text{k}}^{(+)}\left(0\right)\right|}^2}{{\left|{\psi }_{\text{k}}\right|}^2}=\frac{{\left|{\psi }_{\text{k}}^{(-)}\left(0\right)\right|}^2}{{\left|{\psi }_{\text{k}}\right|}^2}=\frac{2\pi }{k\left(e^{2\pi /k}-1\right)},$

and for an attractive field,

(136.11) $\frac{{\left|{\psi }_{\text{k}}^{(+)}\left(0\right)\right|}^2}{{\left|{\psi }_{\text{k}}\right|}^2}=\frac{{\left|{\psi }_{\text{k}}^{(-)}\left(0\right)\right|}^2}{{\left|{\psi }_{\text{k}}\right|}^2}=\frac{2\pi }{k\left(1-e^{-2\pi /k}\right)}.$

The functions ψ _k ⁽⁺⁾ and ψ _k ⁽⁻⁾ play an important part in problems relating to the application of perturbation theory in the continuous spectrum. Let us suppose that, as a result of some perturbation $\widehat{V}$ , the particle makes a transition between states of the continuous spectrum. The transition probability is determined by the matrix element

(136.12) $\int {\psi }_f*\hat{V}{\psi }_i\text{d}V.$

The question arises of which solutions of the wave equation are to be taken as the initial (ψ _i ) and final (ψ _f ) wave functions, in order to obtain the amplitude for a transition of the particle from a state with momentum ħk to one with momentum ħk′ at infinity.† We shall show that this requires that

(136.13) $\begin{array}{cc}{\psi }_i={\psi }_{\text{k}}^{(+)},& {\psi }_f={\psi }_{\text{k}}^{(-)}\end{array}$

(A. Sommerfeld 1931).

This becomes clear if we consider how the problem would be solved by perturbation theory applied not only as regards the perturbation $\widehat{V}$ but also as regards the field U (r) in which the particle is moving. In the zero-order approximation (with respect to U), the matrix element (136.12) is

$V_{{\text{k}}^{\prime }\text{k}}=\int e^{-i\text{k}\prime ·\text{r}}\hat{V}{e}^{i\text{k}·\text{r}}\text{d}V.$

In subsequent approximations with respect to U, this integral is replaced by a series of which each term is an integral

$\int \frac{V_{{\text{k}}^{\prime }{\text{k}}_{\text{1}}}{U}_{{\text{k}}_{\text{1}}{\text{k}}_{\text{2}}}\dots U_{\text{k}\eta \text{k}}}{\left(E_{\text{k}}-E_{{\text{k}}_{\text{1}}}+i0\right)\dots \left(E_{\text{k}}-E_{{\text{k}}_n}+i0\right)}{\text{d}}^{\text{3}}{k}_1\dots {\text{d}}^{\text{3}}{k}_n;$

cf. §§43 and 130. The numerator contains the matrix elements (in varying order) with respect to the unperturbed plane waves, and all poles are avoided in the integrations, according to one fixed rule. On the other hand, this series can be obtained as the matrix element (136.12) with the wave functions ψ _i and ψ _f as perturbation-theory series with respect to the field U. The fact that the result must be a sum of integrals in which all poles are avoided by the same rule means, therefore, that the poles in the terms of the series representing ψ _i and ψ _f ∗ must be avoided by a similar rule. But if the wave equation is solved by perturbation theory with this avoidance rule, we necessarily obtain a solution whose asymptotic form includes an outgoing (as well as a plane) wave. In other words, the wave functions, which in the zero-order approximation (with respect to U) have the form

$\begin{array}{cc}{\psi }_i=e^{i\text{k}·\text{r}}& {\psi }_f*=e^{-i{\text{k}}^{\prime }·\text{r}},\end{array}$

must be replaced by exact solutions of the wave equation, respectively ψ _k ⁽⁺⁾ and ${\psi }_{-\text{k}}^{(+)}+\left({\psi }_{\text{k}}^{(-)}\right)\ast$ . This proves the rule (136.13).

The choice of ψ _k′ ⁽⁻⁾ as the final wave function applies also to transitions from the discrete to the continuous spectrum; here there is, of course, no problem of choosing ψ _i .

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