5.28 Infinite Region and Continuous Spectrum of Eigen-values. Radiation-condition: As the size of the region increases, the eigen-values move closer together; when the extension becomes infinite, they lie everywhere densely; we are dealing with a continuous spectrum of Eigen-values.
Consider now the interior of a sphere of radius a for different boundary values. For purely radial oscillations, its eigen-values are given by
Hence, kna = np and the difference of neighbouring eigen-values is
Accordingly, one can view the everywhere regular and at infinity vanishing function y0(kr) as the eigen-function of the infinite space. Thus, if one has to solve an acoustic or optical task with given sources (discrete or continuous), located in the finite part of space, for a give wave number, one could every time add this function y0 to the solution. Hence, vibration problems (in contrast to potential problems) are not uniquely defined by their sources, prescribed in the finite part of space. This paradox conclusion shows already that it is insufficient to impose just at infinity the condition of vanishing, but that it must be replaced by a sharper condition, which relates to infinity. We call it the radiation condition: The sources are to be sources, not sinks of energy. The energy radiated by the sources must disperse at infinity: Energy cannot be radiated from infinity into the prescribed singularities of a field (plane waves are excluded, because for them the condition u = 0 at infinity is not satisfied).
In our special case of the eigen-function
the conditions are easily surveyed: With the time dependence e-iw t is eikt /r a radiating, e-ikt /r an infusing, y0 (kr) a standing wave (nodal surfaces kr = np). By prohibiting radiation from infinity, we make addition of the eigen-function y0 (kr) impossible. Hence, the admissible singularities have the form
Then, as is readily confirmed, holds the condition
which we will now transfer as general radiation condition onto all acoustic and electro-dynamic oscillation problems in infinite space in as far as they are excited by sources which lie entirely in the finite part of space.
Indeed, Condition (2) is not only valid in the case of the spherical wave radiating from r = 0, but, as one likewise confirms, also in the case of an excitation at x = x0, y = y0, z = z0
and therefore, for example, during a continuous excitation of the space density r = r(x0, y0, z0):
It does not only apply in unlimited space, but also when there exist in a finite region surfaces s on which there are prescribed any linear boundary conditions, whether homogeneous, for example, u=0, or inhomogeneous, for example, u = U. In the first case, one is concerned with reflected or dispersed radiation, which emits from the surface s, in the second case with excitation by radiation through the pulsating surface s itself.
So to say, we can confront the radiation condition (2) with the irradation condition
We prove the general validity of the radiation condition by showing that it guarantees the uniqueness of the just described vibration problem. Accordingly, the only solution of the mathematical problem occurs, as we can be convinced, with Nature's solution. Our problem is as follows:
a) Let u satisfy outside a surface s, which can consist of several partial surfaces s1, s2, ···, the differential equation
The function r measures the strength of the sources, which can be continuously distributed. r is given and should vanish sufficiently strongly at infinity.
b) u satisfies on s the condition u = U. U is given as a function of the location of s and, in particular, for example, can vanish. All s lie entirely in the finite part of space.
c) At infinity, u satisfies (2). The quantity r in (2) denotes the distance from any fixed point r = 0 in space. About this point, we place a sphere S with radius which therefore does not intersect s. Let its area element be dS = r²dw, where dw is the solid angle, seen from r = 0. The region between S s and s is called S.
d) u satisfies the continuity conditions, which were already assumed during the formulation of the differential equation, except at possibly prescribed sources.
We will assume that there are two solutions u1 and u2 of this problem and form, as usually,
as well as the conjugate function w*. They satisfy the conditions a) to d) with r = 0 and U = 0. Hence, there vanishes in Green's theorem
the integral on the left hand side and the first integral on the right hand side, whence also the integral extended over the sphere S must vanish.
For the discussion of this integral, we look at
of the sufficient generality of which we can convince ourselves as follows:
We imagine that w has been expanded in terms of spherical functions . By 4.24.1, its coefficients must have the form
where the z are related to the half-integer Hankel functions by (21.15). However, we must have here Dn = 0 due to the behaviour of z²n for large values of the argument (4.21.4). We find at the quoted location that the z 1n involve a finite number of terms of the form eikr/(kr)m, m < n. By ordering the thus arising spherical function expansion according to the powers of r -n, there arises indeed (5), where, by the way, the turn out to be finite aggregates of spherical surface functions.
There exists between these fn a simple recursion formula. In fact, the differential equation Dw+k²w=0, written in terms of , demands, by (4.22.4) with the differential symbol D introduced in (4.23.15b), the angle co-ordinates :
Applied to (5), (6) yields after omission of two equal terms with opposite signs
If one interchanges here in the first term of the brackets the summation index n with n + 1, one obtains
and concludes that there exists the recursion formula
Hence, if f0 = 0, also all f0 = f1= ··· = 0.
Next, we examine the integral remaining in (4) after dw and replace in it, since it relates to , w by the first term of (5), with neglect of the higher powers of 1/r, i.e.,
Thus, we obtain
The integrand in this integral is positive as long as . However, we have seen by (4) that this integral must vanish. Thus,
w = 0 and u2 = u1.
The author's initial proof was given in Frank-Mises II, Chapt. XIX, Section 5. For the form of the following proof, I am obliged to Mr. F. Sauter
This proof assumed apart from the earlier conditions a), b), c) for u the existence of the Green function for the outside of the surface s and an additional finality condition. Mr. F. Rellich has proved with all mathematical exactness that this condition is superfluous, even for any number of dimensions h, where the radiation condition is
(Annual report of the German Math. Union 53 (1943) 57, where also the case is treated, when the surface s extends to infinity.)
In the two-dimensional case, h = 2, when, as we know, the cylindrical wave H10(kr) takes the place of the spherical wave eikr/r, (7) becomes
which is indeed truly asymptotic for H10(kr). The same holds in the one-dimensional case, when the radiating wave type if given by eik|x| and (7) yields
Also with Mr. Relich, we emphasize that there cannot exist a radiation solution u of the wave equation, which vanishes at infinity more than like 1/r. If (5) is transferred to such a u, one would have f0 = 0; however, that would, as we have seen, cause u to vanish identically. In this respect, the wave equation differs from the potential equation. In the latter case, there exist solutions, which with growing r decrease more rapidly than 1/r, the so-called dipol-, quadrupol, octupol-fields of 4.24.3. In contrast, there arises for the wave equation such an r-dependence, which then for r = 0 induces a stronger singularity than 1/r, but only in the so=called proximity zone (r < l, l = wave length); in the distance zone (r > l), every solution of the wave equation behaves like the spherical wave eikr/r. The potential theory takes here its place as the limiting case as in that case we can say that the proximity zone extends to infinity.
We now come to the problem of Green's function for a continuous system. To start with, we study it in detail for the very simplest one-dimensional example in which there is prescribed no other than the radiation condition at infinity. Green's function is then identical with the principal solution and has like it at any given point, say, x = x0, a unit source (Exercise 2.3). It must meet the conditions
As is readily verified, the solution is
At first, we will compare it with (27.5) for an, at first, finite region -l < x <+l, at the ends of which, however, there should not apply one of the ordinary boundary conditions, but the radiation condition. We will denote the eigen-values kn of this domain, as preparation for the transition to the continuous spectrum, by l; the associated eigen-function u = ul is then defined by
If we write now, by a),
then, by b),
whence one has for l the equation
It splits into
Equation (9a) yields in first and second approximation, respectively, where m is an integer,
corresponding to (9b)
As one sees, the l-values, computed from (9a) and (9b), become a sequence of points (+ in Fig. 26), which, starting with l = 0, at first descend along a straight line into the negative-imaginary half-plane and then, for large values of l (large m), approach asymptotically from below the real l-axis. By (9a,b), the successive points belong intermittently to the cos- and sin-eigen-functions.
Obviously, the fact that we find, in contradiction to an earlier theorem, complex eigen-values is due to our present boundary condition which, in contrast to the earlier one, is itself complex.
Normalized to 1, the cos- and sin-eigen-functions are
In the limit , our l-points fill everywhere densely half of the curve on the right hand side, denoted by W1. The difference of consecutive points in the sequence (10a) or (10b) thus throughout becomes
After these preparations, we return to our representation (27.5) of Green's function. We substitute for u(P) the expressions (11), for u(Q) the same expressions with x0 instead of x and combine two consecutive terms, i.e., those cos- and sin- terms which belong to neighbouring values of l. Thus, we obtain for the numerator of (27.5)
However, by (11) and (11a), one has for
Thus, the numerator, referred to, becomes cos l(x - x0)dl/p , while the denominator in our present notation is k² - l², whence (27.5) becomes
In the last term, W denotes the solid line in Fig. 26 W1 + W2. The fact that the integration along W1 equals half the integral, executed along the entire path W, follows in that in the integral along W1 the numerator and denominator are even functions of l. On the other hand, the fact that in the last term of (12) cos could be replaced by the exponential function follows in that the sin-part of the latter, being odd in l, drops out during the integration. The path W is of much greater advantage than W1, because it can be detached from the origin and deformed according to the methods of complex integration.
It also follows from Fig. 26 above how that has to be done. For positive x - x0, the path W can be drawn into the positive-imaginary, for negative x - x0, into the negative imaginary l-half-plane. The first time, it gets stuck at the pole l = +k of the integrand of (12), the second time, at the pole l = -k. By forming the residues, (12) becomes on combination of the two cases
This agrees completely with (8).
Thus, we see: Our general presentation (27.5) of Green's function is also applicable in the case of a continuous eigen-value spectrum, provided we displace the integration path in correspondence with the radiation condition. If we had used instead of this condition the irradiation condition [inverted sign of i in (1a) and (2)], we would have had to employ instead of the path W the path, reflected in the real l-axis; in that case, Equation (13) would use a negative sign with i.
When we go from the one-dimensional region to two or three dimensions and use instead of x the polar co-ordinates r, j or , respectively, the spectrum of the eigen-values becomes continuous only in the r-co-ordinate, but remains discrete with respect to the angle-co-ordinates. From our point of view, we would start, for example, in the three-dimensional unbounded case from the representation of the Green function
As in 5.27, P and Y are the spherical and Bessel functions, normalized to 1; in the same way, the functions Z1and Z2 , to be used below, correspond to the earlier Hankel functions z1 ,z2. The factor 2p on the left hand side is due to the normalization of The path W1 runs, as in Fig.26, in the complex part of the l-plane from to during which it avoids the pole l = k. We will now briefly indicate how one has to go about with this representation analogously to the one-dimensional case and then is led to the already known representation of the spherical and cylinder functions.
In order to transform the path W1 into the at infinity closed path W of Fig. 26, one sets
where , due to the convergence difficulties arising with its normalization, we must refer the reader to Appendix 5.1. According to the circulation relations of the Hankel functions [Exercise 4.2, especially (12) of its hints, and the later discussion concerning (32.13)], the integral W1 can be transferred with the meaning of F from (14a) into the integral W by
respectively. Since the integrand ½F1,2(k² - l²) in both cases vanishes at infinity of the positive-imaginary l-plane, the integral in (14) reduces to the residue at the pole l = k :
If one now applies the addition theorem for spherical functions in the form (22.34) with the meaning of Q given there, one finds from (14)
For reasons of symmetry, G(P, Q) is in an unbounded space a pure function of the distance of P from Q
in fact, with respect to the definition of the unit source
the last thanks to the meaning of z0 in (21.15a). If one now changes also on the right hand side in (16) from P, Y, Z to P,y, z [cf. Y, Z, cf. 5.A.2, (9a)], one obtains exactly the earlier expansions (24.9, (24.9a), also called addition theorems.
The corresponding and correspondingly to be obtained series in the two-dimensional case are also contained in (21.3).
More important than the derivation of these well-known functions is their generalization to the case when the space is not unbounded, but is bounded in its finite part by a closed surface s (two-dimensionally by a curve s) with prescribed boundary conditions. In that case, we have to deal with the actual problem of Green's function: Since a function G(P, Q), which has at Q a unit source, satisfies at infinity the radiation condition and on s (s, respectively) a prescribed boundary condition.
We will choose especially a sphere with radius r = a as surface and
as boundary condition. The point Q lies on the half-ray
The eigen-function, which belongs to l is now not, as before, yn(l r), but may be set up in the un-normalized form
In this case, A is determined from (17)
For the derivation of the Green function, we will not follow the general method of (14), but can proceed more directly, but also less systematically, while we rely on the just again derived result (24.9) for the unlimited space:
Naturally, Equation (19b) does not yet satisfy Condition (17) for r = a; in order to adjust it to (17), we complement its right hand side by
whereby it is indeed made to vanish and becomes, by (18),
If we make the same extension to (19a), then the steady transition between (19a, b) for r = r0 as well as the fulfilment of the radiation condition for is assured. As a result, (19a) yields
Hence (20) and (21) yield directly the required Green function, if, as in (16a), by addition of the factor k/4p i, one makes allowance for the condition of the unit source. Thus,
The adopted mode of presentation allows already to recognize the connection with our general method in (14). In fact, the function F in (14a), if we disregard constant normalization factors, is now represented by
correspondingly, the functions F1, F2, which arise during the transfer of the path W1 to W, are given by
formation of the residues for l = k shows then obviously the formation of (25).
In Appendix II, we will present a new kind of method for the derivation of Green's function, which does not only improve the convergence of the series, representing them in the practically most important cases, but also opens up new vistas into future possibilities of application of this method.
Moreover, in the Appendix to Chapter VI, we will show that this method solves at the same time the problem of wireless telegraphy on the sphere-like Earth (for well conducting soil and a vertical dipole antenna - more correctly, would solve if not the ionosphere were to have a decisive role).
Finally, we emphasize: A representation of the form (14) remains valid when we employ instead of the sphere an ellipsoid and, correspondingly, instead of employ the co-ordinate system of confocal ellipsoids and hyperboloids. The eigen-value spectrum of the external space of the ellipsoids would then remain discrete in the parameters of the one- and two sheeted hyperboloids, but become continuous in that of the ellipsoids; after execution of the integration over the latter parameters, it would admit a similar simplification as has been given by (22). Also, in the most general cases, when there do not exist separating co-ordinates, in which the eigen-functions can be split into products, Equation (27.5) could still serve as the starting point for the representation of the Green function.
5.29 The Eigen-value Spectrum of Wave Mechanics. The Balmer Term: In the simple case of the hydrogen atom, the wave mechanics equation of Erwin Schrödinger (1887 - 1961) is
This is our earlier equation (7.15) with the difference that the energy symbol W of the there assumed force-free motion is replaced by the difference of the total energy W and the potential energy V, i.e., in mechanical terms, by the kinetic energy. As it is known, Rutherford's model of the H-atom comprises a nucleus, the proton, the charge +e and one electron with the charge -e, which moves in the proton's field. Its potential (Coulomb) energy, measured in ordinary electro-static units, is
where r is its distance from the proton and V is normalized, so that V = 0 at infinity. The total energy then does not contain the mass energy m0c² of the electron at rest. In the sequel, one has to envisage the proton being at rest at r = 0.
Equation (1) differs from the so far treated wave equation by the fact that the expression, which has taken the place of the hitherto constant k², depends on the location and is singular at the point r=0. While we have so far called k an eigen-value, we will now treat W as eigen-value parameter, i.e., we will seek such values of W for which (1) admits through the entire space a continuous solution. These solution are the eigen-functions of our Kepler-Problem, in which the nucleus has the role of the Sun and the electron that of the planets. Since in this problem the electron disposes of the unlimited space, we understand that its eigen-value spectrum, as in (18), is continuous with respect to the r- co-ordinate. However, it is more important for us that this spectrum at the same time has discrete components.
In a spectral apparatus, we obtain indications from a discrete spectrum by measurement of the line-spectrum, which is represented in the case of hydrogen visually by the Balmer Series Ha, Hb, Hg, Hd, ···. Its lines accumulate at a border, given by the Rydberg Constant R. The adjoint continuum lies in the nearby ultra-violet. Schrödinger's equation accounts for the discrete as well as continuous spectrum. Thus, it reduces the riddle of the spectral series with their point of accumulation in the finite part of space, which deviates strongly from the behaviour of all mechanical systems, to a simple mathematical formula .
Indeed, as it is known, Niels Bohr (1885 - 1962) has already given a general explanation of the Balmer series as well as of their bounding frequencies 12 years before Schrödinger by equipping Rutherford's model with certain quantum-theoretical features. However, the orbit concepts used led to diverse contradictions and had to be relinquished in favour of the analytical model, contained in Equation (1). Its involvement of Planck's constant bears witness that in the end Equation (1) rests on the quantum theory.
The reader will now ask: What is the physical meaning of the eigen-function y? The answer demonstrates the complete change of the concept of Nature, which the quantum theory has brought with it: |y|²dxdydz means the probability, with which we can expect that the hydrogen-electron can be found at the point x, y, z with the action-space dx, dy, dz! Thus, the concept of probability takes in wave mechanics the place of the strict determinateness which rules in classical mechanics. The measure of the indeterminateness in the atomic event is directly Planck's h (Heisenberg).
Thus, the normalization of the eigen-functions to 1, which we have hitherto introduced for the sake of mathematical simplification, has a fundamental meaning. In fact,
guarantees the certainty of finding the electron somewhere in space, whence this condition is necessary from a wave-mechanical point of view. It applies in this form to the discrete spectrum; it must be modified for the continuum according to the rules in Appendix 5.1.
We now proceed to the integration of (1) and introduce for this purpose the polar co-ordinates . From the wave equation, written as in (22.4), and
follows with respect to the differential equation (22.13) from Pml
At first, we are interested in the case in which the electron is bound to the nucleus. Then, W must be negative: The energy of the electron, at rest at infinity, is normalized to zero. When it is captured by the nucleus and becomes stably bound, its energy has dropped. In contrast, if W > 0, the electron has also at infinite distance from the nucleus positive kinetic energy and describes, speaking mechanically, a hyperbolic orbit.
The asymptotic behaviour of c for follows from (5), if we erase all terms with 1/r and 1/r²:
Hence, we write for negative W
The other solution c = e+r/2 of (5a) must be dismissed, since c is to be finite everywhere.
In order to solve (5) exactly, we set
and obtain for v the differential equation
with the, for the present, still quite non-commital abbreviation
We have the method of (19.36) for our examination of this equation. Let
and find for l, corresponding to (19.37),
The other root of (7a) l = -l - 1 is of no interest, because v just as c must remain finite for r = 0. On the other hand, the recursion formula for the ak is obtained, corresponding to (19.37a), by setting to zero the coefficient of rl+k-1 in the power series, following from (6a) and (7). Thus:
If one sets here the factor ak to zero by choosing
then vanishes also ak+1 as well as all further terms in w: The series is truncated, w becomes a polynomial of degree k, the closer study of which will employ us soon. In advance, we emphasize two aspects:
1. The function c, due to the factor e-r/2 in (6), vanishes for any degree of the polynomial w for so strongly that, following (3), y becomes normalizable, which we must demand.
2. In the case of a non-terminating series, there would result from (7b) an asymptotic behaviour of the ak for , which results in w becoming infinite like e+r for and makes impossible the normalization of y. Hence, the demand that w must truncate is necessary for wave mechanics.
To start with, our main interest concerns Equation (8). We denote the here relevant value of k by nr (radial quantum number) and substitute for l from (7b) the value l (azimuthal quantum number). Hence n is, by (8), an integer:
We call n the main-quantum number. By (6b), we obtain for W from n the value
We set W equal to the energy quantum hn and obtain
with the abbreviation
Here, R is the Rydberg frequency which has already been referred to above. Its numerical value can be determined spectroscopically with extraordinary accuracy and can therefore serve to refine our knowledge of the fundamental constants e, m and h. The frequency v in (9) is called the Balmer term.
The observable frequency of a spectral line arises during the transition of the atom from an initial state 1 to a final state 2 and is computed from the difference of the two associated terms n2 and n1. Hence, in the hydrogen spectrum,
The Balmer Series corresponds to the transition to the final state n2 = 2, the Lyman-Series, located in the ultra-violet, to the transition to the basic state of the hydrogen atom n2 = 1, in both cases at an arbitrary initial state n1 > n2. Hence
The series with n2 = 3, n3 = 4, ··· lie in the infrared.
Having found the eigen-values of the H-atom, we will inspect more closely the analytical character of their eigen-functions. Using (7), (7a) and (8a), one obtains by reorganisation of (6a) the differential equation for w
This equation arises by (2l +1)-fold differentiation from the simpler differential equation
This equation has for every integer m one and only one polynomial solution of degree m; with suitable normalization, one has, as is readily verified,
These are the same expressions as we have computed in Exercise 1.6 as Laguerre polynomials; (12) is the Laguerre differential equation as is already indicated by the notation L. It is identical with the differential equation (24.29) of the confluent hypergeometrical function for the values a=-m=-n-1 and g =1, whence we have
Hence, (4), (6), (7) and (7a) yield, if we add with respect to (3) a normalization factor N, the representation of the hydrogen eigen-function
We find still from (5a) and (8b) the expression for r, where we employ the since the start of atomic physics customary notation a for the hydrogen radius
In order to justify this term and as a sole special application of the preceding work, we compute the probability density |y|² at the basic state n = 1 of the H-atom. With n = 1, one has, by (8a), l=m=0, nn = 0 and (14) yields
where N1 is readily computed from (3b) at (p a³)-½. Hence, the residence-probability of the electron is distributed in spherical shape about the nucleus. It has for r = 0 its maximum N21, is for r = a only still (N1/e)² and vanishes only at infinity. The fraction is the charge density. Seen wave-mechanically-statistically, we do not have a point-shaped concentrated electron, but a charge-cloud, the principal part of which lies inside a sphere of radius a.
In contrast, from an older point of view of the concept of orbit, one had to ascribe to the H-atom a disk shape. A circular disc corresponded to the base state (circular orbit of radius a). In a magnetic field, all circular discs of an H-atom gas had to align parallel and perpendicular to the magnetic lines of force; a ray of light, sent through this gas, would have to exhibit magnetic double refraction. Exact measurements by Schütz, which, by the way, were performed not with H-atom gas, but with , as we will see below, analogous Na-vapour, did not exhibit a trace of it. This is one of the contradictions, which wave mechanics has cleared up.
The same which applies to the ground state of the H-atom applies to all atom states with l = 0, the so-called s-terms of the spectroscoper. In fact, with l = 0, one obtains from (14)
i.e., again spherical symmetry. Such s-terms are the base states of the alkali-atoms Li. Na, K, ···. However, for interest's sake, the same applies also to all fully occupied shells, for example, the eight-shells of the noble gases. Here, the proof rests on the addition theorem of the spherical functions. For all chemical applications, this spherical symmetry of the closed shells is apparently of great importance.
We must still add some details regarding the continuous spectrum of hydrogen, i.e., the states W>0 (the hyperbola orbits of the older theory). The electron is then no more bound to the nucleus; however, it yet remains in the field of the proton.
Due to W> 0, by (5a) and (6b), r and n become purely imaginary. In the asymptotic solution (5a), both signs of r have equal rights, i.e., both solutions e±r/2 can be used. It is unnecessary and also, due to the imaginary character of r, impossible to let the series truncate, whence every value of W is possible. The W-spectrum becomes continuous and reaches from Since, by (8), W = 0 corresponds to the boundary of the discrete spectrum, each of these spectra is continued, as we have said already above, by a continuous spectrum towards the short wave side. As regards its analytic form, the representation (14) is retained; however, L is no longer a Laguerre polynomial, but a non-truncating, confluent hypergeometric series, because the parameter a = -n - 1 in (13) is now not a negative integer, but a general complex number.
5.30 The Green function of the wave-mechanical scattering problem. Rutherford's formula of nuclear physics: As it is known, there stands at the cradle of nuclear physics Rutherford's scatter experiments of a-rays at heavy atoms. Since the electronic shroud of the atoms is inconsequential for the fast a-rays, we can attach the arising scatter problem to the circle of formulae of the continuous H-spectrum. Indeed, one is concerned here as there with a two body problem: A nucleus (here of the charge Ze, Z = order number of the atom, in the case of the H-spectrum was Z = 1) and a particle in interaction (here an a-particle of mass m and the charge Z'e with Z' = 2, before an electron of mass m, and charge -e, corresponding to the charge number Z'=-1). We seek, first of all, that position of the continuous spectrum which corresponds to the energy constant Wa of the entering a-rays. It is simplest to compute it from its kinetic energy at infinite distance from the a-particle and nucleus
where is the momentum of the a-particle.
If we now change over from the hitherto corpuscular concept of the a-rays to the with it complimentary wave concept, then is at the same time the wave number k of the a-rays, i.e.,
Indeed, the formula is nothing else but the equation of L. de Broglie h times the reciprocal wave length equal momentum,which on its part represents the relativistic complement to Planck's equation h times reciprocal period equals energy.
Hence, the variable r, defined in (29.5a), can be transformed into
We employ, instead of ka, for an arbitrary place in the continuous spectrum (i.e., for any value W other than Wa), the letter l, in order to achieve the connection with 5.28, . Equations (1) and (2) then become generalized to
If we now assume that the nucleus, as there the proton, is at rest, the wave equation (29.1) becomes
Instead of (3), we shall now write temporarily
especially at l = k, the spectrum then corresponds to
We make with respect to this the important observation that in the difference K² - K²a the potential term, which depends on the location, drops out so that this difference becomes independent of the location:
The reader should confirm that all earlier conclusions from Green's theorem, which in 5.26, for example, led to the orthogonality of the eigen-functions or in 5.27 to the representation of the Green function with constant k², can be transferred without change to our more general wave equation with the local dependence of K², stated in (3a, b).
After these preparations, we proceed to the theory of Rutherford's scattering experiment. If we view the source of the a-rays (radium specimen) in the form of a point and located in the finite part of space, we have to deal with a spherical wave of corpuscular rays, which is modified in the manner prescribed by the wave equation (3) by the presence of the nucleus. However, if we shift, what appears to be more natural and at the same time simpler, the source to infinity, we have then to deal with the same problem for the plane wave. In both cases, the solution is given by our Green function in 5.28, in the first case with general location of the source point Q, in the last case with the limit Since we had to sum in our Green function over the complete system of eigen-functions, we must take into account for a finite location of Q beside the continuous also the discrete spectrum of eigen-values. However, at the border , there vanishes u(Q) for all eigen-values of the discrete spectrum, whence one has only to integrate over the continuous spectrum. Also, we can now retain for the relevant eigen-functions u(P) the expression (29.14) of the discrete hydrogen eigen-function, provided we replace there r, according to (2a), by 2ilr; moreover, if we shift the direction to the extension of the line joining Q and the location of the nucleus, our scattering problem becomes symmetrical with respect to the axis , i.e., independent of j; so that also the eigen-function u(P) must be independent of j. We then have, by (29.14),
The related expression for u(Q) follows then from (5) after replacement of P1(cos q) by P1(cosq0)=P1(-1) = (-1)l, cl(r) by cl(r0) and going with r0 to the limit . As representation of the standing wave, one obtains then from (28.14), after integrating along the path W of Fig. 26 and forming the residue at the pole l = ka , a series representation of the form
in which the coefficients Cl are determined in a somewhat long-winded manner from the normalization factors of c and P and the asymptotic behaviour of c(r0) for . It has been derived first by W. Gordon (S. Physik 48 (1928). Cf. also the excellent book of Mott and Massey: The theory of atomic collisions, Oxford 1933, Chapter III).
However, much simpler than a closed representation is obtained when one does not work in polar co-ordinates but in parabolic co-ordinates x, h. It yields as wave function of the scattering process (cf. Appendix A5.3)
where k is the wave number, given in (1),
n is the principal quantum number, which, as we have already emphasized in 5.29, becomes purely imaginary in the continuous spectrum. It follows from (29.6.b), where one, however, has to replace e², as in (3), by -ZZ'e²,
Ln is the in (29.13) defined confluent hypergeometric function for l = 0;
h is the parabolic co-ordinate, defined in (7), the other one is will be referred to again as scatter angle.
For , follows asymptotically from (7)
with the abbreviations
The first term in (8) gives the incident plane wave, the second the spherical wave, scattered by the nucleus. In fact, C1 and C2 are not constants, but depend still on h, but only in their phase component, since n is purely imaginary. We are only interested in the absolute value of the ratio C2/C1, which is independent of h and therefore also of r and only depends on the scatter angle . In fact, (8a) and (7a, b) yield
Its square yields, according to the wave mechanical definition of the location-probability (29.3), the number of scattered particles per unit area of a plane, positioned perpendicularly to the entrance direction. This law was initially derived by Rutherford by geometrical consideration of the hyperbolic orbits without reference to quantum theory. The fact that this was possible is linked to the circumstance that the constant in (9) has dropped out. Rutherford's law is obviously valid for a-rays, but also, with a correspondingly changed meaning of Z' and ma, for any other particles (protons, electrons, ···) which interact with the nucleus according to Coulomb's law. We cannot treat here the interesting exchange effect, which occurs when impacting and ejected particles are equal. At very large velocities of the colliding particles, one would naturally have to take account of the theory of relativity.