The Dreams That Stuff is Made of
Page 85
where S is determined by the equationSH (p0q 0) S−1 = W.
The commutation relations and the equations of motion are then evidently satisfied and the desired proof is thus now complete.
The commutation relations are also invariant with respect to linear orthogonal transformations of qk and pk . For if we place
(10)
Then
and similarly for the other relations. Therefore, if we postulate that our fundamental relations hold for one cartesian system, they hold for any such system.
Proceeding systematically, we have now to study degenerate systems, that is, systems such that several of the Wn-values are equal, therefore several of the frequencies ν(nm) are zero. The constancy of the energy H = 0 can still be deduced from the equations of motion and the commutation relations, but it no longer follows in general from H = 0 that H is a diagonal matrix and therefore the proof of the frequency theorem cannot be carried through. The equations of motion and the commutation relations alone are here insufficient for a unique determination of the properties of the system and a further restriction of the fundamental equations is necessary. It is obvious that this restriction must be as follows: As fundamental equations, the commutation relations and
(11)
shall hold. Then the validity of the frequency condition is also assured for degenerate systems.
Although, except in singular cases, the energy is uniquely determined by these conditions, the coördinates qk are not uniquely determined. In non-degenerate systems, as was seen already in the example of the harmonic oscillator, only certain phase constants are arbitrary; one for each stationary state. In degenerate systems a much greater indetermination exists, which evidently is related to a sort of lability which allows arbitrarily small external perturbations to produce finite changes in the coördinates. But it can be shown that even then those properties of the system on which the polarization of the emitted light depends vary only continuously, a fact which Heisenberg has called “spectroscopic stability.” We shall not discuss this question further.
LECTURE 16
Conservation of angular momentum—Axial symmetrical systems and the quantization of the axial component of angular momentum.
The applications of the basic principles, which have been considered so far, suppose that several specially simple systems which are used as starting points in the calculus of perturbations are completely known. For this purpose we have until now studied the particular example of the harmonic oscillator. We must now develop general methods for the direct integration of the fundamental equations. These methods are the same as those used in classical mechanics, i.e., general properties of the energy-function H are used to find integrals. The conservation of energy has been thus derived as a consequence of the property of H not to depend explicitly on the time. The conservation of momentum and moment of momentum will now be developed, making the same hypotheses on H as in ordinary mechanics. The integration method is quite similar to that used in the derivation of the conservation of energy. The equations of motion, considered for the elements of the matrices, form an infinite system for an infinite number of variables: in general each equation contains an infinite number of variables. To begin with, a function A(pq), constant according to the fundamental equations, and therefore a diagonal matrix for non-degenerate systems is determined. If φ(pq) is any function, the differenceφ A—Aφ = ψ
can be calculated from our commutation rules. But as A is a diagonal matrix each of the equations for the elements contains only one of the elements of φ and ψ, besides two diagonal terms of A.
In Galilean-Newtonian mechanics, as well as in Einstein’s (“relativistic”) mechanics:
(1)
The components of momentum are
(2)
and the components of moment of momentum
(3)
If derivatives with respect to time are now taken and it is noted that, because of our assumption on H, all ṗkx ... etc. depend only on qkx ... etc., and all qkx etc. on pkx ... etc., it is seen that all these derivatives have the form φ(q) + ψ(p). Now since all the q’s and all the p’s are interchangeable among themselves, these expressions all vanish under the same conditions as in classical mechanics. The theorems on uniform motion of the center of gravity and on the conservation of angular momentum (law of areas) therefore hold exactly as in classical theory.
Let us now build up the expression
therefore
(4)
whence it is seen that the law of areas, as in classical mechanics, holds either only for one or for all three axes.
It will now be assumed that the system consists only of discrete energy levels, further that it is not degenerate and that the law of areas holds for one of the momenta, for instance . This is e.g. the case if the external forces acting on the atom are symmetrical with respect to the z-axis. Then Mz is a diagonal matrix and the individual elements Mxn are to be interpreted as the angular momenta of the atom around the z-axis for the corresponding individual states.
From the definition of Mx, My, Mz, and the commutation rules follow the matrix equations
(5)
AsMz (nm) = δnm Mzn,
these expressions can be rewritten
(6)
Equations (6) express, in the ordinary language of Bohr’s theory, the following: For a quantum jump in which the angular momentum Mzn changes, qlz(nm) = 0 and the plane of oscillation of the emitted light wave is therefore perpendicular to the z-axis. For jumps in which Mzn does not change qlx (nm ) = 0, qly (nm) = 0 and the emitted light therefore vibrates parallel to the z-axis. Moreover, in the former case,
(7)
That is: for every quantum jump Mzn changes by 0 or by . In the former case the emitted light is linearly polarized parallel to the z-axis, in the latter case it is circularly polarized around this axis. Mzn can therefore be represented by
(8)
If states existed the angular momentum of which did not find a place in this series, there could be no jumps or interactions between them and those belonging to the above series.
From these results it is seen that the index n can be split up into two components, one of which is the number n1 which has already been introduced, while the other, n2, numbers the different n’s with the same n1. Our matrices become four-dimensional and the “polarization rules” already derived are equivalent to the following expressions:
(9)
All these relations hold if qlx, qly, qlz are replaced by plx , ply , plz or by Mx , My , Mz.
In particular we note that
Further we need the following derived commutation relations: If,
then simple calculations give
(10)
which means that and M2 are diagonal matrices with respect to the quantum number n1.
The two components Mx , My may also be constant, but can never be diagonal matrices. For fromMy Mz − Mz My = − ∈ Mx
orMy (nm ) ( Mzn − Mzm ) = − ε Mx (nm)
it would follow that, for My (nm ) = Myn δnm, Mx would vanish identically, and therefore that My , Mz would also vanish identically. Such a system with a constant vector M, for instance a system moving freely in space, is hence necessarily degenerate.
Consider now a system the energy function of which isH = H0 + λ H1 + · · ·
under the following assumptions: For λ = 0 the law of areas holds for all three directions. For λ ≠ 0 the system is not degenerate, but Mz is constant. The energy H0 does not depend on n1. A system of this kind is, for instance, an atom in an axially symmetrical field of strength proportional to λ. This investigation leads also to definite information about the degenerate system with the energy-function H0, for every property of the perturbed system which is independent of λ or the choice of the privileged direction z must remain valid for λ = 0.
According to our hypothesis that for λ = 0 the law of areas along all three directions holds, Mx, My and therefore also d/dt (M2) have no terms without λ. Therefore
As it has been further supposed that H0 = W0 is independent of the quantum number n1, we have
whence
(12)
It was shown earlier that M2 is quite in general a diagonal matrix with respect to n1; it is now shown that (M0)2 is a diagonal matrix with respect to both quantum numbers n1, n2. The same holds for . Now
and
(13)
The diagonal terms of are therefore always positive; and since (M0)2 does not depend on n1 , the number of possible values of for a given n2 , therefore for a given , is finite. In other words the number of values n1 for a given n2 is finite. Hence the sum
has only a finite number of terms. This sum is an element of . If we now form in the same way and sum the equations
over n1 for fixed n2, this sum is zero on the right-hand side because in general, for finite matrices, the diagonal sum of ab is equal to that of ba:
Therefore
(14)
This holds for every complete series of n1. Therefore the possible values of n1 which go with a fixed n2 always form a symmetrical series with respect to the origin. Hence (n1 + C) runs through a finite series of whole numbers ...−2,−, 0, 1, 2 ...or of “half-numbers” ...,...
In the literature m (magnetic quantum number) is used in place of n1 + C. It has therefore been shown that the quantum number m defined by the diagonal term of is either a whole or a half number and that the selection rule
(15)
holds.
This result does not seem to lead much further than that which was obtained from the classical theory of multiple-periodic systems, but it must be borne in mind that in classical theory certain orbits frequently had to be ruled out by additional excluding rules. For instance, in the theory of the hydrogen atom, orbits leading to a collision between the electron and the nucleus were excluded. In the present theory no such additional rules are necessary, a fact which must be regarded as an essential step forward. To this must be added the full justification of half and whole quantum numbers, which so far could not be explained theoretically, while the empirical facts necessarily led to the introduction of the former, as already shown.
LECTURE 17
Free systems as limiting cases of axially symmetrical systems—Quantization of the total angular momentum—Comparison with the theory of directional quantization—Intensities of the Zeeman components of a spectral line—Remarks on the theory of Zeeman separation.
The detailed presentation of the derivations in the preceding lecture are, I believe, sufficient to show the method clearly. From now on I shall mainly emphasize results. Proceeding along the same line of reasoning we arrive at a new quantum number j which determines, in the limit λ → 0, the diagonal terms of M2, as follows,
(1)
Further j is always equal to the maximum value of the quantum number m and therefore is a whole or a half number. The selection rule
(2)
holds. The proof is quite similar to that in classical theory. In the latter a new rectangular system of coördinates is introduced whose z-axis coincides, for λ = 0, with the fixed direction of the angular momentum. Considerations concerning the total angular momentum are quite similar, for this system, to those for Mz in the case of axial symmetry. A linear combination of the coördinate matrices is formed which corresponds formally to a rotation of the system of coördinates into the desired position (z-axis parallel to the angular momentum). The equations obtained from these expressions have a finite number of matrix elements of a type similar to that obtained previously for the coördinates themselves except that M2 occurs instead of Mz. We find from Mz and M by means of the identity
and the above relations for Mx and My [Equations (3), (4), Lecture (16)]
(3)
It is also possible to express explicitly the coördinates qlx , qly , qlz in terms of the quantum numbers m, j. The result can be most clearly stated if written separately for the three possible jumps of j.
(4)
(5)
(6)
where A, B, C, depend in some way on the other quantum numbers of the system.
These expressions, as the perturbation and dispersion formulas given above (Lecture 14), had been found previously by considerations of correspondence before they were derived by the methods of our theory. This is seen most easily from formula (4) for j → j by going over to the limiting case of large quantum numbers m, j. Then 1 can be neglected compared with m and j and we find for the ratio of the intensities of the two circular vibrations and the linear oscillation:
(7)
where M, Mx , My , Mz denote the total angular momentum and its components in the quantized state in question m, j. These formulas can, however, be obtained classically as follows:
Consider the motion of the electrons as represented by the motion of their electric center of gravity S. The motion of S is decomposed into a linear component parallel to the angular momentum M and two circular components rotating in opposite directions perpendicular to M; the first component corresponds alone to the jump j → j, the two others correspond to jumps j → j ± 1. The linear oscillation parallel to M is given by
where φ, θ are the polar coördinates of the direction of M in a fixed system of coördinates. The motion in the xy-plane can be decomposed into two rotations in opposite directions,
where
To these correspond the jumps m → m ± 1, while the component qz corresponds to jumps m → m. The intensities are proportional to
(8)
If a weak outer field is now established in the z-direction, the whole atom rotates slowly around this direction. The circular frequencies of the two circular component oscillations are thereby changed slightly as well as their intensities, but in the limiting case of an infinitely weak field these changes can be neglected. Then we have
If these are introduced in the above relations the formula (8) given above is shown to be the limiting value of the rigorous formula (4), obtained through the new quantum theory. The cases j → j ± 1 can be interpreted in a precisely similar manner.
Historically, in fact, the opposite has been done. Starting from the classical motion, intensity formulas have been found, which should be correct for large quantum numbers, but which need a correction for smaller m and j. This correction has been found in a number of ways. Goudsmit and Kronig have used the so-called “boundary principle,” that is the requirement that the intensities must vanish when either of the states disappears. It was noted above that j is the maximum value of m; therefore, the intensities of all jumps for which j remains unchanged but to changes vanish if we place m = j + 1. Our formula is seen to satisfy this condition.
The first incentive to the investigation of these intensity laws was the experimental researches on the relative brightness of the components of the Zeeman effect. These, under the direction of Ornstein, were carried out by Moll, Burgers, Dorgelo and others. These investigators at Utrecht first found empirical whole-number laws for the intensities in the Zeeman effect and gave simple rules for their calculation. The theory then developed as outlined above.
Our formulas fit exactly the case of an atom in a weak magnetic field. The number and position of the components into which a line is split up by the field cannot as yet be calculated theoretically. We shall return to this question later. But, if we consider the system of split-up lines given experimentally, we can read off from it the value of j . Thus, starting from the middle, the Zeeman components vibrating parallel to the field (m → m) are assigned half or whole numbers and the largest value of m is equal to j. We have therefrom the corresponding values of m and j for each line and hence can calculate the relative intensities by our formula. Comparisons with observations have in all cases verified the theory.
As already stated, the actual magnetic split of spectral lines is not given by the present theory as it stands. For if linear terms in the field strength, such as
are alone taken into account in the magnetic additional terms of th
e energy, as usual, then we obtain, on account of our formula for Mz, the equidistant term sequence
with the normal separation νm corresponding to the classical Larmor precession. Experimentally, however, the separation is found to be gνm; the numbers g have been determined empirically by Landé as functions of the quantum numbers characterizing the corresponding spectral lines. All attempts to derive these g-formulas from classical models led to similar formulas, but never the correct ones. In these formulas the square of the angular momentum M2 enters, among other quantities. The former, of course, has always been replaced so far by (jh/2π)2, but Landé’s empirical formulas always require the expression j(j + 1) instead of j 2. Our new quantum theory gives in fact M2 = (h/2π)2 j(j + 1), a circumstance which encourages us to further researches.