The Resultant Angular Momentum

If we now change to polar co-ordinates, we see (cf. Appendix XVIII p. 298) that 

This simply means that for every state with aisimuthal quantum number Z, has the proper value —- Z(Z + ) The magnitude of the 

Further, in polar co-ordinates with polar axis we have 27ri 02/ dx/ 27ri dcj) 

Applying this operator to the one-valued function (where m = 0, +1, +2, c.), we have 

Deduction of Eutherford s Scattering Formula by Wave Mechanics (p. 131). 

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Let us consider au atom with two s electrons, with different total quantum numbers for example, a beryllium atom with one valence electron in a 2s orbital and the other in a 3s orbital, in addition to the two electrons in the K shell. The orbital angular momenta of the two valence electrons are zero (h = 0, k = 0), and accordingly the resultant angular momentum is zero (L = 0). Each of the two electrons has spin quantum number (si = sz = ), and each spin angular mo-... 

It follows that once the total angular momentum of an ion, atom or molecule is known, so too is its magnetic moment. Most free atoms possess net angular momentum and therefore have magnetic moments, but when atoms combine to form molecules or solids, the electrons interact so that the resultant angular momentum is nearly always zero. Exceptions are atoms of the elements of the three transition series which, because of their incomplete inner electron shells, have a resultant magnetic moment. 

Briefly, the derivation is as follows. In general, in an atom which contains several electrons, the orbital momenta 11( 12,13,... and the spins s1( Sg, s3,. .. of the individual electrons are strongly coupled among themselves. (This is called Russell-Saunders coupling.) By quantum-theoretical addition of angular momentum vectors the addition of the 1< gives the resultant angular momentum vector L. For 4 + 12,... 

The atomic terms are characterized by the orbital angular momentum which is reflected in the value of the total orbital quantum number L. Just as the individual angular momenta are quantized, so are the resultants. For the example above, with = 1 for two p electrons, the maximum value of the resultant angular momentum is represented by the sum of the quantum numbers, and the minimum value is represented by the difference. Intermediate values are allowed too, provided they differ from the extremes by a whole number ... 

Here is the component of the angular momentum p in the direction of the field, so that this is quantised. Consequently the resultant angular momentum, the magnitude p of which must be an integral multiple of hj Tr, can have only a finite number of possible inclinations to the direction of H, in fact 2 +1 inclinations (fig. 10), corresponding to the 2ifc + 1 possible values for its component p ... 

If an external field is apphed, however, this number is considerably increased owing to the various settings of the resultant angular momentum relative to the special direction a term with resultant angular momentum j has 2j -f 1 possible settings in the field. We thus obtain the following scheme ... 

Now it is important to know that the same formula results from calculations on the basis of the quantum theory, i.e. if we take into account only a finite number of positions of the moment. We assume that j8 has small values and the resultant angular momentum large... 

Since we have in addition the principle of the conservation of the total angular momentum at our disposal the integration can be completely carried out. We can take the hitherto arbitrary axis of i in the direction of the resultant angular momentum. Since the nodal line is perpendicular to this, the angular momentum about the nodal line will be... 

The motion under no forces of the symmetrical top therefore consists of a uniform rotation about the axis of symmetry, together with a uniform precession of this axis about the direction of the resultant angular momentum. 

For an arbitrary external field, on the other hand, the resultant angular momentum p is not in general an integral of the equations of motion and cannot therefore be quantised, but it may happen, in special cases, that p is constant and is an action variable. The relations (8) and (9) will then be true at the same time but pt is the projection of p in the direction of the field and, if a denotes the angle between the angular momentum and the direction of the field, wo have... 

This angle is therefore not only constant (regular precession of the resultant angular momentum about the direction of the field), but is also restricted by the quantum condition to discrete values. One... 

We consider now the case of a system which is subject to internal forces only. The above considerations are then applicable to the axis of the resultant angular momentum, where, in place of , the angle denoted above by ifi appears and the quantum condition (8) applies. The polarisation of the light cannot be observed, however, since the atoms or molecules of a gas have all possible orientations. The case mentioned above, where all the particles of the system move in planes perpendicular to the axis, is of frequent occurrence, e.g. in the case of the two-body problem (atom with one electron) and in that of the rigid rotator (dumb-bell model of the molecule) the transition j->j is then impossible. 

The conception of molecules as rigid bodies must, of course, be founded on the electron theory for, actually, the molecule is a complicated system made up of several nuclei and a large number of electrons. It can in fact be shown 1 that the nuclei move, to a close approximation, like a rigid system, but the resultant angular momentum of the molecules will not be identical with the angular momentum of the nuclear motion, because the electron system itself possesses, relatively to the nuclei, an angular momentum of the same... 

Since the constant angular momentum can have an arbitrary direction in space, the motion is degenerate and we can reduce the number of degrees of freedom by 1. We can, for example, without loss of generality, choose the fixed polar axis 9 =0 of the Kulerian co-ordinate system in the direction of the resultant angular momentum D, in which case we get ... 

In the case of the top we do not denote the quantum number of the resultant angular momentum by j, an in the general theory, but by m, because this letter is used to denote the terms of a molecular rotation spectrum (see Rotator, 12). 

If in this case also wc take the fixed polar axis in the direction of the resultant angular momentum, the relations (8) are again valid and we have... 

The stationary motions are obtained when mhj2tt is put for the resultant angular momentum D and the values of E so chosen that the ellipsoid represented by (22), whose centre is at the point 12a, cuts from the sphere D =const. a surface whose ratio to that of the hemisphere is njm we shall return later to the consideration of the significance of Q and the question whether this quantity is to be subjected to a quantum condition. 

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