Spherical HarmonicsYlm(θ,ϕ): Positive and Negative Integer Representations ofsu(1,1)forl-mandl+m

The azimuthal and magnetic quantum numbers of spherical harmonicsYlm(θ,ϕ)describe quantization corresponding to the magnitude andz-component of angular momentum operator in the framework of realization ofsu(2)Lie algebra symmetry. The azimuthal quantum numberlallocates to itself an additional ladder symmetry by the operators which are written in terms ofl. Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative(l-m)- and(l+m)-integer discrete irreducible representations forsu(1,1)Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation ofsu(2)compact Lie algebra via theYlm(θ,ϕ)’s for a givenl, we can also representsu(1,1)noncompact Lie algebra by spherical harmonics for given values ofl-mandl+m.

L-DEPENDENCE OF PARTICLE RADIATION IN MAGNETIC-SOLENOID FIELD AS AHARONOV-BOHM EFFECT

Aharonov-Bohm solenoid changes the energy spectrum of charge particles in pure magnetic field. In particular, the degeneracy with respect to azimuthal quantum number l is partially lifted. In turn, this complicates the radiation spectrum of a charged particle in magnetic field in the presence of the solenoid (Aharonov-Bohm effect). In particular, the degeneracy of the radiation intensity with respect to the azimuthal quantum number is lifted completely. In the present work we study l-dependence (induced by Aharonov-Bohm solenoid) of synchrotron radiation intensity in semiclassical approximation.

Some Properties of Dipole and Quadripole Radiation from Nuclei

1. The purpose of this paper is to examine in detail the types of radiation emitted by a quantum mechanical system (representing a nucleus) when the radiating particle changes its azimuthal quantum number l by two, one, or nought. We assume that the radiating particle is spinless* and that it moves in a field possessing central symmetry, so that the particle may be described by a wave function ψn,l,m which factorizes into radial and angular parts,We assume secondly that the nucleus has a finite radius aN of the order of 10−12 cm., that is to say that π(r) = 0 for r > aN.

The Stark effect for mercury

The Stark effect in the mercury spectrum has already received some attention. The first observations were made by Wendt and Wetzel, who used a canal-ray method. They reported small displacements in the diffuse series groups 2 3 P i — n 3 D i , n = 4, 5, 6, and in the sharp series line 2 3 P 1 —4 3 S. Ritter investigated the effect, using a Lo Surdo source, and noted small displacements toward the red in the lines 2 3 P 0 —5 3 D 1 , 2 3 P 1 —6 3 D 2 in the fields up to 26,000 volts per centimetre. Using an absorption method, Terenin detected a definite effect in the diffuse series triplets 2 3 P i — 3 3 D i . A far more extensive and valuable investigation of the effect for mercury was made by Hansen, Takamine and Werner at the Institute for Theoretical Physics, Copenhagen. A fine column of mercury in a quartz tube served as the cathode of a discharge tube. The light immediately above the mercury surface was analysed with a Hilger E 2 spectrograph. Observations were limited mainly to the diffuse series triplets 2 3 P i — n 3 D i , and the associated combination lines which appeared even in low electric fields. For values of n lower than five no definite Stark effect was observed, but in the higher members of the three series the 3 D lines were displaced to the red by an amount which increased with increasing term number. Grouped about the 3 D lines new combination lines of the types 3 P— 3 F, 3 P— 3 G, etc., were photographed and interpreted on the Bohr theory. In these mercury line groups, the restriction regard­ing the change in the azimuthal quantum number (∆ k = ± 1) is removed by low external fields, hence the new lines may be traced to a point very near their zero field position. No analysis of the patterns formed by individual lines could be attempted, however, since the electric fields which were established in the source were too weak to permit the necessary resolutions.

The spectrum of H 2 .-The bands ending on 2p 3 II levels

Rather more than a year ago it was announced that the bands which go down to the two 2 p 3 II ab levels had been found, but owing to the inclusion of a considerable number of wrong lines little progress in understanding them has been made until quite recently. The discovery of these bands is important for several reasons, of which we shall mention one at this stage. It proves that a system of triplet states analogous to the states of the orthohelium line spectrum really exists in the spectrum of H 2 and to that extent confirms the view we have taken of the structure of this spectrum. This follows since the singlet 2 p 1 II ab levels have now been firmly identified with the C level of Dieke and Hopfield; the final levels of the present band systems are undoubtedly 2 p II ab levels, and there is no room for any other 2 p II level in the singlet system. The notation here used is that proposed by Mulliken. It has been described by one of us in the 'Transactions of the Faraday Society,’ vol. 25, p. 628. It is assumed that in all the electronic states of H 2 with which these bands are concerned only one electron gets excited, the other being in an s state ( l = 0). Thus the resultant azimuthal quantum number L of the two electrons is equal to that of the azimuthal quantum number l of the excited electron. The magnitude of both these quantities is thus expressed by the letters s (for l = 0), p (for l == 1), d (for l ==2), etc., in such symbols as 2 p 3 II. In addition we have to specify A the resolved part of L about the molecular axis. This is indicated by the symbols Σ for Ʌ = 0, II for Ʌ = 1, Ʌ for Ʌ = 2. etc. The first number, such as 2 in 2 p 3 II ab , indicates the principal quantum number n and the second such as 3 shows that the level is believed to be a triplet level. The suffixes ab distinguish the double character of II, Δ, etc., levels which arise according to whether the value of Ʌ is positive or negative.

The relativistic theory of an atom with many electrons

This paper derives the ordinary classification of multiplets, and the selection and summation rules, from Dirac's relativistic equation. The non-relativistic theory of the inner quantum number j and the magnetic quantum number u , and their selection rules, was worked out for an atom with any number of point-electrons by Born, Heisenberg and Jordan, using matrices, and by Dirac, using q -numbers. The two methods are equivalent, and depend principally upon the properties of the total angular momentum. 2 points out that the total angular momentum has the same properties in the new theory, so that the previous work can be taken over with scarcely any amendment. 3. deals with a selection rule that has received little theoretical attention. The azimuthal quantum number for a single electron is denoted by k , and Σ k is the sum for all the orbits involved in a given state. It is known empirically that Σ k always changes by an odd number. This is the basis of the distinction between S, P, D, ... and S', P', D', ... terms. The rule is proved rigorously in the absence of external fields. A practical consequence is that the O ++ lines of nebular spectra, if rightly identified, can occur only in electric or non-uniform magnetic fields, for they have ∆Σ k = 0.

Application of quantum mechanics to the stark effect in helium

Many interesting features of the Stark-effect may be seen with unusual clearness in the arc spectra of helium, and these we shall mention very briefly before referring to the theory. The fact that an electric field applied to the source will bring out the combination lines 2 p — mp was discovered by Koch in 1915. Since that time many investigators have shown by Lo Surdo’s method that a moderate external field is sufficient to remove all restrictions with regard to changes in the azimuthal quantum number.