Half-quantum numbers

Wc must add the amount of energy at the bottom of the bowl" in Fig. 10-lb to the siirri fix)rri Exercise 10-5. This energy is one-half a quantum at the wavenumber extrapr)lated rnie-half quantum number below n 0 (see Pn)blerns). 

The experimental figures, with one exception, were obtained from oscillation-rotation spectra with the use of integral rotational quantum numbers by Kratzer, Z. f. Physik, vol. 3, p. 289 (1920). The second figure for hydrogen chloride was calculated by Colby, Astrophys. Journ., vol. 58, p. 303 (1923), from the same data, with the use of half quantum numbers, and by Czerny,... 

Z. f. Physik, vol. 34, p. 227 (1925), from pure rotation spectra with half quantum numbers. 

For systems of the type under discussion, the second term introduces half-quantum numbers i.e., with y = y0 +... 

The value can arise by the angular momentum of the electrons being A/2tt and making an angle of 30° with the nuclear axis. This assumption leads, however, to difficulties in connection with the intensities given by the correspondence principle.1 For this reason Kramers and Pauli return to the assumption = , =0, in other words to an electron momentum (with a half quantum number) perpendicular to the nuclear axis. 

In connection with half-quantum numbers, we will also consider the adiabatic transition in the gradual separation of the two nuclei. From (22) and (21) we see that the quantum condition for J can be written ... 

Complications included the occurrence of half quantum numbers, the problem of the anomalous Zeeman effect, and the doublet riddle. P. Forman, The Doublet Riddle and Atomic Physics circa 1924, Isis, 59, 156—174,1968. 

V l5ini7iand S = I, respectively.. STmist be positive and can assume either integral or half-integral values, and the quantum numbers lie in the mterval... 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

The presence of the half-odd quantum number j in Eq. (69) is potentially a physically measurable consequence of geomehic phase, which was first claimed to have been detected in the spectrum of Naa [16]. The situation is, however, quite complicated and the first unambiguous evidence for geometric phase in Nas was reported only in 1999 [17],... 

Thus, we can use the approximate quantum number m to label such levels. Moreover, it may be shown [11] that (1) 3/m is one-half of an integer for the case with consideration of the GP effect, while it is an integer or zero for the case without consideration of the GP effect (2) the lowest level must have m = 0 and be a singlet with Ai symmetry in 53 when the GP effect is not taken into consideration, while the first excited level has m = 1 and corresponds to a doublet E conversely, with consideration of the GP effect, the lowest level must have m = j and be a doublet with E symmetry in S, while the first excited level corresponds to m = and is a singlet Ai. Note that such a reversal in the ordering of the levels was discovered previously by Hancock et al. [59]. Note further thatj = 3/m has a meaning similar to thej quantum numbers described after Eq. (59). The full set of quantum numbers would then be... 

Beeause Pij obeys Pij Pij = 1, the eigenvalues of the Pij operators must be +1 or -1. Eleetrons are Fermions (i.e., they have half-integral spin), and they have wavefunetions whieh are odd under permutation of any pair Pij P = - P. Bosons sueh as photons or deuterium nuelei (i.e., speeies with integral spin quantum numbers) have wavefunetions whieh obey Pij P = + P. 

Fermions are particles that have the properties of antisymmetry and a half-integral spin quantum number, among others. 

Phosphorus has only one stable isotope, J P, and accordingly (p. 17) its atomic weight is known with extreme accuracy, 30.973 762(4). Sixteen radioactive isotopes are known, of which P is by far the most important il is made on the multikilogram scale by the neutron irradiation of S(n,p) or P(n,y) in a nuclear reactor, and is a pure -emitter of half life 14.26 days, 1.7()9MeV, rntan 0.69MeV. It finds extensive use in tracer and mechanistic studies. The stable isotope has a nuclear spin quantum number of and this is much used in nmr spectroscopy. Chemical shifts and coupling constants can both be used diagnostically to determine structural information. 

The number of energy levels found to date, with the aid of the Zeeman effect and the isotope shift data, is 605 even and 586 odd levels for Pu I and 252 even and 746 odd for Pu II. The quantum number J has been determined for all these levels, the Lande g-factor for most of them, and the isotope shift for almost all of the Pu I levels and for half of those of Pu II. Over 31000 lines have been observed of which 52% have been classified as transitions between pairs of the above levels. These represent 23 distinct electron configurations. 

The integer n labels the wavefunctions and is called a quantum number. In general, a quantum number is an integer (or, in some cases, Section 1.10, a half-integer) that labels a wavefunction, specifies a state, and can be used to calculate the value of a property of the system. For example, we can use n to find an expression for the energy corresponding to each wavefunction. 

According to the old quantum theory, the orbit of an electron moving in such a field consists of a number of elliptical segments. Each segment can be characterized by a segmentary quantum number n, in addition to the azimuthal quantum number Ic, which is the same for all segments. In all cases it is found that about half of the entire orbit lies in the outermost (j.th) region. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

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