Quantum number parity

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

In these equations, J and M are quantum numbers associated with the angular momentum operators and J, respectively. The number II = 0, 1 is a parity quantum number that specifies the symmetry or antisymmetry of the column vector with respect to the inversion of the nuclei through G. Note that the same parity quantum number II appears for and Also, the... 

Nuclear spin quantum number of ground (g) and excited (e) state (the sign refers to the parity)... 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

Since 72 = 1, the operator has two eigenvalues, 1. In summary, the photon state may be uniquely specified by giving four quantum numbers to quantify the energy u>, the angular momentum j, the component of angular momentum M and the parity A. The normalized wave function is of the form... 

With the prospect for realizing a symmetry restriction based on - S established, it is interesting to consider the possible magnitude of the effect. For 02, J - 5" is a well-defined quantum number with odd and even values corresponding to e and/parity label states respectively. The e/f notation refers to the... 

The identity operator within the space of functions with a hxed value of J and the parity (denoted by p), and that are associated assymptotically with a quantum number K of the body-fixed z component of the total angular momentum, is... 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

Hence, = I + 1 if k > 0 and = I — 1 if k < 0. Consequently, in the Dirac-Pauli representation and have definite parity, (—1) and (—1) respectively. It is customary in atomic physics to assign the orbital angular momentum label I to the state fnkm.j- Then, we have states lsi/2, 2si/2) 2ri/2, 2p3/2, , if the large component orbital angular momentum quantum numbers are, respectively, 0,0,1, ,... while the corresponding small components are eigenfunctions of to the eigenvalues 1,1,0,2,. 

One can see that E is a vector, whereas B is a pseudovector, that is, E changes sign upon inversion of the coordinate system, while B remains unchanged. As a consequence, electric-field-induced interactions couple states of different parity, while interactions induced by the magnetic field conserve parity. Thus, parity remains a good quantum number for quantum systems in a magnetic field. 

Now the parity of a multielectron, ionic, or atomic wave function is given by TT ( — l)/fc, where lk (the angular-momentum quantum number)... 

If the rare-earth ion is immersed in a crystal field, the perfect symmetry of the free ion is destroyed, leaving parity in some cases not quite a good quantum number. Under this circumstance, electric-dipole transitions become quite possible. It was Van Vleck (25), in his classic 1937 paper The Puzzle of Rare Earth Spectra, who first pointed out that the weak electric dipole emission was due to this mixing of states of opposite parity by the crystal field. 

Another example is the particle in a box. With the origin at the center of the box, the potential energy is an even function, and the wave functions are of definite parity, determined by whether the quantum number is odd or even. Hence for electric-dipole transitions, the quantum number must go from even to odd, or vice versa, as concluded previously. 

The integral < vib vib) maY vanish because of symmetry considerations. For example, the C02 normal mode v3 in Fig. 6.2 has eigenvalue — 1 for the inversion operation. Hence (Section 6.4), the v3 factor in the vibrational wave function is an even or odd function of the normal coordinate Q3, depending on whether v3 is even or odd. For a change of 1,3,5,... in the vibrational quantum number v3, the functions p vib and p"ib have different parities and their product is an odd function, so that ( ibl vib) vanishes. Thus we have the selection rule Ac3 = 0,2,4,... for electronic transitions in... 

For rotational excitation of HC1 by Ar at room temperature, the maximum rotational angular momentum quantum number coupled during collision is about 12. The maximum number of coupled j,m states is Nc = (jmax + 1)(jmax + 2)/2 = since HC1 is a heterodiatomic molecule, and thus all states of the same total parity are coupled. With 91 channels, the quantum scattering calculations are feasible, but rather expensive. A further complication of the... 

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