Quantum number I

The above definitions must be qualified by stating that for principal quantum number I there are only s orbitals for principal quantum number 2 there are only s and p orbitals for principal quantum number 3 there are only s, p and d orbitals for higher principal quantum numbers there are s, p, d and f orbitals. 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

It follows that the only possible values for la + Ip are S A and the computation of vibronic levels can be carried out for each K block separately. Matrix elements of the electronic operator diagonal with respect to the electronic basis [first of Eqs. (60)], and the matrix elements of T are diagonal with respect to the quantum number I = la + Ip. The off-diagonal elements of [second and third of Eqs. (60)] connect the basis functions with I — la + Ip and I — l + l — l 2A. 

The isotope has a nuclear spin quantum number I and so is potentially useful in nmr experiments (receptivity to nmr detection 17 X 10 that of the proton). The resonance was first observed in 1951 but the low natural abundance i>i S(0.75%) and the quadrupolar broadening of many of the signals has so far restricted the amount of chemically significant work appearing on this rcsonance, However, more results are expected now that pulsed fourier-transform techniques have become generally available. 

The bound states (where < 0) are characterized by the three quantum numbers n (the principal quantum number), I (the azimuthal quantum number) and mi (the magnetic quantum number). 

We often say that an electron is a spin-1/2 particle. Many nuclei also have a corresponding internal angular momentum which we refer to as nuclear spin, and we use the symbol I to represent the vector. The nuclear spin quantum number I is not restricted to the value of 1/2 it can have both integral and halfintegral values depending on the particular isotope of a particular element. All nuclei for which 7 1 also posses a nuclear quadrupole moment. It is usually given the symbol Qn and it is related to the nuclear charge density Pn(t) in much the same way as the electric quadrupole discussed earlier ... 

This term describes a shift in energy by Acim rn, for an orbital with quantum numbers I — 2, mi and that is proportional to the average orbital angular momentum (/z) for the TOj-spin subsystem and the so-called Racah parameters Bm, that in turn can be represented by the Coulomb integrals and The operator that corresponds to this energy shift is given by... 

In order to evaluate the above expression, solutions were found for the Schrodinger equation using the Morse potential for rotational quantum number i not equal to zero ... 

Quadmpole interaction lifts the degeneracy of nuclear states with spin quantum numbers I > 1/2, and is manifested in the Mossbauer spectmm as quadmpole splitting A q (as will be further discussed in Sect. 4.3). According to (4.5), the classical electric monopole and quadmpole interaction energies Ei and q are additive, that is, = E + Eq. 

The Hamiltonian operator for the electric quadrupole interaction, 7/q, given in (4.29), coimects the spin of the nucleus with quantum number I with the EFG. In the simplest case, when the EFG is axial (y = Vyy, i.e. rf = 0), the Schrddinger equation can be solved on the basis of the spin functions I,mi), with magnetic quantum numbers m/ = 7, 7—1,. .., —7. The Hamilton matrix is diagonal, because... 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

Although the expectation value r )ni cannot be obtained from equation (6.70), it can be evaluated by regarding the azimuthal quantum number I as the parameter in the Hellmann-Feynman theorem (equation (3.71)). Thus, we have... 

Angular momentum quantum number i 0 (n - 1) in integer steps... 

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

The diabatic LHSFs are not allowed to diverge anywhere on the half-sphere of fixed radius p. This boundary condition furnishes the quantum numbers >i x and 2 , each of which is 2D since the reference Hamiltonian h has two angular degrees of freedom. The superscripts n(, in Eq. (95), with n refering to the union of n. and, indicate that the number of linearly independent solutions of Eqs. (94) is equal to the number of diabatic LHSFs used in the expansions of Eq. (95). 

For high quantum numbers, i.e., high R, the number of distinct energy levels with energy not greater than s is given by (note that only the first octant of the sphere of quantum numbers, without the origin, is to be taken into account)... 

Show that in a hydrogen-like atom of nuclear charge Ze the average distance of the electron from the nucleus, in the stale described by quantum numbers /, >i, is... 

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