Angular momentum component quantum number

The quantum number K gives the rotational angular-momentum component along a molecule-fixed axis of the spherical top. From (5.34) and... 

Here y is the component of the transition-dipole operator in the direction of the light s electric field vector E, Jj, M, and p are the energy, total angular momentum, its space-fixed projection, and the parity of the initial bound state k, v, j, and irij are the relative momentum, vibrational quantum number, rotational angular momentum, and its space-fixed projection for the scattering state. 

This is the Wigner-Eckart theorem, a very important result which underpins most applications of angular momentum theory to quantum mechanics. It states that the required matrix element can be written as the product of a 3- j symbol and a phase factor, which expresses all the angular dependence, and the reduced matrix element (rj, j T/ (d) if. j ) which is independent of component quantum numbers and hence of orientation. Thus one quantity is sufficient to determine all (2j + 1) x (2k + 1) x (2/ + 1) possible matrix elements (rj, j, mfIkq(A) rj, jf m ). The phase factor arises because the bra (rj, j, m transforms in the same way as the ket (— y m rj, j, —m). The definition of the reduced matrix element in equation (5.123), which is due to Edmonds [1] and also favoured by Zare [4], is the one we shall use throughout this book. The alternative definition, promoted by Brink and Satchler [3],... 

For describing the photons with the angular set of quantum numbers jlM the vector spherical functions are introduced. The three-component photon spin wave function can be considered as the three-dimensional vector x. Then the spherical wave function YjiM(f) in momentum space is defined as ... 

K i the quantum number of the angular momentum component in the direction of the... 

List the possible values of L, S,Ms,J, and /Wj for the following (a) two coupled p electrons in different shells, (b) two coupled f electrons in different shells, (c) two coupled electrons, one a p electron and one a d electron. Remember that the z-component quantum numbers depend on the values of the total angular momentum quantum numbers. 

The terms of an atom with a valence electron are characterised by four quantum numbers, n, Z, j, m. Here I corresponds to the angular momentum of the electronic orbit in the model, j to the total angular momentum of the atom, and m to its component in the direction of an externed field. For the formal representation we can ascribe the angular momentum component which contributes to j apart from... 

For two angular momentum vectors, Ji and Jj, with associated quantum numbers and both equal to 1, there are (2 i + 1)(2 2 + 1) = 9 different combinations of their possible orientations wUh respect to the z-axis. These 9 ways are shown here, and for each a resultant vector from the sum of Jj and J2 has been drawn. In each case, the z-component of the resultant vector can be obtained from the sum of the z-component quantum numbers... 

The vector L is so strongly coupled to the electrostatic field and the consequent frequency of precession about the intemuclear axis is so high that the magnitude of L is not defined in other words L is not a good quantum number. Only the component H of the orbital angular momentum along the intemuclear axis is defined, where the quantum number A can take the values... 

The component of the total (orbital plus electron spin) angular momentum along the intemuclear axis is Qfi, shown in Figure 7.16(a), where the quantum number Q is given by... 

According to quantum mechanics, the maximum observable component of the angular momentum is Ih/lir, where h is Planck s constant. A nucleus can assume only 21+1 energy states. Associated with each of these states is a magnetic quantum number m. where m has the values I, I — I, I —2,, —1+ 1, —I. 

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