Angular momentum three-dimensional rotation

To obtain the selection rules on AJ and AM, we exploit the properties of vector operators. All quantities that transform like vectors under three-dimensional rotations have operators exhibiting commutation rules that are identical to those shown by the space-fixed angular momentum operators f,c->... 

The molecule also has angular momentum, which you would expect it to have because it is rotating. The quantum number / is used to define the total angular momentum of the molecule rotating in three dimensions. The total angular momentum of a molecule is given by the same eigenvalue equation from three-dimensional rotational motion ... 

The angular momentum for the particle can now be determined. When the particle is confined to rotate in only two-dimensions (i.e. confined to rotate on a ring), the angular momentum is parallel to the z-axis and is fully determined by the value of m,. In three-dimensional rotation, the angular momentum need not be parallel to the z-axis and may also have components in the x and y-axes. The operators for the components of the angular momentum L in Cartesian coordinates are as follows ... 

However, as mentioned above, T c)3) will be orthogonal to all the k states, and T ) is nonzero. This implies that the number of total states of the same eigenvalue E is (k + 1), which contradicts our initial hypothesis. Thus, we conclude that k must be even, and hence proved the generalized Kramers theorem for total angular momentum. The implication is that we can use double groups as a powerful means to study the molecular systems including the rotational spectra of molecules. In analyses of the symmetry of the rotational wave function for molecules, the three-dimensional (3D) rotation group SO(3) will be used. 

Dynamical symmetries for three-dimensional problems can be studied by the usual method of considering all the possible subalgebras of U(4). In the present case, since one wants states to have good angular momentum quantum numbers, one must always include the rotation algebra, 0(3), as a subalgebra. One can show then that there are only two possibilities, corresponding to the chains... 

Components of the angular momentum operator are connected with the infinitesimal operators of the group of rotations in three-dimensional space [11]. 

The metric coefficient in the theory of gravitation [110] is locally diagonal, but in order to develop a metric for vacuum electromagnetism, the antisymmetry of the field must be considered. The electromagnetic field tensor on the U(l) level is an angular momentum tensor in four dimensions, made up of rotation and boost generators of the Poincare group. An ordinary axial vector in three-dimensional space can always be expressed as the sum of cross-products of unit vectors... 

Much of the beauty of high-resolution molecular spectroscopy arises from the patterns formed by the fine and hyperfine structure associated with a given transition. All of this structure involves angular momentum in some sense or other and its interpretation depends heavily on the proper description of such motion. Angular momentum theory is very powerful and general. It applies equally to rotations in spin or vibrational coordinate space as to rotations in ordinary three-dimensional space. 

Orbital angular momentum is associated with rotational motion in three-dimensional space. In terms of the operators representing the position r and linear momentum p of a particle, we have the important expression for the orbital angular momentum... 

The rigid rotor model assumes that the intemuclear distance Risa constant. This is not a bad approximation, since the amplitude of vibration is generally of the order of 1% of i . The Schrbdinger equation for nuclear motion then involves the three-dimensional angular-momentum operator, written J rather than L when it refers to molecular rotation. The solutions to this equation are already known, and we can write... 

---