Angular momentum cartesian components

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

The spherical harmonics in real form therefore exhibit a directional dependence and behave like simple functions of Cartesian coordinates. Orbitals using real spherical harmonics for their angular part are therefore particularly convenient to discuss properties such as the directed valencies of chemical bonds. The linear combinations still have the quantum numbers n and l, but they are no longer eigenfunctions for the z component of the angular momentum, so that this quantum number is lost. 

Polarization functions may be optionally chosen to be of pure or Cartesian form (by another keyword). In the former case, one includes the expected number of angular-momentum components (i.e., five d orbitals, seven f orbitals, etc.), whereas in the latter case some additional component(s) of lower angular momentum are included (e.g., a Cartesian d set includes five d orbitals plus one s orbital, a Cartesian f set includes seven f orbitals plus three p orbitals, and so forth). 

Equation (13) shows that the complete temperature and field dependence of the strains can be calculated from static correlation functions (J Jj )7-,h (y, y — 1.2,3 label the cartesian components of the angular momentum J) where O7- h denote thermal expectation values (Callen and Callen 1965). As already mentioned above, a mean field theory may be used to evaluate (13) and calculate the magnetostriction. 

The angular momentum operator is a first-rank tensor whose Cartesian and spherical components transform into each other as... 

The matrix elements of the angular momentum operators in Cartesian directions a = x,y, z form (complex) matrices La. Now we can proceed with an evaluation of the /1-tensor components in three steps. 

The connection between the covariant cyclic and cartesian coordinates of the vector J yields Eq. (A.6), whilst (A.5) makes it possible to form the vector itself out of the components (J)q. As follows from (2.18), the components of the multipole moment pq characterize the preferred orientation of the angular momentum J in the molecular ensemble. Fig. 2.3(a, b) shows the probability density p(0, [Pg.30]

Figure 2.12 Definition of the components of angular momentum in cartesian and in spherical polar coordinates.

It has the same form as the cartesian components and the solution, = ke tmf describes rotation about the polar axis in terms of the orbital angular momentum vector LZ1 specified by the eigenvalue equation... 

The commutation relations of the orbital angular momentum operators can be derived from those between the components of r and p. If we denote the Cartesian components by the subindices i, k, and /, we can use the short-hand notation... 

For the determination of matrix elements, it is often more convenient to use linear combinations of the Cartesian components of the angular momentum operator instead of the Cartesian components themselves. In the literature, two different kinds of operators are employed. The first type is defined by... 

As for the orbital angular momentum, the commutation relations between the Cartesian components of a general angular momentum / and its square modulus A read... 

A first-rank tensor operator 3 V) is also called a vector operator. It has three components, 2T and jH j. Operators of this type are the angular momentum operators, for instance. Relations between spherical and Cartesian components of first-rank tensor operators are given in Eqs. [36] and [37], Operating with the components of an arbitrary vector operator ( 11 on an eigenfunction u1fF) of the corresponding operators and 3 yields... 

After the Hamiltonian, the operators for angular momenta are probably the most important in quantum mechanics. The definition of angular momentum in classical mechanics is L = r x p. In terms of its Cartesian components. 

The quantities relevant to the rotationally averaged situation of randomly oriented species in solution or the gas phase must necessarily be invariants of the rotational symmetry. Accordingly, they must transform under the irreducible representations of the rotation group in three dimensions (without inversion), R3, just like the angular momentum functions of an atom. The polarisability, po, is a second-rank cartesian tensor and gives rise to three irreducible tensors (5J), (o), a(i),o(2), corresponding in rotational behaviour to the spherical harmonics, with / = 0,1,2 respectively. The components W, - / < m < /, of the irreducible tensors are given below. 

Here, we call the attention of the reader that our Eq. (24) in the previous interlude correspond to Eqs. (32) in Ref. [3] for the cartesian components of the angular momentum in the body frame and the inertial frame, respectively, in terms of the Euler angles. Notice that the angles if and

commutation rules from Eq. (22) in Ref. [3], for the analysis of the rotations of asymmetric molecules are as follows ... 

The work [5], to be reviewed in this section, makes use of the asymmetric distribution Hamiltonian, as well as the cartesian component, ladder and square of the angular momentum operators and their actions on the chosen spherical harmonic basis lnij, for (L/,k) = cyc x,y,z). Here, we start from its Eqs. (26-28) for the matrix elements of H in the alternative bases ... 

The linear momentum operator and its connection with the angular momentum operator in spheroconal and cartesian components ... 

For spinors the time reversal operation is not just complex conjugation. To find the effect of the time reversal operator T on a general angular momentum state j, m) we note that T anticommutes with any cartesian component of the angular momentum operator ... 

The nonnegative integers i,j, and k in Eq. (19) are related to the angular momentum of an electron in this AO as I = i+j+k. Gaussian functions are nearly always added in full shells (i.e., for a given orbital exponent a and a given l, all components i+j+k = l are included in the basis simultaneously, thereby treating all Cartesian directions equivalently. 

Theories of chemical bonding based on the properties of degenerate states with fixed I assume independent behaviour of the electrons in these states. In particular, for three electrons in the three-fold degenerate /i-state with 1=1, they are assumed to have distinct values of m, without mutual interference. To make this distinction it is necessary to identify some preferred direction in which the components of angular momentum are quantized. By convention this direction is labeled as Cartesian Z. If the electrons share the degenerate p-state with parallel spins, they must share the same direction of quantization. This being the case, only one of the electrons can have the quantum number m = 0, characteristic of the real function (7). 

As shown already, the linear combinations that define and Py are simple rotations of the coordinate axes, as, for instance, defined by (25). It follows that the functions p and Py are both characterized by the quantum number m = 0 that defines zero component of angular momentum, now directed along the Cartesian X and Y axes respectively. 

The quantum number I specifies the angular momentum of the electron in units of h (h bar), known as Planck s constant. In the presence of an applied magnetic field the component of angular momenmm in the direction of the field is quantified by Ml as = mih. The subscript z refers to the convention of defining a right-handed set of Cartesian laboratory axes such that Z coincides with the direction of the magnetic field. 

An important property of quantum-mechanical angular momentum is the prediction and demonstration that the Cartesian components thereof do not commute. This property is responsible for the fact that two components, such as and... 

Since many of the operators that appear in the exact Hamiltonian or in the effective Hamiltonian involve products of angular momenta, some elementary angular momentum properties are summarized in the next section. Matrix elements of angular momentum products are frequently difficult to calculate. A tremendous simplification is obtained by working with spherical tensor operator components and, in this way, making use of the Wigner-Eckart Theorem (Section 3.4.5). A more elementary but cumbersome treatment, based on Cartesian operator components, is presented in Section 2.3. 

An angular momentum operator may be defined by the commutation behavior of its space-fixed Cartesian components,... 

Choosing initial Cartesian coordinates for the polyatomic reactants follows the procedures outlined above for an atom + diatom collision and for normal-mode sampling. If the cross section is calculated as a function of the rotational quantum numbers J and K, the components of the angular momentum are found from... 

Cartesian coordinates and momenta for a polyatomic reactant are found from the energies of its normal modes [Eq. (2.33)] and the components of its angular momentum. The procedure is given by Eqn. (210) and steps 1-3 for microcanonical normal-mode sampling in Section II.A.3.a, and is applied to both reactants. Each reactant is randomly rotated through its Euler angles, as described by Eqs. (3.24) and (3.25), and the impact parameter, center-of-mass separation, and relative velocity added as described by Eqs. (3.26) and (3.27). 

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