Matrix trigonometric function

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

The coordinate map given by the variables (cD+, cD ) is a significant improvement as compared to eq. (3.25). Nevertheless, an explicit expression for an h matrix in its terms is still a clumsy combination of the trigonometric functions of two triples of reparametrizing angles w . It is known however that in the case of the SO(3) group [8] its quaternion [27] parameterization has the advantage that the matrix elements of SO(3) rotation matrices, when expressed in terms of the components of the normalized quaternion, are quadratic functions of these components. 

If we develop the torsional solutions of equation (113) on the basis of double products of trigonometric functions (built up with 16 cosine and 15 sine ones), the Hamiltonian matrix to be diagonalized would be of order 31 x 31 = 961. 

One sees from the functional dependence of the energy, eq (44), on trigonometric functions that the energies of the bands are of the standard form. Because of the one-dimensional nature of the problem and the 2x2 dimensions of the secular matrix, it is possible simply to include account of overlap automatically without specifically using methods of orthonormalization, such as Lowdin s method [42-44]. 

Note that this is a replacement, not an equation the sum in K-matrix theory is over ail the bound states which actually occur, whereas the trigonometric functions used to represent resonances in MQDT continue to repeat even outside the range of physical validity. They therefore include poles in an energy range below the physical resonances, whereas K-matrix theory does not. So the replacement should only be used when one is above the lowest resonances in the channel, which is anyway the range of validity of MQDT... 

A disadvantage of the Euler angle approach is that the rotation matrix contains a total of six trigonometric functions (sine and cosine for each of the three Euler angles). These trigonometric functions are computationally expensive to calculate. An alternative is to use quaternions. A quaternion is a four-dimensional vector such that its components sum to 1 0 + 1 + <72 + = 1- quaternion components are related to the Euler angles as follows ... 

In this way the coordinates of P2 are obtained as functions of the coordinates of Pi and the rotation angle. This derivation depends simply on the trigonometric relationships for sums and differences of angles. We may also express this result in the form of a matrix transformation. For this, we put the coordinates in a column vector... 

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