Coordinate rotation

The quantity in square brackets looks Hke an ordinary stationary state, suggesting that the real part of the energy (Eg) is the resonance energy. The imaginary part of the energy (—T/2) contributes an envelope function that decays exponentially on a timescale t h/r. This is consistent, up to factors of order unity, with the time-energy uncertainty principle in Eq. [19], if we take A T. It is therefore not surprising that the quantity T is known as the resonance width. The lifetime of the metastable resonance is t h/Y. 

The underlying idea behind the complex coordinate rotation (CCR) method that is suggested by the Balslev-Combes theorem is a complex scaling of the Cartesian coordinates in the Hamiltonian operator, each by the same complex phase factor x xe. This transformation defines a new, complex-scaled Hamiltonian, H H 0). In one dimension (for simplicity), the complex-scaled Hamiltonian is 

This idea is readily extended to the Born-Oppenheimer electronic Hamiltonian by noting that x -r xe implies that interparticle coordinates should be scaled asr re . For 0 0, the operator H 6) is non-Hermitian and therefore admits complex eigenvalues. In its simplest form, the CCR method consists of determining these eigenvalues. 

The scattering wave functions, on the other hand, will behave something like at long range. On scaling r re, these continuum functions will not remain finite as r oo unless k - which also makes sense in terms 

With these examples in hand, a pedagogical version of the Balslev-Combes theorem can be stated as follows. 

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Fig. 3 The effect of a counter-clockwise coordinate rotation about the z axis.

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

Since our spin functions are eigenfunctions of S2, we can drop the last term in Eq. (70), because it contributes a constant term to all energy levels and hence drops out when we compute energy differences. Many workers prefer to add a term — 5(5+ 1) to the spin Hamiltonian to get a Hamiltonian which transforms readily under a coordinate rotation. We thus have forJt ... 

Fig. 2.16 In general, the stress state represented on a differential element in a cylindrical coordinate system has nine stress components. The same stress state can be represented as its principal components via a coordinate rotation.

Just as there arc many types of fluids, so there arc. partly as a result, many types of fluid flow. Uniform flow is steady in lime, or the same at all points in space. Steady flow is flow of which the velocity at a point fixed with respect to a fixed system of coordinates is independent of lime. Many common types of flow can be made steady by a suitable choice of coordinates. Rotational flows have appreciable vorticily, and they cannot he described mathematically by a velocity potential function. Turbulent flow is flow in which the fluid velocity at a fixed point fluctuates with lime in a nearly random way. The motion is essentially rotational, and is... 

As with Ps , there is only one bound state of Ps2 but there exist Rydberg series of autodissociating states arising from the attractive interaction between one of the positrons and the residual Ps- (or between one of the electrons and the charge conjugate of Ps ). The positions and widths of several of these states were determined by Ho (1989) using the complex coordinate rotation method. To date Ps2 has not been observed in the laboratory. 

J. Simons, The complex coordinate rotation method and exterior scaling A simple example, Int. J. Quant. Chem. 14 (1980) 113. 

Y.K. Ho, The method of complex coordinate rotation and its applications to atomic collision processes, Phys. Rep. 99 (1983) 1. 

G.D. Doolen, J. Nuttal, R.W. Stagat, Electron-hydrogen resonance calculation by the coordinate-rotation method, Phys. Rev. A 10 (5) (1974) 1612. 

Fano, U. (1960). Real representations of coordinate rotations, J. Math. Phys., 1, 417-423. 

To first order, any function T(ra) is transformed under this coordinate rotation into... 

We can employ an analog of the spatial transformation operator [138] for analyzing the transformation properties of a spinor under coordinate rotations. The evaluation of the corresponding 2D transformation matrices is simplified if we rewrite... 

The above description of the excited states in terms of excitation amplitudes is frame and basis set dependent. A more convenient description is in terms of state multipoles. It can be generalised to excited states of different orbital angular momentum and provides more physical insight into the dynamics of the excitation process and the subsequent nature of the excited ensemble. The angular distribution and polarisation of the emitted photons are closely related to the multipole parameters (Blum, 1981). The representation in terms of state multipoles exploits the inherent symmetry of the excited state, leads to simple transformations under coordinate rotations, and allows for easy separation of the dynamical and geometric factors associated with the radiation decay. 

Redi M. H. (1982) Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr. 12, 1154—1158. 

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