Polar co-ordinates

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide... 

In Figure 6.3 it is convenient to use polar co-ordinates, as they are the modulus and phase angle as depicted in Figure 6.2. From Figure 6.3, the polar co-ordinates are... 

The electrons within the atom are actually not quantised in parabolic coordinates, but instead, on account of the central field of the atom core, in polar co-ordinates. It would, then, not be logical to attempt to select favoured values of m and n3. Instead, we shall calculate the quantity... 

The averages maybe written in spherical polar co-ordinates as angular integrals, which are simple to evaluate numerically, and in some cases have analytic forms. 

General Solution of the LPBE. From the preceding discussion it is apparent that a general solution of the LPBE in spherical polar co-ordinates is needed, subject to the usual boundary conditions. Equation 1 can be easily separated into a product solution (19) which may be written in the form... 

Before proceeding with the discussion of the general case, it is interesting to consider the case of a tube of circular cross-section and to note how the present treatment agrees with Taylor s. In this case we take Ox to be the axis and a to be the radius of the tube and transform to polar co-ordinates... 

To calculate the integral, it is convenient to refer to polar co-ordinates and... 

In view of the trend to more controlled and stereoselective reactions with readily available, less expensive and environmentally non-problematic reagents, the light-induced inner-sphere electron transfer between M-C bonds of less polar co-ordinating organometallics (Zn, Al) and the organic substrate seems to be a particularly attractive alternative to thermal reactions from organolithium or -magnesium compounds. 

Diffusion on a spherical surface. Introducing the usual spherical polar co-ordinates we obtain the uniform probability density in solid angle 0) = (47t) - when 6 ranges from 0 to tt and from 0 to In. In equilibrimn with a field E... 

The distortion in the QeQs co-ordinate space is conveniently expressed in polar co-ordinates,... 

Here also the proper value parameter A depends on the nature of the plate and in effect represents the square of the frequency. The differential equation can be easily solved in polar co-ordinates (Appendix XVII, p. 297). In this case again we obtain possible forms of vibration... 

The solution can also be obtained in polar co-ordinates for the hydrogen atom, as a three-dimensional quantum problem this is shown in Appendix XVIII, p. 298). In this problem, it should be added, we cannot speak of ordinary boundary conditions, since the domain over which tlui independent variables range is the whole of three-dimcnisional space. Instead of boundary conditions, we have now a rule with regiird to the liehaviour of the wave function at infinity. The natural condition to impose is that the wave function should vanish at infinity more strongly than 1/r. This follows from the statistical intfvrprotation of the square of tlio amplitude of the wave function, as the f)r()ba,bility of the electron being found at a definite point of space. T,h( . (amdition is equivalent to this, that the electron must always be at a finite distance. 

Por such motions about a fixed centre the conservation of angular momentum (theorem of areas) holds as well as the conservation of energy. The former implies in the first place that the motion is all in one plane. We take this plane as the ajy-plane and transform to polar co-ordinates by means of the equations x = r coscf), y — r sin. The momentum theorem then gives... 

We see that the solution may be taken as the product of a function R depending on r only and a function > depending on cp only i.e. in polar co-ordinates the variables are separable. The differential equation can then be split up into two equations with the single independent variables r, respectively, by means of a separation parameter, which we shall call mh... 

Changing to three-dimensional polar co-ordinates r, 0, cf), we obtain the equation... 

If we now change to polar co-ordinates, we see (cf. Appendix XVIII p. 298) that... 

Further, in polar co-ordinates with polar axis we have 27ri 02/ dx/ 27ri dcj)... 

On introducing polar co-ordinates a, round the vector nQ — n as axis, we find... 

The Fourier transform of the electron density distribution e(r, [Pg.35]

Transformations of the co-ordinates among themselves which are frequently employed are those which transform rectangular coordinates into cylindrical or polar co-ordinates, and also those which correspond to rotations of the co-ordinate system. 

It may be shown quite generally, that such a two-body problem may be reduced to a one-body problem. We choose the centre of gravity of the two particles as the origin of co-ordinates 0 and determine the direction of the line joining m2 and m1 by the polar coordinates 9, tf). If then jq and r2 are the distances of the particles from 0, their polar co-ordinates will be rlt 9, and r2, tt—9, ir+ and further, r1+r2=r. The Hamiltonian function becomes... 

So far as the calculation is concerned it is immaterial whether we consider our problem as a one-body or as a two-body problem. In the first case we have a fixed centre of force, and the potential of the field of force is a function U(r) of the distance from the centre. In the second case we have two masses, whose mutual potential energy U(r) depends only on their distance apart they move about the common centre of gravity. As wc have shown generally in 20, the Hamiltonian function in polar co-ordinates is precisely the same for the two cases, if, in the one-body problem, the mass /x of the moving... 

We work with polar co-ordinates r, 0, and . Making use of the canonical transformation (13), 7, which transforms rectangular into polar co-ordinates, we obtain for the kinetic energy,... 

It is easy to see that the Hamilton-Jacobi differential equation is separable neither in rectangular nor in polar co-ordinates. It may, however, be made separable by introducing parabolic co-ordinates. We put... 

For E =0 the motion of the Stark effect passes over into the simple Kepler motion. This is separable in polar co-ordinates as well as in parabolic co-ordinates. From the separation in polar co-ordinates ( 22) we obtain the action variables Jr, Je, J0, and the quantum condition... 

If now we calculate the Kepler motion in parabolic co-ordinates, we have only to put E=0 in the above calculations. We obtain the action variables J(, J, and (the last has the same significance as in polar co-ordinates) and the quantum condition... 

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