Spherical polar coordinates state functions

Suppose that the electron is in a 2p state with angular momentum proportional to cos 6 in spherical polar coordinates. The probability density (a ) 2 of such a state would be concentrated near the z-axis, where the length of the radius vector is proportional to cos2 9. Now suppose that the whole system is physically rotated, e.g. by the application of a magnetic field (active rotation) - alternatively the axes may be thought of as rotated in the opposite direction (passive rotation). After rotation the system has a new wave function ip (x) with ip x) 2 concentrated around a displaced axis, but the value of the new wave function at a rotated point must be the same as that of the old wave function at the original point,... 

Since the interaction (4.304) is central, the associate wave equation may be separated in spherical polar coordinates to produce the normalized radial function. For the bound states hydrogenic atoms in the case of an infinitely heavy nucleus it looks like (Bransden Joachain, 1983) ... 

Consider the collision of two particles initially in internal states described by an index i. To simplify notation, it is convenient to use a single index to specify the states of both particles. The angle between the initial and final relative velocities v and 1/ is given by spherical polar coordinates and , where 0 is the deflection angle in the center of mass frame. We start with a well-defined beam of particles with a flux li (number of particles per unit area per unit time). After the collision, the flux Ij (number of particles per unit solid angle per unit time) is a function of the deflection angle 0 and is different for each possible set of final internal states j. We define the differential cross-section as... 

The three diagonal elements of the Cartesian polarization matrix can be simplified to two in spherical polar coordinates in a similar way to the d-orbital functions considered in Section 5.8. So, in the character tables, functions used for the d-orbitals can also be used to identify Raman-active vibrational modes. Thus, the Raman selection rule can be stated... 

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