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The interaction between an electron and the nucleus of an atom depends only on the distance between the electron and the nucleus. This means that it makes sense not to work with Cartesian coordinates x, y, and z for the electron, but with the distance r to the nucleus and some additional coordinates to completely specify an electron's position. The remaining coordinates are two angles q and j. They form together with r the spherical coordinates. If r is the vector from the nucleus to the electron, then q is the angle between r and the z-axis. If r¢ is the projection of r on the xy-plane, then j is the angle over which the positive x-axis has to be rotated around the positive z-axis to point in the same direction as r¢. The following figure should make the definitions of the angles clearer.
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A reference frame and the definition of the spherical coordinates r,
q, and j.
The relation between the Cartesian coordinates and the spherical coordinates is as follows
Expressing the spherical coordinates in terms of the Cartesian coordinates is more difficult. The distance r given by
and the angle q by
| q = arccos(z/ |
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Öx2+y2+z2
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The angle j is tricky. We have
but this does not completely specify j because tan(j) = tan(j+p). To determine the correct value of j it is therefore necessary to look at the signs of x and y to see in which quadrant j has to be. The value for j obtained from tanj = y/x may need to be corrected by adding or subtracting p.
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