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The solutions of the Schrödinger equation for the hydrogen atom can be written as
| Ynlm(r,q,j) = Rnl(r)Slm(q,j). |
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The subscripts are so-called quantum numbers. The function Rnl describes how the wave function varies with the distance of the electron to the nucleus. This function is characteristic for the hydrogen atom. The function Slm describes the angular dependence of the wave function. It can be shown that this function is determined completely be the fact that the hydrogen atom is spherical symmetric. Because all other atoms are spherical symmetric as well, the electrons in all atoms have the same angular dependence; i.e., we can use the same function Slm.
Because the subscripts l and m can take on more than one value, Slm stands really for many different functions. It can be shown that l is always a non-negative integer; i.e., l = 0,1,2,.... It's called the angular quantum number . The number m is also always an integer, but its value range from -l to l; i.e., m = -l,-l+1,-l+2,...,l-2,l-1,l. It is called the magnetic quantum number . Combining l and m gives the angular functions S0,0, S1,-1, S1,0, S1,1, S2,-2, S2,-1, S2,0, S2,1, S2,2, etc. The functions are always grouped. The functions with l = 0 are called s functions, those with l = 1 are called p functions, those with l = 2 are called d functions, and those with l = 3 are called f function. After f we follow the alphabet; i.e., we get g, h, etc. (The names of the s, p, d and f functions stand for ``sharp,'' ``principal,'' ``diffuse,'' and ``fundamental,'' and refer to a description of the shape of the lines of the hydrogen spectrum.)
The function S00 is given by
We see that this function is the same for all values of the angles. This means that the function is spherical symmetric. For l = 1, the p functions, we get
These function do depend on the angles. They seem different, but they really have the same shape. This can be seen by writing them in a combination of spherical and Cartesian coordinates.
These expression show that they only have a different orientation. The next figure shows a contour plot of S1,0. The contours are drawn on a sphere, because the angles can be viewed as describing a point on the sphere; q is the latitude, and j is the longitude. The function S1,0 has a maximum at the north pole, is zero on the equator, and has a minimum at the south pole. |
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For higher values of l the functions rapidly become quite difficult. For l = 2, the d functions, we get
Expressions in spherical coordinates can be obtained by substituting the proper expressions for x, y, and z in spherical coordinates. These function have again the same shape, except S2,0. A contour plot of this function is shown in the next figure. |
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