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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 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. The function Rnl describes how the wave function varies with the distance of the electron to the nucleus; i.e., the radial part of the wave function. This function is characteristic for the hydrogen atom.
Because the subscripts n and l can take on more than one value, Rnl stands really for many different functions. It can be shown that n is always a positive integer; i.e., n = 1,2,.... It's called the principal quantum number . The quantum number l is always a non-negative integer. It's called the angular quantum number . For a given n it is restricted to l = 0,1,2,...,n-1. Combining n and l gives the functions R1,0, R2,0, R2,1, R3,0, R3,1, R3,2, etc. The functions are always grouped. The functions with n = 1 are called the first shell or the K shell, those with n = 2 are called the second shell or the L shell, those with n = 3 are called third shell or the M shell, etc.
The function R10 is given by
| R10(r) = 2 |
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The constant a0 is called the Bohr radius. It is introduced to simplify the expression for the R functions. It is defined as
and it is equal to the radius of the orbit of the electron in the Bohr model of the hydrogen atom when the electron has the lowest possible energy.
The function R20 is given by
| R20(r) = 2 |
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2a0
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Compared with R10 this function has not only an exponentially decreasing factor, but also a polynomial factor. This factor becomes zero for r = 2a0. This is important for the energy of the wave function.
The function R21 is given by
| R21(r) = |
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Ö3
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2a0
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This function too has also a polynomial factor. This factor becomes zero for r = 0. In general, the functions Rnl with l > 0 are all zero for r = 0.
For higher values of n the polynomial factor gets more and more terms, and there are more values for r for which Rnl will be zero.
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