3.5: Some Useful Expressions
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Listed below are a number of equations which give the dependence of 
, 
and 
on the quantum numbers n, l and m. They refer not only to the hydrogen atom but also to any one-electron ion in general with a nuclear charge of Z. Thus He+ is a one-electron ion with Z = 2, Li+2 another example with Z = 3.
The average distance between the electron and the nucleus expressed in atomic units of length is:
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Note that 
is proportional to n2 for l = 0 orbitals, and deviates only slightly from this for l ¹ 0. The value of 
decreases as Z increases because the nuclear attractive force is greater. Thus 
for He+ would be only one half as large as 
for H.
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