The linear operator
T(f) = -fxx-fyy-fzz-2/r f
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on smooth functions in R3 has eigenfunctions
fn,l,m to the eigenvalues -n-2. The eigenfunctions
are of the form
fn,l,m = Rn,l(r) Yl,m(t,s)
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with so called spherical harmonics Yl,m(t,s)=Pl|m|(cos(t)) exp(i m s).
The functions Plm(z) are called Legendre spherical functions
and Rn,l(r) are called Laguerre polynomials.
T is the energy operator of the hydrogen atom. For each n,
there are n2 eigenfunctions. The function f describes an electron
with energy -n-2. The energy differences
1/n12 - 1/n22
can literally be "seen". The number n is called the
principal quantum number, the number l is related to the total angular momentum,
and m is related to the z-component of the angular momentum.
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| Lyman | Balmer | Paschen | Bracket
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| n1=1 | n1=2 | n1=3 | n1=4 |
| Ultraviolet | visible | infrared | infrared |
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