The Rydberg equation is true in the strictest sense only for the so-called central
field in which a single electron moves around a spatially fixed nucleus with the
charge Z×e.However,for the analysis of all other atomic or molecular
spectra, Bohr`s atomic model is insufficient,because of the effects of the
numerous electrons in atoms with a higher atomic number than hydrogen.
Moreover,the Bohr model requires only a single quantum number n,essentially
taking only one spatial dimension into account,while in reality we expect at least
three quantum numbers,corresponding to the Cartesian coordinate systems.
The assumption of plane orbitals for moving electrons does not reflect the
three-dimensional properties of space.Bohr himself did not believe in plane
orbitals for long.
In 1926 the Austrian Erwin Schrodinger formulated quantum mechanics for
atomic systems with his famous equation as he took(following the concept of
de Broglie)the wave properties of matter into account.Here we will describe
only the essential ideas that allow us to understand the spectroscopic concepts.
We will encounter some statements that seriously contradict our macroscopic
view of nature,a problem which in principle cannot be avoided.Firstly, what
is it that vibrates in de Broglie’s concept of the de Broglie wave of an electron?
For electrons in an atom we describe the probability, ψ2,of finding an electron in
a given volume element dV.ψ is the eigenfunction,or wavefunction and is commonly
referred to as an orbital.From the mathematical viewpoint ψ is a complex
function with a real and an imaginary part,which describes any(microscopic) system
completely in space and time,an entity which carries a sign( however ,only in
a few exceptional cases is ψ known exactly).The density of probability is,according
to the laws of physics,defined by the square ψ2 ( i.e.,a real number),the
probability of finding an electron in the volume element dV is obtained by
ψ2dV . The probability of finding the electron somewhere in space is of course
100%,i.e.,∫ψ2dV=1(normalization of the eigenfunction).The Schrodinger
equation is analogous to the equation of motion in classical mechanics which,
for example,describes the vibration of a one-dimensional string with its fundamental
and over-tones.The equation is a partial difierential equation that connects ψ,
V, kinetic(Ek)and potential energy(Ep)of a system.
The short form of the Schrodinger equation does not tell the beginner
much:H ψ=E ψ,where H is the Hamiltonian operator, E the energy
and ψ the wavefunction or eigenfunction.The Hamiltonian operator,as
a mathematical entity,“extracts”the energy value out of the corresponding
eigenfunctions ψ.