Radial Equation Separation

Substituting the form

into the hydrogen
Schrodinger equation allows all the partial
derivatives to be expressed as ordinary
derivatives. Direct substitution and a bit of rearranging gives

Note that the first two terms contain only variable r, and the
remaining terms contain the angular variables. Since the equation must
hold for all values of these variables, the collection of radial terms
must be equal to a constant. This separates out the radial
equation. The sum of the angular terms must therefore be equal to the
negative of that constant. Also, the terms in the two angles must
separately equal a constant since they must hold for all angle values.
This separates the two angular parts into the colatitude
and azimuthal
equations.

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Schrodinger
equation concepts

Hydrogen
concepts

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The Radial Equation

Upon separation
of the Schrodinger
equation for the hydrogen
atom, the radial equation is:

In order to separate the equations, the radial part is set equal to a
constant, and the form of the constant on the right above reflects the
nature of the solution of the colatitude
equation which yields the orbital
quantum number.

Solution of these equations under the constraints
placed on the wavefunction leads to series solutions in the form of
polynomials called the associated Laguerre functions. In order to fit the
physical boundary conditions, these solutions contain a parameter n which
can take only positive integer values; this parameter is called the principal
quantum number. The form of the radial solutions is

where
is the associated Laguerre function. The first few radial wavefunctions R
are shown as part of the hydrogen
wavefunctions.

Index

Schrodinger
equation concepts

Hydrogen
concepts

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The Principal Quantum Number

The principal quantum number or total quantum number n arises from the
solution of the radial
part of the Schrodinger
equation for the hydrogen atom. The bound state energies of the
electron in the hydrogen atom are given by

In the notation of the periodic table, the main shells of electrons are
labled K(n=1),L(n=2),M(n=3), etc. based on the principal quantum number.

Quantum
numbers for atomic electrons

Index

Schrodinger
equation concepts

Hydrogen
concepts

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Quantum Physics

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