The Hydrogen Atom

Spherical
polar coordinates

The solution of the Schrodinger
equation for the hydrogen atom is a formidable mathematical
problem, but is of such fundamental importance that it will be
treated in outline here. The solution is managed by separating
the variables so that the wavefunction is represented by the
product:

The separation leads to three equations for the three spatial
variables, and their solutions give rise to three
quantum numbers associated with the hydrogen energy levels.

Index

Schrodinger
equation concepts

Hydrogen
concepts

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Hydrogen Schrodinger Equation

The electron in the hydrogen
atom sees a spherically symmetric potential, so it is logical to
use spherical
polar coordinates to develop the Schrodinger
equation. The potential energy is simply that of a point
charge:

The expanded form of the Schrodinger equation is shown below.
Solving it involves separating
the variables into the form

The
starting point is the form of the Schrodinger equation:

Index

Schrodinger
equation concepts

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Quantum Numbers from Hydrogen Equations
The hydrogen
atom solution requires finding solutions to the separated
equations which obey the constraints
on the wavefunction. The solution to the radial
equation can exist only when a constant which arises in the solution
is restricted to integer values. This gives the
principal quantum number:

Similarly, a constant arises in the colatitude
equation which gives the orbital
quantum number:
Finally,
constraints on the azimuthal
equation give what is called the magnetic
quantum number:

Discussion
of concepts

Index

Schrodinger
equation concepts

Hydrogen
concepts

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Separating the Hydrogen Equation

One of the approaches to solving a partial differential equation is to
separate it into individual equations for each variable involved. The hydrogen
Schrodinger equation is separable, and collecting all the
radius-dependent terms and setting them equal to a constant gives the radial
equation:

Then the angular parts of the equation can be separated into a colatitude
equation:

and an azimuthal
equation.

Index

Schrodinger
equation concepts

Hydrogen
concepts

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