Schrödinger Wave Equation for Hydrogen Atom

Getting from the Schrödinger Wave Equation, to a better model than the Bohr Model.

The Potential energy of the electron-proton system is electrostatic.

Negative means potential is attractive

Transform the Time-dependent Equation into spherical coordinates. Also use reduced mass in the equation, because it gives a more accurate result for free.

Assume , so it is a separable function. This makes the above equation simplify a lot!

Azimuthal-angle equation

Sines and cosines satisfy this equations, but its convenient to choose the complex exponential.

Quantizing

Because and are the same angle, you can set g of each equal and you solve for this:

must be an integer for this to be true.

Polar Equation for H

See Polar-Angle Equation

Radial Equation for H

See Radial Wave Equation

Complete Hydrogen Wave Functions

The result is a wave; states have some angular momentum proportional to . However, such motion can only be inferred indirectly, so we opt for the spatial wave-function instead, which is a product of the radial wave function and the spherical harmonics :

See also

Quantum Number