Values of the agular momentum quantum number

[fkliment2000]

Hello!

Why is it the case, that the principle quantum number limits the values of the angular momentum quantum number like

l <= n-1

How is it possible to derive this?


Thank you for your answers.

 

[Hootenanny]

It falls out the the Schrödinger equation (SE) when solving the Hydrogen atom. On approach to solving the SE is to separate the variables into different equations. In order to solve the SE, we separate the radial part and set it equal to a constant. This yields what is known as the Colatitude equation. We then separate the Azimuthal Equation (which coincidentaly is the easiest to solve) and solve. This gives the value of the constant in the Colatitude equation and the azimuthal quantum number. We then solve the Colatitude equation using ploynomial expansion and obtain solutions of the form $C = l(l+1)$ and the solution only exists if l = 0,1,2,3,...,n-1

More information is available at http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2

 

[fkliment2000]

Thanks very much!

 

[masudr]

Also, note that this restriction only holds for hydrogenic systems.