Quantum Chemistry/Example 26

The square of the angular momentum of a hydrogen atom is measured to be .  What are the possible values of the z-component of the orbital angular momentum, , that could be measured for this atom?


Solution:

The eigenvalues of the square of the angular momentum operator (2) for a quantum mechanical system, such as an electron in a hydrogen atom, are given by:



where are the spherical harmonics, which are eigenfunctions of 2, is the orbital quantum number, and is the magnetic quantum number. The given value for = 20 ħ2 , so we set up the equation:



Dividing by and simplifying, we get:



In this quadratic equation, can be factored to get:



Since must be a non-negative integer. The magnetic quantum number can take on any integer value from to , thus for , can be:



The z-component of the angular momentum, , is quantized in units of and given by:


.


Therefore, the possible values of the z-component of the orbital angular momentum, , that could be measured for the atom with a given are .