# Quantum Chemistry/Example 26

The square of the angular momentum of a hydrogen atom is measured to be ${\displaystyle L^{2}}$${\displaystyle =20\hbar ^{2}}$.  What are the possible values of the z-component of the orbital angular momentum, ${\displaystyle L_{z}}$, that could be measured for this atom?

Solution:

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

${\displaystyle {\hat {L}}^{2}Y_{l}^{m}(\theta ,\phi )=h^{2}l(l+1)Y_{l}^{m}(\theta ,\phi )}$

where ${\displaystyle Y_{l}^{m}(\theta ,\phi )}$ are the spherical harmonics, which are eigenfunctions of ${\displaystyle {\hat {L}}}$2, ${\displaystyle l}$ is the orbital quantum number, and ${\displaystyle m}$ is the magnetic quantum number. The given value for ${\displaystyle L^{2}}$ = 20 ħ2 , so we set up the equation:

${\displaystyle \hbar ^{2}l(l+1)=20\hbar ^{2}}$

Dividing by ${\displaystyle \hbar ^{2}}$ and simplifying, we get:

${\displaystyle l(l+1)=20}$

${\displaystyle l^{2}+l-20=0}$

In this quadratic equation, ${\displaystyle l}$ can be factored to get:

${\displaystyle (l-4)(l+5)=0}$

${\displaystyle l=4}$

Since ${\displaystyle l}$ must be a non-negative integer. The magnetic quantum number ${\displaystyle m}$ can take on any integer value from ${\displaystyle -l}$ to ${\displaystyle l}$, thus for ${\displaystyle l=4}$, ${\displaystyle m}$ can be:

${\displaystyle m=-4,-3,-2,-1,0,1,2,3,4}$

The z-component of the angular momentum, ${\displaystyle L_{z}}$, is quantized in units of ${\displaystyle \hbar ^{2}}$and given by:

${\displaystyle L_{z}=m\hbar }$

${\displaystyle L_{z}=-4\hbar ,-3\hbar ,-2\hbar ,-1\hbar ,0\hbar ,1\hbar ,2\hbar ,3\hbar ,4\hbar }$.

Therefore, the possible values of the z-component of the orbital angular momentum, ${\displaystyle L_{z}}$, that could be measured for the atom with a given ${\displaystyle L^{2}}$${\displaystyle =20\hbar ^{2}}$ are ${\displaystyle L_{z}=-4\hbar ,-3\hbar ,-2\hbar ,-1\hbar ,0\hbar ,1\hbar ,2\hbar ,3\hbar ,4\hbar }$.