Quantum Chemistry/Example 26

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

Solution

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

${\hat{L}}^2{Y}_l^m(\theta ,\varphi )=h^{2}l(l+1)Y_l^m(\theta ,\varphi )$

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

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

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

$l(l+1)=20$

$l^2+l-20=0$

In this quadratic equation, $l$ can be factored to get:

$(l-4)(l+5)=0$

$l=4$

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

$m=-4,-3,-2,-1,0,1,2,3,4$

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

$L_z=m\hslash$

$L_z=-4\hslash ,-3\hslash ,-2\hslash ,-1\hslash ,0\hslash ,1\hslash ,2\hslash ,3\hslash ,4\hslash$.

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