Question

If an electron in an atom has orbital angular momentum with values limited by 3, how many values of (a) $L_{orb,z}$and (b) ${\mu }_{orb,z}$can the electron have? In terms of h, m, and e, what is the greatest allowed magnitude for (c)$L_{orb,z}$and (d)${\mu }_{orb,z}$? (e) What is the greatest allowed magnitude for the z component of the electron’s net angular momentum (orbital plus spin)? (f) How many values (signs included) are allowed for the z component of its net angular momentum?

Short answer

1. Number of values of $L_{orb,z}$that an electron can have is seven
2. Number of values of${\mu }_{orb,z}$ that an electron can have is seven
3. Greatest allowed magnitude of$L_{orb,z}$is$\frac{3h}{2\pi }$
4. Greatest allowed magnitude of${\mu }_{orb,z}$is$\frac{3eh}{4\pi m_e}$
5. Greatest allowed magnitude for z component of the electron’s net angular momentumis$\frac{3.5h}{2\pi }$
6. Number of values allowed to magnitude for z component of the electron’s net angular momentum is 8.

Step-by-step solution

Listing the given quantities

Angular orbital momentum has value,$m_2=\pm 3$

Understanding the concepts of magnetic field

By using the concept of quantum numbers, we can find the number of values and greatest allowed values.

Formulas:

Number of different values of$m_1$is given byrole="math" localid="1663138105073" $\left(2l+1\right)$

$L_{orb,z}=\frac{m_{l}h}{2\pi }$

(a) Calculations of Number of values of Lorb,z

For the given value of $l,m$varies from $-lto+l.$ Thus, in this case, $1=3$, and the number of different values of $m_1$is

$2l+1=2\left(3\right)+1=7$

So, there are 7 different values of $L_{orb,z}$

(b) Calculations of number of valuesμorb,z of μorb,z

Similarly, since ${\mu }_{orb,z}$is directly proportional to$m_1$, there are total 7 different values of ${\mu }_{orb,z}$.

(c) Calculations of greatest allowed magnitude of Lorb,z

As we know,$L_{orb,z}=\frac{m_{l}h}{2\pi }$. So, the greatest allowed value of$L_{orb,z}$is given by

$\frac{{\left|m_l\right|}_{maximum}h}{2\pi }=\frac{3h}{2\pi }$

Greatest allowed magnitude of $L_{orb,z}$islocalid="1663139848998" $\frac{3h}{2\pi }$

(d) Calculations of greatest allowed magnitude of μorb,z

Since${\mu }_{orb,z}=-m_l\mu B$, the greatest allowed value of${\mu }_{orb,z}$is given by

${\left|m_l\right|}_{maximum}\mu B=\frac{3eh}{4\pi m_e}$

Greatest allowed magnitude of${\mu }_{orb,z}$ is$\frac{3eh}{4\pi m_e}$

(e) Calculations of greatest allowed magnitude for z component of the electron’s net angular momentum

The z component of the net angular momentum of the electron is given by

$$\begin{gathered} L_{net,z}=L_{orb,Z}+L_{s,Z} \\ =\frac{m_{l}h}{2\pi }+\frac{m_{s}h}{2\pi } \end{gathered}$$

From the given value,$m_l=3\&m_s=\frac{1}{2}.$ Thus,

role="math" localid="1663140942371" $$\begin{gathered} {\left[L_{net,z}\right]}_{maximum}=\frac{\left(3+\frac{1}{2}\right)h}{2\pi } \\ =\frac{3.5h}{2\pi } \end{gathered}$$

Greatest allowed magnitude for z component of the electron’s net angular momentum is$\frac{3.5h}{2\pi }$

(f) Calculations of number of values allowed to magnitude for z component of the electron’s net angular momentum

Since, the values of$L_{\left(net,z\right)}$ are given by$\frac{{\left[m_j\right]}_{maximum}h}{2\pi }$with${\left[m_j\right]}_{maximum}=3.5$

The number of allowed values for z component of$L_{net,z}$ is given as

role="math" localid="1663141133934" $$\begin{gathered} 2{\left[m_j\right]}_{maximum}=2\left(3.5\right)+1 \\ =8 \end{gathered}$$

Number of values allowed to magnitude for z component of the electron’s net angular momentum is 8.