Q9P - Fundamentals Of Physics

Question

An electron is in a state with I = 3 . (a) What multiple of  gives the magnitude of $\overrightarrow{\mathbf{L}}$? (b) What multiple of ${\mathbf{\mu }}_{\mathbf{B}}$ gives the magnitude of $\overrightarrow{\mathbf{\mu }}$ ? (c) What is the largest possible value of ${\mathbf{m}}_{\mathbf{1}}$ , (d) what multiple of $\mathit{ħ}$ gives the corresponding value of$\overrightarrow{\mathbf{L}}$ , and (e) what multiple of ${\mathbf{\mu }}_{\mathbf{B}}$ gives the corresponding value of  ${\mathbf{\mu }}_{\mathbf{orb}\mathbf{,}\mathbf{z}}$? (f) What is the value of the semi-classical angle $\mathbf{\theta }$ between the directions of ${\mathbf{L}}_{\mathbf{z}}\mathbf{ }\mathbf{and}\mathbf{ }\overrightarrow{\mathbf{L}}$ ? What is the value of angle $\mathbf{\theta }$ for (g) the second largest possible value of  ${\mathbf{m}}_{\mathbf{I}}$ and (h) the smallest (that is, most negative) possible value of ${\mathbf{m}}_{\mathbf{I}}$ ?

Step-by-Step Solution

Answer

a)   The$ħ$ multiple of magnitude of $\overrightarrow{\mathrm{L}} \mathrm{is} 3.46$ .

b)   The${\mathrm{\mu }}_{\mathrm{B}}$ multiple of magnitude of $\overrightarrow{\mathrm{\mu }} \mathrm{is} 3.46$  .

c)   The largest possible value of${\mathrm{m}}_{\mathrm{I}} \mathrm{is} 3$ .

d)   The corresponding value of${\mathrm{L}}_{\mathrm{z}} \mathrm{is} 3$.

e)  The corresponding value of${\mathrm{\mu }}_{\mathrm{orb},\mathrm{z} }\mathrm{is} -3$ .

f)   The value of the semi-classical angle $\theta$ between the directions ${\mathrm{L}}_{\mathrm{z}} \mathrm{and} \overrightarrow{\mathrm{L}} \mathrm{is} 30°$  .

g)  The value of angle $\theta$for the second largest possible value of ${\mathrm{m}}_{\mathrm{I}} \mathrm{is} 54.7°$  .

h)  The value of the angle $\theta$ for the smallest possible value of ${\mathrm{m}}_1 \mathrm{is} 150°$  .

Step 1: The given data:

An electron is in state with value I = 3.

Step 2: Understanding the concept of orbital angular quantum values:

Orbital angular momentum quantum number is 'I' . The orbital angular momentum quantum number determines the shape of the orbital and hence the angular distribution. The value of I is dependent on the principal quantum number n .

 

Using the given value of the orbital angular quantum number, get the magnitude of the orbital angular momentum and dipole moment. As the range of m varies from -I to +I, the maximum value is given by the most positive value of I . Similarly using the concept of the orbital momentum and dipole moment projection along the z-axis, define the required values using the given value in the formula. The semi-classical angle between two vectors can be given using the cosine angle between them.

Formulas:

The magnitude of the orbital angular momentum in terms of $ħ$ is,

$\overrightarrow{\mathrm{L}}=\sqrt{\mathrm{I}\left(\mathrm{I}+1\right)}\mathrm{ħ}$                                                                                                              ….. (1)

 

The magnitude of the orbital dipole moment in terms of Bohr’s magnetron ${\mathrm{\mu }}_{\mathrm{B}}$ ,

$\overrightarrow{{\mathrm{\mu }}_{\mathrm{orb}}}=\sqrt{\mathrm{I}\left(\mathrm{I}+1\right)}{\mathrm{\mu }}_{\mathrm{B}}$                                                                                                         ….. (2)

 

The z-component of the orbital angular momentum is,

$L_z=m\mathcal{l}ħ$                                                                                                                     ….. (3)

 

The z-component of the orbital magnetic dipole moment is,

${\mathrm{\mu }}_{\mathrm{orb},\mathrm{z}}=-{\mathrm{m}}_{\mathrm{I}}{\mathrm{\mu }}_{\mathrm{B}}$                                                                                                             ….. (4)

 

The semi-classical angle between a vector and its z-component is,

 $\mathrm{\theta }={\cos }^{-1}\left(\frac{{\mathrm{a}}_{\mathrm{z}}}{\mathrm{a}}\right)$                                                                                                             ….. (5)

Step 3: (a) Calculation of the magnitude of orbital angular momentum:

Define the value of the magnitude of the orbital angular momentum by substituting 3 for I in equation (1) as below.

$$\begin{gathered} \overrightarrow{\mathrm{L}}=\sqrt{3\left(3+1\right)}ħ \\ =\sqrt{12 }ħ \\ =3.46 ħ \end{gathered}$$ 

Hence, the $ħ$ multiple value of the orbital angular momentum is 3.46.

Step 4: (b) Calculation of the orbital magnetic dipole moment:

Calculate the value of the orbital magnetic dipole moment ny substituting 3 for I into equation (2) as below.

 $$\begin{gathered} \overrightarrow{{\mathrm{\mu }}_{\mathrm{orb}}}=\sqrt{3\left(3+1\right)}{\mathrm{\mu }}_{\mathrm{B}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ } =\sqrt{12 }{\mathrm{\mu }}_{\mathrm{B}} \\ =3.46 {\mathrm{\mu }}_{\mathrm{B}} \end{gathered}$$

 

 Hence, the ${\mathrm{\mu }}_{\mathrm{B}}$ multiple value of the dipole moment is 3.46.

Step 5: (c) Calculation of the largest value of m I :

Using the concept of range of ${\mathrm{L}}_{\mathrm{z}}={\mathrm{m}}_{\mathrm{I}}\mathrm{ħ}$ for I values, get the largest value of the ${\mathrm{m}}_{\mathrm{I}}$ as follows:

 $$\begin{gathered} {\mathrm{m}}_{\mathrm{I}}=+\mathrm{I} \\ =3 \end{gathered}$$

Hence, the largest value of ${\mathrm{m}}_{\mathrm{I}}$ is 3 .

Step 6: (d) Calculation of the z-component of the orbital angular momentum:

Using the largest value of ${\mathrm{m}}_{\mathrm{I}}=3$ in equation (3), obtain the $ħ$ multiple value of z-component of the orbital angular momentum as follows:

${\mathrm{L}}_{\mathrm{z}}=3ħ$ 

 

Hence, the $ħ$ multiple value of the z-component is 3 .

Step 7: (e) Calculation of the z-component of the orbital magnetic dipole moment:

Using the largest value of ${\mathrm{m}}_{\mathrm{I}}=3$ in equation (3), determine  the ${\mathrm{\mu }}_{\mathrm{B}}$ multiple value of z-component of the orbital angular momentum as follows:

${\mathrm{\mu }}_{\mathrm{orb}.\mathrm{z}}=-3{\mathrm{\mu }}_{\mathrm{B}}$ 

 

Hence, the ${\mathrm{\mu }}_{\mathrm{B}}$ multiple value of the z-component is -3.

Step 8: (f) Calculation of the angle between the orbital angular momentum and its z-component:

Using the value of $\overrightarrow{\mathrm{L}}$ and from part (a) and (d) calculations, get the semi-classical angle between the two directions ${\mathrm{L}}_{\mathrm{z}}$ and $\overrightarrow{\mathrm{L}}$ using equation (5) as follows:

 $$\begin{gathered} \mathrm{\theta }={\cos }^{-1}\left(\frac{3\mathrm{ħ}}{\sqrt{12}\mathrm{ħ}}\right) \\ ={\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right) \\ =30° \end{gathered}$$

 

Hence, the value of the angle is $30°$ .

Step 9: (g) Calculation of the angle for the second largest value of m 1 :

Using the concept, we can say that the second largest value of ${\mathrm{m}}_1 \mathrm{is} +2$ .

Thus, using this value in equation (3), determine the component value as below.

${\mathrm{L}}_{\mathrm{z}}=2ħ$ 

 

Now, the angle between the orbital angular momentum and the second largest value of   is given using the above value and value of angular momentum from part (a) calculations in equation (5) as follows:

$$\begin{gathered} \mathrm{\theta }={\cos }^{-1}\left(\frac{2\mathrm{ħ}}{\sqrt{12}\mathrm{ħ}}\right) \\ ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right) \\ =54.7° \end{gathered}$$ 

 

Hence, the value of the angle is $54.7°$ .

Step 10: (h) Calculation of the angle for the smallest value of :

Using the concept, we can say that the smallest value of ${\mathrm{m}}_{\mathrm{I}} \mathrm{is} -3$  .

Thus, using this value in equation (3), get the component value as:

${\mathrm{L}}_{\mathrm{z}}=-3ħ$ 

 

Now, the angle between the orbital angular momentum and the second largest value of   is given using the above value and value of angular momentum from part (a) calculations in equation (5) as follows:

 $$\begin{gathered} \mathrm{\theta }={\cos }^{-1}\left(\frac{-3\mathrm{ħ}}{\sqrt{12}\mathrm{ħ}}\right) \\ ={\cos }^{-1}\left(\frac{-\sqrt{3}}{2}\right) \\ =150° \end{gathered}$$

 

 

Hence, the value of the angle is $150°$ .