Chapter 11, Problem 11.58E

In exercise 11.57 regarding $C_{60}$, what are the numerical values of the total angular momenta of the electron for each state having quantum number $l$? What are the z components of the angular momentum for each state?

Interpretation Introduction

Interpretation:

The total angular momenta of the electron of the given state that has quantum number $l$ and the z-components of the angular momentum for the state are to be calculated.

Concept introduction:

The total angular momentum for the 3-Dimensional system is given by,

$L=\sqrt{l\left(l+1\right)}\hslash$

The total angular momentum does not depend on the mass of the particle, radius of the rotation and also the magnetic quantum number.

The z-component of the three dimensional angular momentum that has components in x, y and z direction is quantized. Its value depends on the magnetic quantum number and it is given by,

${\widehat{L}}_z=m_l\hslash$

Answer to Problem 11.58E

The total angular momenta of the electron of the given state that has quantum number $l=5$, $l=6$, $l=7$ and $l=8$ is $5.78\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$, $6.84\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$, $7.89\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$ and $8.95\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$ respectively.

The z-components of the angular momentum for the state $l=5$ is $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$ and $+5\hslash$.

The z-components of the angular momentum for the state $l=6$ is $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$ and $+6\hslash$.

The z-components of the angular momentum for the state $l=7$ is $-7\hslash$, $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$, $+6\hslash$ and $+7\hslash$.

The z-components of the angular momentum for the state $l=7$ is $-8\hslash$, $-7\hslash$, $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$, $+6\hslash$, $+7\hslash$ and $+8\hslash$.

Explanation of Solution

The moment of inertia for the electron in the spherical $C_{60}$ molecule is $1.12\times {10}^{-49}\text{kg}\cdot {\text{m}}^2$

The total angular momentum for the 3-D rotational wavefunction is calculated using the formula,

$L=\sqrt{l\left(l+1\right)}\hslash$

Where,

• $L$ is the total angular momentum.

• $l$ is the quantum number.

• $\hslash$ has the value $\frac{h}{2\pi }$, $h$ is the Planck’s constant.

For $l=5$, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

$\begin{array}{rll}L& =& \sqrt{5\left(5+1\right)}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& \sqrt{30}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& 5.78\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}\end{array}$

Thus, the total angular momentum of the electron in spherical $C_{60}$ molecule for $l=5$ is $5.78\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$.

For $l=6$, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

$\begin{array}{rll}L& =& \sqrt{6\left(6+1\right)}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& \sqrt{42}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& 6.84\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}\end{array}$

Thus, the total angular momentum of the electron in spherical $C_{60}$ molecule for $l=6$ is $6.84\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$.

For $l=7$, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

$\begin{array}{rll}L& =& \sqrt{7\left(7+1\right)}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& \sqrt{56}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& 7.89\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}\end{array}$

Thus, the total angular momentum of the electron in spherical $C_{60}$ molecule for $l=7$ is $7.89\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$.

For $l=8$, substitute the values of quantum number, moment of inertia and Planck’s constant in the given formula.

$\begin{array}{rll}L& =& \sqrt{8\left(8+1\right)}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& \sqrt{72}\left(\frac{6.626\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}}{2\pi }\right)\\ L& =& 8.95\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}\end{array}$

Thus, the total angular momentum of the electron in spherical $C_{60}$ molecule for $l=8$ is $8.95\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$.

The relation between the Z-component of angular momentum and magnetic quantum number for 3-dimensional rotation is

${\widehat{L}}_z=m_l\hslash$.

Where,

• $m_l$ is the magnetic quantum number.

• $\hslash$ has the value $\frac{h}{2\pi }$, $h$ is the Planck’s constant.

The possible vales of $m_l$ for quantum number $l=5$ is $-5$, $-4$$-3$, $-2$, $-1$, $0$, $+1$, $+2$, $+3$, $+4$ and $+5$.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$ and $+5\hslash$ respectively.

The possible vales of $m_l$ for quantum number $l=6$ is $-6$, $-5$, $-4$$-3$, $-2$, $-1$, $0$, $+1$, $+2$, $+3$, $+4$, $+5$ and $+6$.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$ and $+6\hslash$ respectively.

The possible vales of $m_l$ for quantum number $l=7$ is $-7$, $-6$, $-5$, $-4$$-3$, $-2$, $-1$, $0$, $+1$, $+2$, $+3$, $+4$, $+5$$+6$ and $+7$.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is $-7\hslash$, $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$, $+6\hslash$ and $+7\hslash$ respectively.

The possible vales of $m_l$ for quantum number $l=8$ is $-8$, $-7$, $-6$, $-5$, $-4$$-3$, $-2$, $-1$, $0$, $+1$, $+2$, $+3$, $+4$, $+5$$+6$, $+7$ and $+8$.

The Z-component of angular momentum to by substituting the values of magnetic quantum number in the given formula is $-8\hslash$, $-7\hslash$, $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$, $+6\hslash$, $+7\hslash$ and $+8\hslash$ respectively.

Conclusion

The total angular momenta of the electron of the given state that has quantum number $l=5$, $l=6$, $l=7$ and $l=8$ is $5.78\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$, $6.84\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$, $7.89\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$ and $8.95\times {10}^{-34}\text{kg}\cdot {\text{m}}^2\cdot {\text{s}}^{-1}$ respectively.

The z-components of the angular momentum for the state $l=5$ is $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$ and $+5\hslash$.

The z-components of the angular momentum for the state $l=6$ is $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$ and $+6\hslash$.

The z-components of the angular momentum for the state $l=7$ is $-7\hslash$, $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$, $+6\hslash$ and $+7\hslash$.

The z-components of the angular momentum for the state $l=7$ is $-8\hslash$, $-7\hslash$, $-6\hslash$, $-5\hslash$, $-4\hslash$$-3\hslash$, $-2\hslash$, $-1\hslash$, $0$, $+1\hslash$, $+2\hslash$, $+3\hslash$, $+4\hslash$, $+5\hslash$, $+6\hslash$, $+7\hslash$ and $+8\hslash$.