Section: 17 | Physics Related |

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10 PHYSICS RELATED

10.1 Clebsch–Gordan Coefficients

The Clebsch–Gordan coefficients arise in the integration of three spherical harmonic functions.

$\begin{array}{l} (\begin{matrix} j_{1} & j_{2} \\ m_{1} & m_{2} \end{matrix} | \begin{matrix} j \\ m \end{matrix}) = δ_{m, m_{1} + m_{2}} \sqrt{\frac{(j_{1} + j_{2} - j)! (j + j_{1} - j_{2})! (j + j_{2} - j_{1})! (2 j + 1)}{(j + j_{1} + j_{2} + 1)!}} \\ \times \sum_{0 \leq k < \infty} \frac{{(- 1)}^{k} \sqrt{(j_{1} + m_{1})! (j_{1} - m_{1})! (j_{2} + m_{2})! (j_{2} - m_{2})! (j + m)! (j - m)!}}{k! (j_{1} + j_{2} - j - k)! (j_{1} - m_{1} - k)! (j_{2} + m_{2} - k)! (j - j_{2} + m_{1} + k)! (j - j_{1} - m_{2} + k)!} . \end{array}$

Conditions:
1. Each of ${j_{1}, j_{2}, j, m_{1}, m_{2}, m}$ may be an integer, or half an integer.
2. $j_{1} + j_{2} - j \geq 0$
3. $j_{1} - j_{2} + j \geq 0$
4. $- j_{1} + j_{2} + j \geq 0$
5. $j > 0$ , $j_{1} > 0$ , $j_{2} > 0$
6. $j + j_{1} + j_{2}$ is an integer
7. $j_{1} + m_{1}$ is an integer
8. $j_{2} + m_{2}$ is an integer
9. $| m_{1} | \leq j_{1}$ , $| m_{2} | \leq j_{2}$ , $| m | \leq j$
Special values:
1. $(\begin{matrix} j_{1} & j_{2} \\ m_{1} & m_{2} \end{matrix} | \begin{matrix} i \\ m \end{matrix}) = 0$ if $m_{1} + m_{2} \neq m$ .
2. $(\begin{matrix} j_{1} & 0 \\ m_{1} & 0 \end{matrix} | \begin{matrix} i \\ m \end{matrix}) = δ_{j_{1}, j} δ_{m_{1}, m}$ .
3. $(\begin{matrix} j_{1} & j_{2} \\ 0 & 0 \end{matrix} | \begin{matrix} j \\ 0 \end{matrix}) = 0$ when $j_{1} + j_{2} + j$ is an odd integer.
4. $(\begin{matrix} j_{1} & j_{1} \\ m_{1} & m_{1} \end{matrix} | \begin{matrix} j \\ m \end{matrix}) = 0$ when $2 j_{1} + j$ is an odd integer.
Symmetry relations: all of the following are equal to (j1j2m1m2|im) :
1. $(\begin{matrix} j_{2} & j_{1} \\ - m_{2} & - m_{1} \end{matrix} | \begin{matrix} j \\ - m \end{matrix})$ ,
2. $(- 1)^{j_{1} + j_{2} - j} (\begin{matrix} j_{2} & j_{1} \\ m_{1} & m_{2} \end{matrix} | \begin{matrix} j \\ m \end{matrix})$ ,
3. $(- 1)^{j_{1} + j_{2} - j} (\begin{matrix} j_{1} & j_{2} \\ - m_{1} & - m_{2} \end{matrix} | \begin{matrix} j \\ - m \end{matrix})$ ,
4. $\sqrt{\frac{2 j + 1}{2 j_{1} + 1}} (- 1)^{j_{2} + m_{2}} (\begin{matrix} j & j_{2} \\ - m & m_{2} \end{matrix} | \begin{matrix} j_{1} \\ - m_{1} \end{matrix})$ ,
5. $\sqrt{\frac{2 j + 1}{2 j_{1} + 1}} (- 1)^{j_{1} - m_{1} + j - m} (\begin{matrix} j & j_{2} \\ m & - m_{2} \end{matrix} | \begin{matrix} j_{1} \\ m_{1} \end{matrix})$ ,
6. $\sqrt{\frac{2 j + 1}{2 j_{1} + 1}} (- 1)^{j - m + j_{1} - m_{1}} (\begin{matrix} j_{2} & j \\ m_{2} & - m \end{matrix} | \begin{matrix} j_{1} \\ - m_{1} \end{matrix})$ ,
7. $\sqrt{\frac{2 j + 1}{2 j_{2} + 1}} (- 1)^{j_{1} - m_{1}} (\begin{matrix} j_{1} & j \\ m_{1} & - m \end{matrix} | \begin{matrix} j_{2} \\ - m_{2} \end{matrix})$ ,
8. $\sqrt{\frac{2 j + 1}{2 j_{2} + 1}} (- 1)^{j_{1} - m_{1}} (\begin{matrix} j & j_{1} \\ m & - m_{1} \end{matrix} | \begin{matrix} j_{2} \\ m_{2} \end{matrix})$ .

Using symmetry relations, Clebsch–Gordan coefficients $(\begin{matrix} j_{1} & j_{2} \\ m_{1} & m_{2} \end{matrix} | \begin{matrix} j \\ m \end{matrix})$ may be put in the standard form $j_{1} \leq j_{2} \leq j$ and $m \geq 0$ .

$j_{1}$	$j_{2}$	$m_{1}$	$m_{2}$	$j$	$m$	$(\begin{matrix} 1 / 2 & 1 / 2 \\ m_{1} & m_{2} \end{matrix} \| \begin{matrix} j \\ m \end{matrix})$
$1 / 2$	$1 / 2$	$- 1 / 2$	$- 1 / 2$	$1$	$- 1$	$1$
$1 / 2$	$1 / 2$	$- 1 / 2$	$1 / 2$	$0$	0	$- {(\frac{1}{2})}^{1 / 2} = - 0.70711$
$1 / 2$	$1 / 2$	$- 1 / 2$	$1 / 2$	$1$	$0$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1 / 2$	$1 / 2$	$1 / 2$	$- 1 / 2$	$0$	$0$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1 / 2$	$1 / 2$	$1 / 2$	$- 1 / 2$	$1$	$0$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1 / 2$	$1 / 2$	$1 / 2$	$1 / 2$	$1$	$1$	$1$

$j_{1}$	$j_{2}$	$m_{1}$	$m_{2}$	$j$	$m$	$(\begin{matrix} 1 & 1 \\ m_{1} & m_{2} \end{matrix} \| \begin{matrix} 1 \\ m \end{matrix})$
$1$	$1$	$- 1$	$0$	$1$	$- 1$	$- {(\frac{1}{2})}^{1 / 2} = - 0.70711$
$1$	$1$	$- 1$	$1$	$1$	$0$	$- {(\frac{1}{2})}^{1 / 2} = - 0.70711$
$1$	$1$	$0$	$- 1$	$1$	$- 1$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1$	$1$	$0$	$1$	$1$	$1$	$- {(\frac{1}{2})}^{1 / 2} = - 0.70711$
$1$	$1$	$1$	$- 1$	$1$	$0$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1$	$1$	$1$	$0$	$1$	$1$	${(\frac{1}{2})}^{1 / 2} = 0.70711$

$j_{1}$	$j_{2}$	$m_{1}$	$m_{2}$	$j$	$m$	$(\begin{matrix} 1 & 1 \\ m_{1} & m_{2} \end{matrix} \| \begin{matrix} 2 \\ m \end{matrix})$
$1$	$1$	$- 1$	$- 1$	$2$	$- 2$	$1$
$1$	$1$	$- 1$	$0$	$2$	$- 1$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1$	$1$	$- 1$	$1$	$2$	$0$	${(\frac{1}{6})}^{1 / 2} = 0.40825$
$1$	$1$	$0$	$- 1$	$2$	$- 1$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1$	$1$	$0$	$0$	$2$	$0$	${(\frac{2}{3})}^{1 / 2} = 0.81650$
$1$	$1$	$0$	$1$	$2$	$1$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1$	$1$	$1$	$- 1$	$2$	$0$	${(\frac{1}{6})}^{1 / 2} = 0.40825$
$1$	$1$	$1$	$0$	$2$	$1$	${(\frac{1}{2})}^{1 / 2} = 0.70711$
$1$	$1$	$1$	$1$	$2$	$2$	$1$

$j_{1}$	$j_{2}$	$m_{1}$	$m_{2}$	$j$	$m$	$(\begin{matrix} 1 & 1 / 2 \\ m_{1} & m_{2} \end{matrix} \| \begin{matrix} 2 \\ m \end{matrix})$
$1$	$1 / 2$	$- 1$	$- 1 / 2$	$3 / 2$	$- 3 / 2$	$1$
$1$	$1 / 2$	$- 1$	$1 / 2$	$1 / 2$	$- 1 / 2$	$- {(\frac{2}{3})}^{1 / 2} = - 0.816497$
$1$	$1 / 2$	$- 1$	$1 / 2$	$3 / 2$	$- 1 / 2$	${(\frac{1}{3})}^{1 / 2} = 0.577350$
$1$	$1 / 2$	$0$	$- 1 / 2$	$3 / 2$	$- 1 / 2$	${(\frac{2}{3})}^{1 / 2} = 0.816497$
$1$	$1 / 2$	$0$	$- 1 / 2$	$1 / 2$	$- 1 / 2$	${(\frac{1}{3})}^{1 / 2} = 0.577350$
$1$	$1 / 2$	$0$	$1 / 2$	$1 / 2$	$1 / 2$	$- {(\frac{1}{3})}^{1 / 2} = - 0.577350$
$1$	$1 / 2$	$0$	$1 / 2$	$3 / 2$	$1 / 2$	${(\frac{2}{3})}^{1 / 2} = 0.816497$
$1$	$1 / 2$	$1$	$- 1 / 2$	$1 / 2$	$1 / 2$	${(\frac{2}{3})}^{1 / 2} = 0.816497$
$1$	$1 / 2$	$1$	$- 1 / 2$	$3 / 2$	$1 / 2$	${(\frac{1}{3})}^{1 / 2} = 0.577350$
$1$	$1 / 2$	$1$	$1 / 2$	$3 / 2$	$3 / 2$	$1$

10.2 MOMENT OF INERTIA FOR DIFFERENT SHAPES

The moment of inertia of a volume $V$ about an axis is $\int_{V} ρ (r) d^{2} (r) d V (r)$ where $ρ$ is the density and $d$ is the distance of a point to the axis. Often the result can be written in terms of the total mass of the body, $m = \int_{V} ρ (r) d V (r)$ .

Body	Axis	Moment of inertia
Uniform thin rod of length $l$	Normal to the length, at one end	$m \frac{1}{3} l^{2}$
Uniform thin rod of length $l$	Normal to the length, at the center	$m \frac{1}{12} l^{2}$
Thin rectangular sheet, sides $a$ and $b$	Through the center parallel to $b$	$m \frac{1}{12} a^{2}$
Thin rectangular sheet, sides $a$ and $b$	Through the center perpendicular to the sheet	$m \frac{1}{12} (a^{2} + b^{2})$
Thin circular sheet of radius $r$	Normal to the plate through the center	$m \frac{1}{2} r^{2}$
Thin circular sheet of radius $r$	Along any diameter	$m \frac{1}{4} r^{2}$
Thin circular ring. Radii $r_{1}$ and $r_{2}$	Through center normal to plane of ring	$m \frac{1}{2} (r_{1}^{2} + r_{2}^{2})$
Thin circular ring. Radii $r_{1}$ and $r_{2}$	Any diameter	$m \frac{1}{4} (r_{1}^{2} + r_{2}^{2})$
Rectangular parallelepiped, edges $a$ , $b$ , and $c$	Through center perpendicular to face $a b$ , (parallelto edge $c)$	$m \frac{1}{12} (a^{2} + b^{2})$
Sphere, radius $r$	Any diameter	$m \frac{2}{5} r^{2}$
Spherical shell, external radius, $r_{1}$ , internal radius $r_{2}$	Any diameter	$m \frac{2}{5} \frac{(r_{1}^{5} - r_{2}^{5})}{(r_{1}^{3} - r_{2}^{3})}$
Spherical shell, very thin, mean radius, $r$	Any diameter	$m \frac{2}{3} r^{2}$
Right circular cylinder of radius $r$ , length $l$	The longitudinal axis of the solid	$m \frac{1}{2} r^{2}$
Right circular cylinder of radius $r$ , length $l$	Transverse diameter	$m (\frac{r^{2}}{4} + \frac{l^{2}}{12})$
Hollow circular cylinder, length $l$ , radii $r_{1}$ and $r_{2}$	The longitudinal axis of the figure	$m \frac{1}{2} (r_{1}^{2} + r_{2}^{2})$
Thin cylindrical shell, length $l$ , mean radius, $r$	The longitudinal axis of the figure	${m r}^{2}$
Hollow circular cylinder, length $l$ , radii $r_{1}$ and $r_{2}$	Transverse diameter	$m (\frac{r_{1}^{2} + r_{2}^{2}}{4} + \frac{t^{2}}{12})$
Elliptic cylinder, length $l$ , transverse semiaxes $a$ and $b$	Longitudinal axis	$m \frac{1}{4} (a^{2} + b^{2})$
Right cone, altitude $h$ , radius of base $r$	Axis of the figure	$m \frac{3}{10} r^{2}$
Spheroid of revolution, equatorial radius $r$	Polar axis	$m \frac{2}{5} r^{2}$
Ellipsoid with axes $2 a$ , $2 b$ , $2 c$	Axis $2 a$	$m \frac{1}{5} (b^{2} + c^{2})$

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