Latex Version of Equations
eq 1
$$\left[\begin{array}{c}
x_{\rm new}^P(x,y,z)\\
y_{\rm new}^P(x,y,z)\\
z_{\rm new}^P(x,y,z)\end{array}\right] = [\quad D(P)\quad ]
\left[\begin{array}{c}
x\\ y\\ z \end{array}\right] ~ , $$
eq 2
$$P \, f (x,y,z) \equiv f \left(x_{\rm new}^P(x,y,z), ~
y_{\rm new}^P(x,y,z), ~ z_{\rm new}^P(x,y,z) \right) = g(x,y,z) ~ . $$
eq 3
$$Q\,P\,f (x,y,z) \equiv Q\,g (x,y,z)= h(x,y,z) ~ , $$
eq 4
\begin{eqnarray*}
\left[ \begin{array}{c}
x_{\rm new}^{QP}(x,y,z)\\ ~\\
y_{\rm new}^{QP}(x,y,z)\\
z_{\rm new}^{QP}(x,y,z) \end{array}\right]
&=& Q\,P \left[\begin{array}{c}
x\\ y \\ z\end{array} \right] = Q[\quad D(P)\quad ]
\left[\begin{array}{c}
x\\ y\\ z \end{array}\right]
= [\quad D(P)\quad ] Q \left[\begin{array}{c}
x\\ y\\ z \end{array}\right] \\
&=& [\quad D(P)\quad ] ~ [\quad D(Q)\quad ] ~
\left[\begin{array}{c}
x\\ y\\ z \end{array}\right] ~ . \end{eqnarray*}
eq 5
\begin{eqnarray*}
\left[ \begin{array}{c}
(yp_z - zp_y)_{\rm new}\\ ~\\
(zp_x - xp_z)_{\rm new}\\
(xp_y - yp_x)_{\rm new} \end{array}\right]
= P \left[\begin{array}{c}
(yp_z - zp_y)\\
(zp_x - xp_z)\\
(xp_y - yp_x)\end{array} \right] = [ ~ D^{F_1}(P) ~ ]
\left[\begin{array}{c}
(yp_z - zp_y)\\
(zp_x - xp_z)\\
(xp_y - yp_x)\end{array}\right] ~ , \end{eqnarray*}
eq 6
\begin{eqnarray*}
\left[ \begin{array}{c}
(2z^2-x^2-y^2)_{\rm new}\\ ~\\
\sqrt{3}(x^2 - y^2)_{\rm new} \end{array}\right]
= P \left[\begin{array}{c}
(2z^2-x^2-y^2)\\
\sqrt{3}(x^2 - y^2) \end{array} \right] = [ D^E(P) ]
\left[\begin{array}{c}
(2z^2-x^2-y^2)\\
\sqrt{3}(x^2 - y^2)\end{array}\right] ~ ,\end{eqnarray*}
eq 7
\begin{eqnarray*}
\left[ \begin{array}{c}
f\\ g\\ h\end{array}\right]
= [ \quad U \quad ] \left[\begin{array}{c}
x\\ y\\ z\end{array}\right] ~ , \end{eqnarray*}
eq 8
$$\left[\begin{array}{c}
(\mbox{\boldmath $R$}_1)_{\rm new} \\
(\mbox{\boldmath $R$}_2)_{\rm new} \\
(\mbox{\boldmath $R$}_3)_{\rm new} \end{array}\right] = (123)
\left[\begin{array}{c}
(\mbox{\boldmath $R$}_1)\\ ~\\
(\mbox{\boldmath $R$}_2)\\ ~\\
(\mbox{\boldmath $R$}_3) \end{array}\right] = \left[\begin{array}{c}
(\mbox{\boldmath $R$}_2)\\ ~\\
(\mbox{\boldmath $R$}_3)\\ ~\\
(\mbox{\boldmath $R$}_1) \end{array}\right] ~ .$$
eq 9
$$\mbox{\boldmath $R$}_i =
\mbox{\boldmath $R$}+S^{-1}(\chi\theta\phi)
(\mbox{\boldmath $a$}_i+\mbox{\boldmath $d$}_i) ~ .$$
eq 10
\begin{eqnarray*}
&~& \qquad\qquad X \qquad\qquad\qquad\qquad Y \qquad\qquad\qquad Z \\
\begin{array}{r}
x\\
S(\chi\theta\phi) = y\\
z \end{array} &~&
\left[\begin{array}{ccc}
c\chi \, c\theta \, c\phi -s\chi \, s\phi
& c\chi \, c\theta \, s\phi +s\chi \, c\phi
& -c\chi \, s\theta\\
-s\chi \, c\theta \, c\phi -c\chi \, s\phi
& -s\chi \, c\theta \, s\phi +c\chi \, c\phi
& s\chi \, s\theta\\
s\theta \, c\phi
& s\theta \, s\phi
& c\theta\end{array} \right] ~ , \end{eqnarray*}
eq 11
\begin{eqnarray*}
\Sigma_i m_i (\mbox{\boldmath $R$}_i - \mbox{\boldmath $R$}) &=&\mbox{\boldmath $0$}\\
\Sigma_i m_i \mbox{\boldmath $a$}_i \times S(\chi\theta\phi) \cdot
(\mbox{\boldmath $R$}_i - \mbox{\boldmath $R$}) &=&\mbox{\boldmath $0$} ~ .
\end{eqnarray*}
eq 12
$$(\mbox{\boldmath $d$}_i)_{\rm new} = M ~\mbox{\boldmath $d$}_j ~ , $$
eq 13
$$\mbox{\boldmath $a$}_j = M^{-1} ~\mbox{\boldmath $a$}_i ~ , $$
eq 14
$$S(\chi_{\rm new} , \theta_{\rm new} , \phi_{\rm new})= M~S(\chi , \theta , \phi) ~ , $$
eq 15
$$\mbox{\boldmath $R$}+S^{-1} (\chi\theta\phi) M^{-1}
(\mbox{\boldmath $a$}_i + M\mbox{\boldmath $d$}_j) =
\mbox{\boldmath $R$}+S^{-1} (\chi\theta\phi)
(\mbox{\boldmath $a$}_j + \mbox{\boldmath $d$}_j) ~ .$$
eq 16
$$(\mbox{\boldmath $d$}_i)_{\rm new} = N~\mbox{\boldmath $d$}_j ~ ,$$
eq 17
$$\mbox{\boldmath $a$}_j = N^{-1}~\mbox{\boldmath $a$}_i ~ , $$
eq 18
$$S(\chi_{\rm new} , \theta_{\rm new} , \phi_{\rm new})= -N~S(\chi , \theta , \phi) ~ , $$
eq 19
$$-\mbox{\boldmath $R$}~-\!S^{-1}(\chi\theta\phi) \,
N^{-1} (\mbox{\boldmath $a$}_i + N \,\mbox{\boldmath $d$}_j) =
-\mbox{\boldmath $R$}~-\!S^{-1}(\chi\theta\phi) \,
(\mbox{\boldmath $a$}_j + \mbox{\boldmath $d$}_j) ~ . $$
eq 20
\begin{eqnarray*}
S_1 &=& 4^{-1/2} (\delta r_1 + \delta r_2 + \delta r_3 + \delta r_4)\\
S_{2a}&=&12^{-1/2} (2\delta\alpha_{12} + 2\delta\alpha_{34} - \delta\alpha_{13}
- \delta\alpha_{24} - \delta\alpha_{23} - \delta\alpha_{14})\\
S_{2b}&=& 4^{-1/2} (\delta\alpha_{13} + \delta\alpha_{24} - \delta\alpha_{23}
- \delta\alpha_{14})\\
S_{3x}&=& 4^{-1/2} (+\delta r_1 - \delta r_2 + \delta r_3 - \delta r_4)\\
S_{3y}&=& 4^{-1/2} (-\delta r_1 + \delta r_2 + \delta r_3 - \delta r_4)\\
S_{3z}&=& 4^{-1/2} (+\delta r_1 + \delta r_2 - \delta r_3 - \delta r_4)\\
S_{4x}&=& 2^{-1/2} (\delta\alpha_{13} - \delta\alpha_{24})\\
S_{4y}&=& 2^{-1/2} (\delta\alpha_{23} - \delta\alpha_{14})\\
S_{4z}&=& 2^{-1/2} (\delta\alpha_{12} - \delta\alpha_{34}) ~ ,
\end{eqnarray*}
eq 21
\begin{eqnarray*}
\delta r_i &=& (\mbox{\boldmath $a$}_i/a) \cdot
(\mbox{\boldmath $d$}_i-\mbox{\boldmath $d$}_{\rm C})\\
\delta\alpha_{ij} &=&- \, {\scriptstyle{1\over4}} \sqrt{2} \,
\left[ \frac{(\mbox{\boldmath $a$}_i/a) \cdot
(\mbox{\boldmath $d$}_i-\mbox{\boldmath $d$}_{\rm C})}{a} +
\frac{(\mbox{\boldmath $a$}_j/a) \cdot
(\mbox{\boldmath $d$}_j-\mbox{\boldmath $d$}_{\rm C})}{a} \right]^{\phantom I}\\
&~&- \, {\scriptstyle{3\over4}} \sqrt{2} \, \left[ \frac{(\mbox{\boldmath $a$}_i/a) \cdot
(\mbox{\boldmath $d$}_j-\mbox{\boldmath $d$}_{\rm C})}{a} +
\frac{(\mbox{\boldmath $a$}_j/a) \cdot
(\mbox{\boldmath $d$}_i-\mbox{\boldmath $d$}_{\rm C})}{a}\right]^{\phantom I} ~ , \end{eqnarray*}
eq 22
$${\mit\Gamma}(v_1, v_2, v_3, v_4) = [A_1^{v_1}] \times [E^{v_2}]
\times [F_2^{v_3}] \times [F_2^{v_4}] ~ , $$
eq 23
$$\left[\begin{array}{c}
J_x\\ J_y\\ J_z \end{array}\right] =
\left[\begin{array}{ccc}
+\cos\chi \cot\theta & \sin\chi & -\cos\chi \csc\theta \\
-\sin\chi \cot\theta & \cos\chi & +\sin\chi \csc\theta \\
+1 & 0 & 0 \end{array}\right]
\left[\begin{array}{c}
p_\chi\\ p_\theta\\ p_\phi \end{array}\right] ~ , $$
eq 24
$$\left[\begin{array}{c}
J_X\\ J_Y\\ J_Z \end{array}\right] = [S^{-1}(\chi\theta\phi)]
\left[\begin{array}{c}
J_x\\ J_y\\ J_z \end{array}\right] ~ . $$
eq 25
$$\left[\begin{array}{c}
J_X\\ J_Y\\ J_Z \end{array}\right] =
\left[\begin{array}{ccc}
+\csc\theta \cos\phi & -\sin\phi & -\cot\theta \cos\phi \\
+\csc\theta \sin\phi & +\cos\phi & -\cot\theta \sin\phi \\
0 & 0 & +1 \end{array}\right]
\left[\begin{array}{c}
p_\chi \\ p_\theta \\ p_\phi \end{array}\right] ~ .$$
eq 26
$|kJm\rangle = [(2J + 1)/8\pi^2]^{1/2} ~
{\cal D}_{km}^{(J)}(\{\chi\theta\phi\}) ~ ,$
eq 27
$$| K^\pm Jm\rangle = 2^{-1/2} \, [\,|KJm\rangle \pm | -KJm\rangle] ~ ,$$
eq 28
$$\left[\begin{array}{c}
x_{\rm new} \\
y_{\rm new} \\
z_{\rm new} \end{array}\right] = [ S(\alpha\beta\gamma) ]
\left[\begin{array}{c}
x\\ y\\ z \end{array}\right] ~ .$$
eq 29
$S(\chi_{\rm new},\theta_{\rm new},\phi_{\rm new}) =
S(\alpha\beta\gamma) \cdot S(\chi\theta\phi) ~ .$
eq 30
${\cal D}^{(1)}(\{\chi\theta\phi\}) = U\cdot S(\chi\theta\phi)
\cdot U^{-1} ~ ,$
eq 31
$$ U = 2^{-1/2} ~ \left[ \begin{array}{rrr}
1 & +{\rm i} & 0 \\
0 & 0 & \sqrt{2} \\
-1 & +{\rm i} & 0 \end{array} \right] ~ . $$
eq 32
${\cal D}^{(J)}(\{\chi_{\rm new},\theta_{\rm new},\phi_{\rm new}\}) =
{\cal D}^{(J)}(\{\alpha\beta\gamma\}) \cdot
{\cal D}^{(J)}(\{\chi\theta\phi\}) ~ ,$
eq 33
$$R_{\alpha\beta\gamma} \, |kJm\rangle = \sum_{k^\prime}~
{\cal D}^{(J)}_{kk^\prime} \, (\{\alpha\beta\gamma\}) \,
|k^\prime Jm\rangle ~ ,$$
eq 34
\begin{eqnarray*}
(234) \, | m_a m_b m_c m_d\rangle &\equiv &
(234) \, | m_a(1), m_b(2), m_c(3), m_d(4)\,\rangle\\
&=& \phantom{(234)} \, | m_a(1), m_b(3), m_c(4), m_d(2)\,\rangle \equiv
| m_a m_d m_b m_c\rangle ~ . \end{eqnarray*}
eq 35
$$ \left[ \begin{array}{c}
\mu_x\\ \mu_y\\ \mu_z \end{array}\right] =
\left[ \begin{array}{c}
~\\ S(\chi\theta\phi) \\
~ \end{array}\right]
\left[\begin{array}{c}
\mu_X \\ \mu_Y \\ \mu_Z \end{array}\right] ~ .$$
eq 36
$$ \left[ \begin{array}{c}
L_{sx} \\ L_{sy} \\ L_{sz} \end{array}\right] =
\left[ \begin{array}{c}
Q_{sy} P_{sz} - Q_{sz} P_{sy}\\
Q_{sz} P_{sx} - Q_{sx} P_{sz}\\
Q_{sx} P_{sy} - Q_{sy} P_{sx}\end{array}\right] ~ , $$
eq 37
$(u^1_{10}) \cdot (u^1_{01}) =
(u^1_{11}) \cdot [(u^1_{00}) - \sqrt{3} (u^0_{00})] ~ .$
eq 38
$$ \left[ \begin{array}{c}
Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] \equiv Q_s
\left[ \begin{array}{l}
\sin\theta_s \, \cos\phi_s\\
\sin\theta_s \, \sin\phi_s\\
\cos\theta_s \end{array}\right] ~ , $$
eq 39
$$ \left[ \begin{array}{c}
L_{sx} \\ L_{sy} \\ L_{sz} \end{array}\right] =
\left[ \begin{array}{ccc}
+\csc\theta_s \, \cos\phi_s & -\sin\phi_s & -\cot\theta_s \, \cos\phi_s \\
+\csc\theta_s \, \sin\phi_s & +\cos\phi_s & -\cot\theta_s \, \sin\phi_s \\
0 & 0 & +1 \end{array}\right] ~
\left[ \begin{array}{c}
0 \\ P_{\theta_s} \\ P_{\phi_s} \end{array}\right] ~ , $$
eq 40
$$|L \, k_L \rangle = [(2L+1)/4\pi]^{1/2} \,
{\cal D}_{0k_L}^{(L)}\, (\{0\, \theta_s \phi_s \}) ~ , $$
eq 41
$$R_{\alpha\beta\gamma} ~ \left[ \begin{array}{c}
Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] =
\left[ \begin{array}{c}
~\\ S(\alpha\beta\gamma) \\~ \end{array}\right] ~
\left[ \begin{array}{c}
Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array}\right] ~ . $$
eq 42
\begin{eqnarray*}
R_{\alpha\beta\gamma}[Q_{sx}, Q_{sy}, Q_{sz}]
&=& R_{\alpha\beta\gamma}[\sin\theta_s\cos\phi_s,\sin\theta_s\sin\phi_s,\cos\theta_s]\,Q_s\\
&=& R_{\alpha\beta\gamma}[0 \, 0 \, 1 ] \cdot S(\chi_s \theta_s \phi_s) \cdot Q_s\\
&=& [0 \, 0 \, 1 ] \cdot S(\chi_s \theta_s \phi_s) \cdot S^{\rm tr}(\alpha\beta\gamma) \cdot Q_s
~ . \end{eqnarray*}
eq 43
$$ R_{\alpha\beta\gamma}\,S(\chi_s\theta_s\phi_s)
= S(\chi_s\theta_s\phi_s) \cdot S^{\rm tr}(\alpha\beta\gamma) ~ ,$$
eq 44
$${\cal D}_{\mu^\prime\mu}^{(j)} (\{\alpha\beta\gamma\}) =
(-1)^{\mu-\mu^\prime} ~ {\cal D}_{-\mu^\prime-\mu}^{(j)*}
(\{\alpha\beta\gamma\}) ~ ,$$
eq 45
$$R_{\alpha\beta\gamma}~
{\cal D}_{k^{\prime\prime}k_L}^{(L)} (\{\chi_s\theta_s\phi_s\})=
\sum_{k_L^\prime}~{\cal D}_{k^{\prime\prime}k_L^\prime}^{(L)}
(\{\chi_s\theta_s\phi_s\}) \, (-1)^{k_L^\prime-k_L} ~
{\cal D}_{-k_L-k_L^\prime}^{(L)} (\{\alpha\beta\gamma\}) ~ . $$
eq 46
$$R_{\alpha\beta\gamma}~ |L\, k_L\rangle = \sum_{k_L^\prime} \,
(-1)^{k_L^\prime-k_L} ~
{\cal D}_{-k_L-k_L^\prime}^{(L)} (\{\alpha\beta\gamma\}) ~
|L\, k_L^\prime\rangle ~ . $$
eq 47
$$|\tilde{L} \, \tilde{k}_L\rangle \equiv (-1)^{k_L} \,
[(2L+1)/4\pi ]^{1/2} ~ {\cal D}_{0\,k_L}^{(L)} (\{0\, \theta_s\, \phi_s\}) =
(-1)^{k_L} \, |L\, k_L\rangle ~ ,$$
eq 48
$$R_{\alpha\beta\gamma}~ |\tilde{L} \, \tilde{k}_L \rangle =
\sum_{\tilde{k}_L^\prime} ~ {\cal D}_{\tilde{k}_L \tilde{k}_L^\prime}^{(L)} \,
(\{\alpha\beta\gamma\}) ~ |\tilde{L} \, \tilde{k}_L^\prime \rangle ~ ,$$
eq 49
\begin{eqnarray*}
(\theta_s)_{\rm new} &=& \pi - \theta_s\\
(\phi_s)_{\rm new} &=& \pi + \phi_s ~ . \end{eqnarray*}
eq 50
$$\mbox{\boldmath $R = J - L$}_s = \mbox{\boldmath $J + \tilde{L}$}_s $$
eq 51
\begin{eqnarray*}
{\cal H}_{\rm r} &=& B[\mbox{\boldmath $J$} - \zeta_s\mbox{\boldmath $L$}_s]^2\\
&=& B[\mbox{\boldmath $J$} + \zeta_s\mbox{\boldmath $\tilde L$}_s]^2\\
&=& B[\mbox{\boldmath $J$}^2 + \zeta_s^2\mbox{\boldmath $\tilde L$}_s^2
+ 2\zeta_s \, \mbox{\boldmath $J \cdot \tilde L$}_s] ~ , \\
s &=& 3 \, {\rm or} \, 4 \end{eqnarray*}
eq 52
$$B \{J(J+1) + \tilde{L}_s(\tilde{L}_s +1) \, \zeta_s^2 +
\zeta_s [R(R+1) - J(J+1) - \tilde{L}_s(\tilde{L}_s +1)\,]\,\} ~ . $$
eq 53
$$\left[\begin{array}{c}
\mu_X \\ \mu_Y \\ \mu_Z \end{array} \right] = (\partial\mu / \partial Q_s )
\left[\begin{array}{c}
~ \\ S^{-1} (\chi \theta \phi ) \\ ~\end{array} \right]
\left[\begin{array}{c}
Q_{sx} \\ Q_{sy} \\ Q_{sz} \end{array} \right] ~ , $$
eq 54
\begin{eqnarray*}
\Delta v_3 ~ {\rm or} ~ \Delta v_4 &=& \pm 1 \\
\Delta R = \Delta k_R &=& 0 ~ . \end{eqnarray*}
eq 55
$$\left[\begin{array}{c}
(R_{iX})_{\rm new} \\ (R_{iY})_{\rm new} \\ (R_{iZ})_{\rm new} \end{array} \right] =
\left[\begin{array}{c}
~\\ S^{-1}(\alpha\beta\gamma) \\ ~\end{array} \right]
\left[\begin{array}{c}
R_{iX} \\ R_{iY} \\R_{iZ} \end{array} \right] ~ . $$
eq 56
$$S^{-1}(\chi_{\rm new},\theta_{\rm new},\phi_{\rm new}) =
S^{-1}(\alpha\beta\gamma) \cdot S^{-1}(\chi\theta\phi) ~ . $$
eq 57
$$R_{\alpha\beta\gamma} \, |kJm\rangle = \sum_{m^\prime}~
{\cal D}^{(J)}_{m^\prime m} \, (\{\alpha\beta\gamma\}) \,
|kJm^\prime\rangle ~ . $$
eq 58
$$(j_1 j_2 m_1 m_2 \, | \, j_1 j_2\,j\,m) \equiv
C (j_1 j_2 j\,;\,m_1 m_2 m) ~ . $$
eq 59
$$| j_1 j_2\,j\,m \rangle = \sum_{m_1 m_2} ~ | j_1 m_1\rangle \,
| j_2 m_2\rangle \, (j_1 j_2 m_1 m_2 \, | \, j_1 j_2\,j\,m) ~ . $$
eq 60
$$| J\,m\,\tilde{L}\,R\,k_R \rangle =
\sum_{k\,\tilde{k}_L} ~ |k\,J\,m\rangle \,
| \tilde{L}\,\tilde{k}_L \rangle \,
(J \tilde{L}\,k\,\tilde{k}_L \, | \,
J \tilde{L}\,R\,k_R) ~ . $$
eq 61
$$| J\,\tilde{L}\,R\,k_R\,\Gamma\,I\,F\,m_F \rangle =
\sum_{m\,m_I} ~ |J\,m\,\tilde{L}\,R\,k_R\,\rangle ~
| \Gamma\,I\,m_I \rangle \,
(J\,I\,m\,m_I \, | \, J\,I\,F\,m_F\,) ~ . $$
eq 62
$$J_i J_j - J_j J_i = -i\hbar \, e_{ijk} J_k ~ . $$
eq 63
\begin{eqnarray*}
{\cal H}_{\rm cent. \, dist.} = -D_J \mbox{\boldmath $J$}^4
&-& {\textstyle{1\over2}} \, D_t \left[ 3\mbox{\boldmath $J$}^4
- 30\mbox{\boldmath $J$}^2 J_z^2 + 35 J_z^4
- 6\mbox{\boldmath $J$}^2 + 25J_z^2 \right .\\
&+& \left .{\textstyle{5\over2}} (J_x+iJ_y)^4
+ {\textstyle{5\over2}} (J_x-iJ_y)^4 \right] ~ . \end{eqnarray*}
eq 64
$$(2 \cdot 5 \cdot 7)^{1/2} \, f_0^{(4)} +
5 \left[f_{+4}^{(4)}+f_{-4}^{(4)} \right] ~ . $$
eq 65
$$h_m^{(k)} = \sum_{m_1 m_2} \, f_{m_1}^{(k_1)} \, g_{m_2}^{(k_2)}
(k_1 k_2 m_1 m_2 \, | \, k_1 k_2 k \, m ) $$
eq 66
$$\langle\Psi_{m_1}^{(j_1)}| \, O_{m_3}^{(j_3)} \, |\Psi_{m_2}^{(j_2)}\rangle
= \langle j_1| |O^{(j_3)}| |j_2\rangle \, (j_2 j_3 m_2 m_3 \, | \, j_2 j_3 j_1 m_1)
~ . $$