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4%,1+,%)#*',5)"1#6"#$%$"&'#()&*')+,"#7#8N9# !>%0*1/'",1#$%$"&'#()&*')+,"#
!L = !r ! !p (1)
H"(r,#) = $(r,#) H"(r,# + #0) = $(r,# + #0) (2)
R(#0)"(r,#) = "(r,# + #0) (3)
HR = RH (4)
#0 (5)
(x, y, z) (6)
(x cos #0 + y sin #0 , "x sin#0 + y cos #0 , z) (7)
[Rz(#0)"](x, y, z) = "(x cos #0 + y sin#0 , "x sin#0 + y cos #0 , z) (8)
#0 #0 # 1 (9)
[Rz(#0)"](x, y, z) $ "(x, y, z) + #0
!y%"
%x" x
%"
%y
"(10)
Rz(#0) $ 1" i#0
hLz (11)
Lx, Ly, Lz (12)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (13)
{H, Lx} {H, Ly} {H, Lz} (14)
{H, Lx, Ly, Lz} (15)
" $ (16)
4/1')(,6"#0%&(',"66"#H#$%$"&'#()&*')+,"#"&#$*(/&)+,"#(6/;;)+,"##
O*E)&)')%&#+,/&')+,"#E%1$"66"$"&'#)5"&')+,"#H#%0*1/'",1#C#'1%);#(%$0%;/&'";##
PQRQ#G&#;>)&'*1";;"#)()#/,#$%$"&'#()&*')+,"#ST%1<)'/6TU#0%,1#6/#0/1')(,6"##
0%&(',"66"#8%&#B"11/#"&;,)'"#6"#$%$"&'#()&*')+,"#ST5"#;0)&TU9#
V(')%&#;,1#,&"#E%&(')%>%&5"##################7##G&#;/)'#+,"#################I#5%&(#H#
!L = !r ! !p (1)
!p =h
i!" (2)
H"(r,#) = $(r,#) H"(r,# + #0) = $(r,# + #0) (3)
R(#0)"(r,#) = "(r,# + #0) (4)
HR = RH (5)
#0 (6)
(x, y, z) (7)
(x cos #0 + y sin #0 , #x sin#0 + y cos #0 , z) (8)
[Rz(#0)"](x, y, z) = "(x cos #0 + y sin#0 , #x sin#0 + y cos #0 , z) (9)
#0 #0 $ 1 (10)
[Rz(#0)"](x, y, z) % "(x, y, z) + #0
!y%"
%x# x
%"
%y
"(11)
Rz(#0) % 1# i#0
hLz (12)
Lx, Ly, Lz (13)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (14)
{H, Lx} {H, Ly} {H, Lz} (15)
{H, Lx, Ly, Lz} (16)
" $ (17)
!L = !r ! !p (1)
!p =h
i!" (2)
"(x, y, z) (3)
H"(r,#) = $(r,#) H"(r,# + #0) = $(r,# + #0) (4)
R(#0)"(r,#) = "(r,# + #0) (5)
HR = RH (6)
#0 (7)
(x, y, z) (8)
(x cos #0 + y sin #0 , #x sin#0 + y cos #0 , z) (9)
[Rz(#0)"](x, y, z) = "(x cos #0 + y sin#0 , #x sin#0 + y cos #0 , z) (10)
#0 #0 $ 1 (11)
[Rz(#0)"](x, y, z) % "(x, y, z) + #0
!y%"
%x# x
%"
%y
"(12)
Rz(#0) % 1# i#0
hLz (13)
Lx, Ly, Lz (14)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (15)
{H, Lx} {H, Ly} {H, Lz} (16)
{H, Lx, Ly, Lz} (17)
" $ (18)
"'#5"#$W$"#0%,1##Lx, Ly (1)
!0 ! 1 (2)
[Rz(!0)"](x, y, z) " "(x + !0y , #!0x + y , z) (3)
y! " #!0x + y (4)
x! " x + !0y (5)
#L = #r $ #p (6)
#p =h
i#% (7)
"(x, y, z) (8)
H"(r,!) = $(r,!) H"(r,! + !0) = $(r,! + !0) (9)
R(!0)"(r,!) = "(r,! + !0) (10)
HR = RH (11)
!0 (12)
(x, y, z) (13)
(x cos !0 + y sin !0 , #x sin!0 + y cos !0 , z) (14)
(x!, y!, z!) (15)
x! = x cos !0 + y sin !0 (16)
y! = #x sin!0 + y cos !0 (17)
z! = z (18)
2Q#
X&B/1)/&("#0/1#1%'/')%&#"'#$%$"&'#()&*')+,"#
!>)&B/1)/&("#0/1#1%'/')%&#
X&B/1)/&("#0/1#1%'/')%&#H#1Y6"#*+,)B/6"&'#0%,1#'%,'";#6";#5)1"(')%&;#5"#6>";0/("#
$%&'()*+,-*.&/01()H#
Z/'.*$/')+,"$"&'I#%*(1)'#("''"#)&B/1)/&("#"&#)&'1%5,);/&'##
6>%0*1/'",1#1%'/')%&###########5>,&#/&@6"#####/,'%,1#5>,&#/A"######5%&&*#H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)#####";'#)&B/1)/&'#0/1#1%'/')%&#;)#
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
[A"$06"#H#1%'/')%&#/,'%,1#5"#6>/A"#z#8(%%15%&&*";#0%6/)1";#C#NO9#
!L = !r ! !p (1)
!p =h
i!" (2)
"(x, y, z) (3)
H"(r,#) = $(r,#) H"(r,# + #0) = $(r,# + #0) (4)
R(#0)"(r,#) = "(r,# + #0) (5)
HR = RH (6)
#0 (7)
(x, y, z) (8)
(x cos #0 + y sin #0 , #x sin#0 + y cos #0 , z) (9)
[Rz(#0)"](x, y, z) = "(x cos #0 + y sin#0 , #x sin#0 + y cos #0 , z) (10)
#0 #0 $ 1 (11)
[Rz(#0)"](x, y, z) % "(x, y, z) + #0
!y%"
%x# x
%"
%y
"(12)
Rz(#0) % 1# i#0
hLz (13)
Lx, Ly, Lz (14)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (15)
{H, Lx} {H, Ly} {H, Lz} (16)
{H, Lx, Ly, Lz} (17)
" $ (18)
V (r) = ! q2
4!"0r
!
x
y
!tournee(r,") = [Rz,!!] (r,") = !(r,"! #)
R!u," !u
!
G0*1/'",1#1%'/')%&#/,'%,1#5"#G\#8(%%15%&&*";#(/1'*;)"&&";9#
:%'/')%>,&#/&@6"#####/,'%,1#5"#6>/A"#G\#H#
y
x
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
!L = !r ! !p (1)
!p =h
i!" (2)
"(x, y, z) (3)
H"(r,#) = $(r,#) H"(r,# + #0) = $(r,# + #0) (4)
R(#0)"(r,#) = "(r,# + #0) (5)
HR = RH (6)
#0 (7)
(x, y, z) (8)
(x cos #0 + y sin #0 , #x sin#0 + y cos #0 , z) (9)
(x!, y!, z!) (10)
[Rz(#0)"](x, y, z) = "(x cos #0 + y sin#0 , #x sin#0 + y cos #0 , z) (11)
#0 #0 $ 1 (12)
[Rz(#0)"](x, y, z) % "(x, y, z) + #0
!y%"
%x# x
%"
%y
"(13)
Rz(#0) % 1# i#0
hLz (14)
Lx, Ly, Lz (15)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (16)
{H, Lx} {H, Ly} {H, Lz} (17)
{H, Lx, Ly, Lz} (18)
!
!
x! = x cos ! + y sin !
y! = !x sin ! + y cos !
= !(x cos " + y sin " , !x sin" + y cos " , z)
!tournee(x, y, z) =
z! = z
[Rz,!!] (x, y, z) = !(x!, y!, z!)
(>";']C]5)1"#H#
H#@*&*1/'",1;#)&E)&)'*;)$/,A#5,#@1%,0"#5";#1%'/')%&;#
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y) (5)
(x cos "0 + y sin"0 , !x sin "0 + y cos "0) (6)
[Rz("0)!](x, y) = !(x cos "0 + y sin "0 , !x sin"0 + y cos "0) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y) # !(x, y) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
:%'/')%&#)&E)&)'*;)$/6"#/,'%,1#5"#3\#H#
y
x
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
!L = !r ! !p (1)
!p =h
i!" (2)
"(x, y, z) (3)
H"(r,#) = $(r,#) H"(r,# + #0) = $(r,# + #0) (4)
R(#0)"(r,#) = "(r,# + #0) (5)
HR = RH (6)
#0 (7)
(x, y, z) (8)
(x cos #0 + y sin #0 , #x sin#0 + y cos #0 , z) (9)
(x!, y!, z!) (10)
[Rz(#0)"](x, y, z) = "(x cos #0 + y sin#0 , #x sin#0 + y cos #0 , z) (11)
#0 #0 $ 1 (12)
[Rz(#0)"](x, y, z) % "(x, y, z) + #0
!y%"
%x# x
%"
%y
"(13)
Rz(#0) % 1# i#0
hLz (14)
Lx, Ly, Lz (15)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (16)
{H, Lx} {H, Ly} {H, Lz} (17)
{H, Lx, Ly, Lz} (18)
V#6>%151"#2#"&##! !
x! ! x + !y
y! ! "!x + y
z! = z
!! 1
[Rz,!!] (x, y, z) ! !(x + "y , ""x + y , z)
! !(x, y, z) + "
!y#!
#x" x
#!
#y
"
Rz,! ! 1" i!
! Lz
C################I#%,#C###############I#%,#C#
X&B/1)/&("#0/1#1%'/')%&#"'#$%$"&'#()&*')+,"#
^)#6>./$)6'%&)">,%<6=$"#DO#";'#)&B/1)/&'#0/1#1%'/')%&I#)6#(%$$,'"#/B"(##
'%,;#6";#%0*1/'",1;#1%'/')%&;I#"�/1')(,6)"1#(",A#/;;%()*;#C#5";#0"')';#/&@6";Q##
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (12)
G�",'#5%&(#(."1(."1#,&"#</;"#01%01"#(%$$,&"##
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (12)
{H, Lx} {H, Ly} {H, Lz} (13)
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (12)
{H, Lx} {H, Ly} {H, Lz} (13)
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (12)
{H, Lx} {H, Ly} {H, Lz} (13)
4",']%&#(."1(."1#,&"#</;"#01%01"#(%$$,&"#/,A#_#%0*1/'",1;###########################7##
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (12)
{H, Lx} {H, Ly} {H, Lz} (13)
{H, Lx, Ly, Lz} (14)
-"6/#;"1/)'#'1%0#;)$06"#F#######################&"#(%$$,'"&'#0/;#"&'1"#",AK#
H!(r,") = #(r,") H!(r," + "0) = #(r," + "0) (1)
R("0)!(r,") = !(r," + "0) (2)
HR = RH (3)
"0 (4)
(x, y, z) (5)
(x cos "0 + y sin "0 , !x sin"0 + y cos "0 , z) (6)
[Rz("0)!](x, y, z) = !(x cos "0 + y sin"0 , !x sin"0 + y cos "0 , z) (7)
"0 "0 " 1 (8)
[Rz("0)!](x, y, z) # !(x, y, z) + "0
!y$!
$x! x
$!
$y
"(9)
Rz("0) # 1! i"0
hLz (10)
Lx, Ly, Lz (11)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (12)
{H, Lx} {H, Ly} {H, Lz} (13)
{H, Lx, Ly, Lz} (14)
! # (15)
!L = !r ! !p (1)
!p =h
i!" (2)
"(x, y, z) (3)
H"(r,#) = $(r,#) H"(r,# + #0) = $(r,# + #0) (4)
R(#0)"(r,#) = "(r,# + #0) (5)
HR = RH (6)
#0 (7)
(x, y, z) (8)
(x cos #0 + y sin #0 , #x sin#0 + y cos #0 , z) (9)
[Rz(#0)"](x, y, z) = "(x cos #0 + y sin#0 , #x sin#0 + y cos #0 , z) (10)
#0 #0 $ 1 (11)
[Rz(#0)"](x, y, z) % "(x, y, z) + #0
!y%"
%x# x
%"
%y
"(12)
Rz(#0) % 1# i#0
hLz (13)
Lx, Ly, Lz (14)
[H, Lx] = 0 [H, Ly] = 0 [H, Lz] = 0 (15)
{H, Lx} {H, Ly} {H, Lz} (16)
{H, Lx, Ly, Lz} (17)
" $ (18)
[Lx, Ly] = ihLz (19)
!";#1"6/')%"#(%$$,'/')%,#$%$"&'#()&*')+,"#
[&#,')6);/&'#6";#5*E)&)')%&;#
[Lx, Ly] = ihLz (19)
Lx =h
i
!y
!
!z! z
!
!y
"(20)
[Lx, Ly] = ihLz (19)
Lx =h
i
!y
!
!z! z
!
!y
"(20)
Ly =h
i
!z
!
!x! x
!
!z
"(21)
%&#$%&'1"#+,"#H##
O"#$W$"#H#
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
X&'"101*'/')%&#@*%$*'1)+,"#(6/)1"#H##
6";#1%'/')%&;#/,'%,1#5";#/A";#x,y#"'#z#&"#(%$$,'"&'#0/;#"&'1"#"66";##
21)1()3(4%)3$5)*6$/&1$765(.)56'47%$18'(1%)*(49)!&'3&5$1%(5)*()
NQ##
`,"6;#*'/';#01%01";#0%,1#
6"#$%$"&'#()&*')+,"#%1<)'/6#7#
G�",'#5%&(#5)/@%&/6);"1#;)$,6'/&*$"&'########"'#,&"#5";#'1%);#%<;"1B/<6";#H#
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R)"&#+,"#######&"#(%$$,'"#0/;#/B"(#######"'#######I#%&#/#H##
C#$%&'1"1#"&#"A"1()("#
4/1#(%&;*+,"&'## %a#%&#/#0%;*#H# L2 = L2x + L2
y + L2z
O)/@%&/6);/')%"################H#6>/001%(."#E%&(')%>%&5"#
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./1$%&)+,";#;0.*1)+,";#
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#&%$<1"#;/&;#5)$"&;)%&#
V''"&')%&#H#
Lz!("r) = # !("r) (1)
d1 d2 M1 M2 (2)
I = M1d21 + M2d
22 (3)
$ = 0 $ = 1 $ = 2 (4)
LzY0,0(%,&) = 0 L+Y0,0(%,&) = 0 (5)
! !
0sin % d%
! 2!
0d& |Y",m(%,&)|2 = 1 (6)
H R(&0) = R(&0) H (7)
R(&0) (8)
= R(&0) '(r,&) = R(&0)"H !(r,&)
#= H
"R(&0)!(r,&)
#(9)
H (10)
'(r,& + &0) = H!(r,& + &0) (11)
'(r,&) = H!(r,&) (12)
H!(r,&) = '(r,&) H!(r,& + &0) = '(r,& + &0) (13)
! ' (14)
Lx, Ly (15)
&0 ! 1 (16)
[Rz(&0)!](x, y, z) " !(x + &0y , #&0x + y , z) (17)
y! " #&0x + y (18)
:"$/1+,"#;,1#6";#B/1)/<6";#0"1')&"&'";#
!/#5*0"&5/&("#"&#r B)"&51/#,6'*1)",1"$"&'I#6%1;+,J%&#/,1/##
01*();*#6J./$)6'%&)"&I#"�/1')(,6)"1#6"#0%'"&')"6#V(r)
!";#%0*1/'",1;##############&"#E%&'#)&'"1B"&)1#+,"#6";#B/1)/<6";#))))))))"'#0/;#rQ#
X$0%;"1#C#####################5JW'1"#*'/'#01%01"#5"###############0"1$"''1/#5"##
5*'"1$)&"1#6/#5*0"&5/&("#"&#))))))))I#$/);#0/;#6/#5*0"&5/&("#"&#r
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DQ#
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!";#B/6",1;#01%01";#5"#######"'#####
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/,A#B/6",1;#01%01";##########################"'############";'#&%'*"#H#
!";#./1$%&)+,";#;0.*1)+,";#
G/#(.%);)'#&%1$*"#H#
V01=;#6/#$*'.%5"#/6@*<1)+,"I#%&#;>/''"&51/)'#C#("#+,"#######
#####"'########0,);;"&'#W'1"#"&')"1;#%,#5"$)]"&')"1;I#$/);KQ#
! !
0sin ! d!
! 2!
0d" |Y",m(!,")|2 = 1 (1)
H R("0) = R("0) H (2)
R("0) (3)
= R("0) #(r,") = R("0)"H $(r,")
#= H
"R("0) $(r,")
#(4)
H (5)
#(r," + "0) = H$(r," + "0) (6)
#(r,") = H$(r,") (7)
H$(r,") = #(r,") H$(r," + "0) = #(r," + "0) (8)
$ # (9)
Lx, Ly (10)
"0 ! 1 (11)
[Rz("0)$](x, y, z) " $(x + "0y , #"0x + y , z) (12)
y! " #"0x + y (13)
x! " x + "0y (14)
%L = %r $ %p (15)
%p =h
i%% (16)
$(x, y, z) (17)
H$(r,") = #(r,") H$(r," + "0) = #(r," + "0) (18)
#"'#))#####5%)B"&'#W'1"#"&')"1;#
#V,#0/1/@1/0."#2I#&%,;#/B%&;#1*;%6,#6>*+,/')%&#/,A#B/6",1;#01%01";#0%,1#######Q##
#P%,;#/B%&;#'1%,B*#6/#5*0"&5/&("#"&######5";#E%&(')%&;#01%01";#5"#6/#E%1$"#H#
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6";#*'/';#5"#$%$"&'#()&*')+,"#&,6#
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LzY0,0(!,") = 0 L+Y0,0(!,") = 0 (1)
! !
0sin ! d!
! 2!
0d" |Y",m(!,")|2 = 1 (2)
H R("0) = R("0) H (3)
R("0) (4)
= R("0) #(r,") = R("0)"H $(r,")
#= H
"R("0) $(r,")
#(5)
H (6)
#(r," + "0) = H$(r," + "0) (7)
#(r,") = H$(r,") (8)
H$(r,") = #(r,") H$(r," + "0) = #(r," + "0) (9)
$ # (10)
Lx, Ly (11)
"0 ! 1 (12)
[Rz("0)$](x, y, z) " $(x + "0y , #"0x + y , z) (13)
y! " #"0x + y (14)
x! " x + "0y (15)
%L = %r $ %p (16)
%p =h
i%% (17)
$(x, y, z) (18)
LzY0,0(!,") = 0 L+Y0,0(!,") = 0 (1)
! !
0sin ! d!
! 2!
0d" |Y",m(!,")|2 = 1 (2)
H R("0) = R("0) H (3)
R("0) (4)
= R("0) #(r,") = R("0)"H $(r,")
#= H
"R("0) $(r,")
#(5)
H (6)
#(r," + "0) = H$(r," + "0) (7)
#(r,") = H$(r,") (8)
H$(r,") = #(r,") H$(r," + "0) = #(r," + "0) (9)
$ # (10)
Lx, Ly (11)
"0 ! 1 (12)
[Rz("0)$](x, y, z) " $(x + "0y , #"0x + y , z) (13)
y! " #"0x + y (14)
x! " x + "0y (15)
%L = %r $ %p (16)
%p =h
i%% (17)
$(x, y, z) (18)
!";#./1$%&)+,";#;0.*1)+,";#8;,)'"9#
Z%$"&'#()&*')+,"#2 :
m
0
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1
2
-2
2 états p
(EQ#%1<)'/6"; px,py,pz 5";#(.)$);'";#
g&#"A"$06"#H#6/#1%'/')%"#$%6*(,6";#5)]/'%$)+,";#
!";#5"@1*;#5"#6)<"1'*#5>,&"#$%6*(,6"#H#
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/B"(#5";#&)B"/,A#*+,)5);'/&';#
:%'/')%"#6/#$%6*(,6"#H#+,"6#;0"('1"#7#
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
h" " 50 a 500 meV (24)
i#
d1 d2 M1 M2 (1)
I = M1d21 + M2d
22 (2)
! = 0 ! = 1 ! = 2 (3)
LzY0,0(",#) = 0 L+Y0,0(",#) = 0 (4)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (5)
H R(#0) = R(#0) H (6)
R(#0) (7)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0)%(r,#)
#(8)
H (9)
$(r,# + #0) = H%(r,# + #0) (10)
$(r,#) = H%(r,#) (11)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (12)
% $ (13)
Lx, Ly (14)
#0 ! 1 (15)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (16)
y! " ##0x + y (17)
x! " x + #0y (18)
`,"6#;0"('1"#0%,1#6"#$%,B"$"&'#5"#1%'/')%##
-6/;;)+,"$"&'I#6>*&"1@)"#5"#1%'/')%&#;>*(1)'#H###
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
h" " 50 a 500 meV (24)
E =L2
2I(25)
H#$%$"&'#5>)&"1')"#5"#6/#$%6*(,6"#
Z%6*(,6"#j#1%'/'",1#1)@)5"#
k/$)6'%&)"&#+,/&')+,"#H##
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
h" " 50 a 500 meV (24)
E =L2
2I(25)
I =12MR2 (26)
H =L2
2I(27)
G&#/''"&5#5%&(#5";#&)B"/,A#5>*&"1@)"#1"0*1*;#0/1#6"#&%$<1"#+,/&')+,"#
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
h" " 50 a 500 meV (24)
E =L2
2I(25)
I =12MR2 (26)
H =L2
2I(27)
# (28)
E! =h2#(# + 1)
2I(29)
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
h" " 50 a 500 meV (24)
E =L2
2I(25)
I =12MR2 (26)
H =L2
2I(27)
# (28)
E! =h2#(# + 1)
2I(29)
i#
I = M1d21 + M2d
22 (1)
! = 0 ! = 1 ! = 2 (2)
LzY0,0(",#) = 0 L+Y0,0(",#) = 0 (3)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (4)
H R(#0) = R(#0) H (5)
R(#0) (6)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0) %(r,#)
#(7)
H (8)
$(r,# + #0) = H%(r,# + #0) (9)
$(r,#) = H%(r,#) (10)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (11)
% $ (12)
Lx, Ly (13)
#0 ! 1 (14)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (15)
y! " ##0x + y (16)
x! " x + #0y (17)
&L = &r $ &p (18)
d1 d2 (1)
I = M1d21 + M2d
22 (2)
! = 0 ! = 1 ! = 2 (3)
LzY0,0(",#) = 0 L+Y0,0(", #) = 0 (4)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (5)
H R(#0) = R(#0) H (6)
R(#0) (7)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0) %(r,#)
#(8)
H (9)
$(r,# + #0) = H%(r,# + #0) (10)
$(r,#) = H%(r,#) (11)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (12)
% $ (13)
Lx, Ly (14)
#0 ! 1 (15)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (16)
y! " ##0x + y (17)
x! " x + #0y (18)
d1 d2 (1)
I = M1d21 + M2d
22 (2)
! = 0 ! = 1 ! = 2 (3)
LzY0,0(",#) = 0 L+Y0,0(", #) = 0 (4)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (5)
H R(#0) = R(#0) H (6)
R(#0) (7)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0) %(r,#)
#(8)
H (9)
$(r,# + #0) = H%(r,# + #0) (10)
$(r,#) = H%(r,#) (11)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (12)
% $ (13)
Lx, Ly (14)
#0 ! 1 (15)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (16)
y! " ##0x + y (17)
x! " x + #0y (18)
d1 d2 M1 M2 (1)
I = M1d21 + M2d
22 (2)
! = 0 ! = 1 ! = 2 (3)
LzY0,0(",#) = 0 L+Y0,0(",#) = 0 (4)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (5)
H R(#0) = R(#0) H (6)
R(#0) (7)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0)%(r,#)
#(8)
H (9)
$(r,# + #0) = H%(r,# + #0) (10)
$(r,#) = H%(r,#) (11)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (12)
% $ (13)
Lx, Ly (14)
#0 ! 1 (15)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (16)
y! " ##0x + y (17)
x! " x + #0y (18)
Z);"#"&#*B)5"&("#5"#("#;0"('1"#
PR^#
V;;"$<6*"#5>/'%$";#1"E1%)5);#0/1#6/;"1#8'"$0*1/',1"#233#$)(1%l9#
V,#;")"#("#@/\#5>/'%$";I#5";#$%6*(,6";#
5)]/'%$)+,";#0",B"&'#;"#E%1$"1#
!/<%1/'%)1"#V)$*#-%''%&I#G1;/?#H#@/\#E1%)5#5>/'%$";#5"#(*;),$#
G*'"('"#6";#$%6*(,6";#"";#)%&);/&'#/B"(#,&##
6/;"1#"'#"&#$";,1/&'#6"#(%,1/&'#5>)%&;#01%5,)';#
[&#(./&@"/&'#6/#E1*+,"&("#5,#6/;"1I#%&#</6/?"#6";##
5)EE*1"&';#&)B"/,A#5>*&"1@)"#0%;;)<6";#5"#6/#$%6*(,6"#
)%&);/')%&#
! = 0 ! = 1 ! = 2 (1)
LzY0,0(",#) = 0 L+Y0,0(", #) = 0 (2)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (3)
H R(#0) = R(#0) H (4)
R(#0) (5)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0) %(r,#)
#(6)
H (7)
$(r,# + #0) = H%(r,# + #0) (8)
$(r,#) = H%(r,#) (9)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (10)
% $ (11)
Lx, Ly (12)
#0 ! 1 (13)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (14)
y! " ##0x + y (15)
x! " x + #0y (16)
&L = &r $ &p (17)
&p =h
i&% (18)
! = 0 ! = 1 ! = 2 (1)
LzY0,0(",#) = 0 L+Y0,0(", #) = 0 (2)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (3)
H R(#0) = R(#0) H (4)
R(#0) (5)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0) %(r,#)
#(6)
H (7)
$(r,# + #0) = H%(r,# + #0) (8)
$(r,#) = H%(r,#) (9)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (10)
% $ (11)
Lx, Ly (12)
#0 ! 1 (13)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (14)
y! " ##0x + y (15)
x! " x + #0y (16)
&L = &r $ &p (17)
&p =h
i&% (18)
! = 0 ! = 1 ! = 2 (1)
LzY0,0(",#) = 0 L+Y0,0(", #) = 0 (2)
! !
0sin " d"
! 2!
0d# |Y",m(",#)|2 = 1 (3)
H R(#0) = R(#0) H (4)
R(#0) (5)
= R(#0) $(r,#) = R(#0)"H %(r,#)
#= H
"R(#0) %(r,#)
#(6)
H (7)
$(r,# + #0) = H%(r,# + #0) (8)
$(r,#) = H%(r,#) (9)
H%(r,#) = $(r,#) H%(r,# + #0) = $(r,# + #0) (10)
% $ (11)
Lx, Ly (12)
#0 ! 1 (13)
[Rz(#0)%](x, y, z) " %(x + #0y , ##0x + y , z) (14)
y! " ##0x + y (15)
x! " x + #0y (16)
&L = &r $ &p (17)
&p =h
i&% (18)
[A"$06"#5"#;0"('1"#5"#1%'/')%$%6*(,6"#-;N9#
^)@&/6#)%&)+,"##
8,&)'*;#/1<)'1/)1";9#G�",'#B*1)E)"1#+,"##
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
h" " 50 a 500 meV (24)
E =L2
2I(25)
I =12MR2 (26)
H =L2
2I(27)
# (28)
E! =h2#(# + 1)
2I(29)
E! ! E!!1 # # (30)(/1/('*1);')+,"#5>,&#;0"('1"#5"#1%'/')%&#
m1*+,"&("#(/1/('*1);')+,"#H#2#ik\#8*&"1@)"#5"#+,"6+,";#$)(1%"h9##
!";#*&"1@)"#6)*";#C#6/#1%'/')%";#$%6*(,6";#;%&'#@*&*1/6"$"&'##
06,;#</;;";#+,"#("66";#/;;%()*";#C#6/#B)<1/')%&#
X()I#6>"EE"'#";'#1"&E%1(*#0/1#6"#E/)'#+,"#6/#$%6*(,6"#";'#01%5,)'"#5/&;#,&##
*'/'#ST"A()'*TUI#/B"(#,&"#@1/&5"#5);'/&("#"&'1"#/'%$";#H##
[Lx, Ly] = ihLz (19)
[Ly, Lz] = ihLx (20)
[Lz, Lx] = ihLy (21)
Lx =h
i
!y
!
!z! z
!
!y
"(22)
Ly =h
i
!z
!
!x! x
!
!z
"(23)
h" " 50 a 500 meV (24)
E =L2
2I(25)
I =12MR2 (26)
H =L2
2I(27)
# (28)
E! =h2#(# + 1)
2I(29)
E! ! E!!1 # # (30)
R = 13 A (31)
!2!(! + 1)2I
! !2(!! 1)!2I
=!2!
I
`,"#0"&;"1#5";#B/6",1;#M#"&')=1";#5"#?#7#
h/6",1;#*6)$)&*";#0%,1#6"#$%$"&'#()&*')+,"#%1<)'/6#"/);%"#H#
V1'"E/('#%,#)&5)(/')%J,&#n$%&5"n#06,;#B/;'"#7#
)7$)3,-56;4()*4)5361)@))
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