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(ρ, φ)]# /+5C#8/03F 9;9)#8QSA)3YC)*)(88)3";/

ρφLd9 #8/8]? Í 3Yx252;/78'X+# 8/W#85

î

(

− h2

2M52 +

1

2Mω2ρ2 + V (z)

)

Ψ(ρ, φ, z) = εΨ(ρ, φ, z).É ë M Ô Ê

0\# #8B9 c# C#-I5ü BH(# Y# B9;]R#8/5l^#8( 39;C Z#8(-I'5nB$;/n;vyz

788/+5 (# ezY]A<#aZ89 2;/ 3# /nB$SAb/Ψ (ρ, φ, z) =

R (ρ) Φ (φ)Z0 (z)]SA

Z0 (z))jp9;ST((L_/78>89 2 8/[AP8/FL52;-I/( 8/+# 9#8-I8/ 3[3F 9;9)#8'MO<; /7U# 9)#83 #8/c;/Z3>?9; /52 3# 93F?8Y52 /#]&'X+# 8/c,

R# /+5

Φ)

− h2

2M

(

∂2

∂ρ2+

1

ρ

∂

∂ρ+

1

ρ2

∂2

∂φ2

)

RΦ +1

2Mω2ρ2RΦ = εRΦ,

É ë M ë ÊSA)3YÌ3# /aBQ(# Y#5 /DI# /Ì#8/789 #8J#8(A# /5Ì#&Y#85 #89+#8('M48sQ#8/789 #8#8(A8/[G+/5

− 1

Φ

∂2Φ

∂φ2= m2,

É ë M ÷Ê# /+5*/8-# 9 ;:5Ì 78/2,/3F 8/A#8

Φ(φ) =1√2πeimφ,

É ë M îDÊSA

m)N#8/Q;/D78]/# -I9 >JXD#8/-ù/D-&BN38'(8/5;/7sJ#8/789 #8-I8-I/D- P#I( /79;Q#8( 39;M4*Y#852)# 9Æ+# /RG/5AQXD# 8/

∂2R

∂ρ2+

1

ρ

∂R

∂ρ+

(

−M2ω2ρ2

h2+

2Mε

h2− m2

ρ2

)

R = 0.É ë M ú Ê

Ã/D2523F /7Qv|96~6;A|Am;68aosc =

√

h/Mω# /+569 ?8È? /7E#P9 ;-I;sSA

ρ→ ∞ ]2 B38-I∂2R

∂ρ2+

(

− 1

a4osc

ρ2 +2Mε

h2

)

R = 0,É ë M ïÊ

SA)3Y 9 #5"6C9;?8È0,#Z89 2 8/R ∝ e−ρ2/2a2

oscMHOn; =#8/# :

R =f(ρ)e−ρ2/2a2

osc /'5Ì /DIÄJXMÉ ë M ú Ês8/QG/+5T,8

f

ρ2f ′′ +

(

ρ− 2ρ3

a2osc

)

f ′ +

(

− 2

a2osc

ρ2 +2Mε

h2ρ2 −m2

)

f = 0.É ë Mñ Ê

Ã_R)R3F9 # [,8-v)[F12( 8/c#RI89 2 8/Z3# /$/9 >WB65/52/DR/\/STYp ρ.

,88]/Q#8(-I'pyG+/# 9Æ89 2 8/6BR ∝ ρ|m|(c0 +

c2ρ2 + ... + c2nρ

2n)e−ρ2/2a2

oscMm7D#;\*+STYA

ρ# */8R#833F2'5=( /3F&?>2()3#89;9 >9 \# /D(9; 8/W#8TB+RG/;6#* 78 /°MÃ/ /7IF12( 8/c,

R /DWÄJXMÉ ë M ú Ê[# /+5=9;?ÈD /7W#RI3F?e:3;/,8* 78'%Ld5s-V7;\#I3F/52; 8/W,8*/7>

εnm = (2n+ |m| + 1)hω.É ë M öDÊ

ú

8W+#Ic/78>^)"9; /#8I;/m

M Ã_ 39;'# "+#I)3#8I527/#3F>];/3R,812#8-I9;R%STST#ª\y,/34;/A#ª\? /7"EXD#8/-V/?-6B+n,m = 0, 2# /+5

n,m = 1, 0#ª\*I# -IE/78>W\8/c78$;Q# /79)# y-I-"/D##852;/DM=)&5278/Y#83F>]U#*SA;9 9JBÌSA/°]U EZ#Z3F#8;/01D/D&B8È/0 /# /<# /#8-I/ 3E+8/D #89²]j8RB?>W";/D2523F 8/$8p;/DY#834;/ /<#Ì(>2(-v8\8#89++# )3F9 MQ8#89ÆST#ª\F,/3F 8/#8

Ψnm =Nnm√

2πHnm(ρ)eimφe−ρ2/2a2

osc ,É ë M þ Ê

SANnm

#8/-#89; :# 8/^3F/%Y# /D*#8/5Hnm(ρ) = ρ|m|(c0 + c2ρ

2 + ... +c2nρ

2n)MPXD#8/D- /?-&BYp# T?pA/?-6B+p ÆY#852)# 92/252']

n]#8/5" 78/?\ª#89;[8UE# /79)# -I8-I/D-W]

mMPÅp89 >/8-# 9 ;:5Æ]?

n = 0ST#ª\,/3F 8/y#

Ψ0m =1

√

π |m|!a2osc

(

ρ

aosc

)|m|eimφe−ρ2/2a2

osc .É ë M Ôÿ Ê

ô 9 # 9 >8]3F 9 9 # 89 /78I)P#ª\8Y# 7sSA)5?"8jA7/5I%Y#EÉn,m = 0

Ê;/79;Ld+# )3F9 AST#ª\F,/3F 8/I /a#E#8-I/ 3D3;9 9)#J3F/2G/-I/D]2# /+5",8FA#8/Ì -I+(Y# /D9 /78c3# 9 [,E>2%-cM1 ¡£¢¥¤¦X§©¨d¤«ª¬¤®1¯ °¤±²¤®o³°´µ3¢·¶s´¡¸X1¹

¡¤®¯Ã_A '#8>""G/5W# /c# /#89;>D 3[F12( 8/,8TQ/78>;À3# 5aB?>a#&ST#8È?9;># /# -I8/)3I+8/ #89J /7=GY%Ld5*B#;/08>8M$W3F3F 8/C 3#89 39)#'5"SAb'(3FpQyS#ª\8T,/3F 8/

Ψ0m ÆÄJXMAÉ ë M Ôÿ ÊU,8

m ≥ 0M3F3F 8/Ì+8/D #89Æ 7;\/WBD>

∆V =λ

2Mω2 ρ4

a2osc

,É ë M Ô8Ô Ê

SAJ52 -"/;/9;'U#8#8-"λ

3F/D89)ÆT(/786 2#8/#8-I8/ 3b%>]# /+5,λ 1

8/*3# /WSA;

〈Ψ0m |∆V |Ψ0m〉 =λ

2hω (|m| + 1) (|m| + 2) .

É ë M Ô'ë ÊR8#89°/7>

ε′0m = (|m| + 1 +λ

2(|m| + 1) (|m| + 2))hω,

É ë M Ô ÷DÊ# /+5*/7>/9 8/7 9 /#8A;/

m]B2yXD#52# )3 M

ï

1º »²X¼¶sX¢i¡¸o¦³4A#6>2%- (\8#89//2Ld /D#34 /76# )3F9 ]D*+'34- 8°R+# -"/)33F 9 9 # 8)AY#B8;/7MJQ Y# 9°/78>ZÉË;/=/bY8

hωÊs,8y#>2%-V8

NB8/A)K%(y&-v 6( /789 FL_#8( 39;Q/78 ]2# [# *9 ;/'# /m

]+# /+5;J8/" 78/'y/7# \8m

É,WB+I-I ;\#5Z;/ZI/1D*34;/+Ê4]°#89 6 Y# 9/78>$)*9 ;/'# * /0I Y# 9J#8/789)# *-I-"/D-L

MÃ_A#8( 39;i

#8*XD#8/-/?-&BY(ni,mi)

]2*F12( 8/a,8*/7>

Etot =

N∑

i=1

(2ni + |mi| + 1) = L+

N∑

i=1

(2ni + 1) .É ë M Ô îDÊ

= Y# 9y/78> ,8$ /52/+52/DÌ8Q$ /52 \? 5# 9R-I8-I/D#08*#8( 39;'M0ST\']SA/$E)[;/D#34;/W'(/']++# )3F9 ASA;Z5½j/DR#8/789 #8-I8-I/DY#E /Y#83F52;j/D9 >8MP,8[/78>I NR8#89j>2%- 52+/58/SAF']D#ª>8]?# 9 9#8( 39;'J3# >&R#8-"[# /79 #8p-I8-I/D-c]DJSAs#89;9# /789)# °-I-"/D- )U3# 5QB?>*#;/79;s# )3F9 8MU UB#8ÈD /78527/#3F>[ 8/A AÈ>&)(';/IA/DJ(52>6 Æ # /7*ÇTFL_Äp;/+% /"38/52/#'M

QOR¾Q ¿Àq]^Áq_\VX]ÂnWe_\Zs`®_Ãp ÄAjs`©WeYvab]1`®_\cµabWeYZÆÅj\VXj\YV!=8 \ª# '5ZB?>=F12;-I/']°ST"S# /DRc /D\( 7# 6IB#ª\? 8Q T;/DY#834;/7B8/IB2K%3FIC8#;/°]pSA;n#Z/#8B9 c389; /7^3F/%Y# /DM c\ª#8)#;/ PE,XD/3F>c8p8#;/$# /5c8P" /Y#83F 8/$(/78$7;\A Ea+#8Y# /+(; 8/°B+%Ss/Q,/5# -I/D#89;9 >Q52½j/DU(# '° /*J7#8],-ÞSA)3YE#A+#852)# 7#8-ü s788/5*(# s s38/52/#sSA;9 9DB+52Y#ªSA/°MX°U-I8J39;D(9;>F1#8-" /Q#3YaË#348T# y#Aj3FQB+#ª\?;A UE>2%-cM1º= Ç´µ=Èu²X³°Xd¡U#8 /7Q/2#89#8-P /#*(-#89;97;/I Æ(#3FA)P/8p;\?)# 9d]8Bpy9;\ª#8/D)aa)EÌ# W AW38/2G/;/7Z+8/D #89²M0W# -I8/)3 /D)# 9T 3F-I-"/ / 12+ -I/D]JB"$# /# -I8/)3/=%52 5 8]sSAb #C(-#89;9XD# )3&- /3F9 52'5Æ]U#55Q\8#89U /D'%;/7Ì,# RWI#(52)# 7#8-cMÍ /3FIST"(+52>Z# /<ej3F \89 >=%ST L52 -"/;/# 9(>2(-W]jF1#34Q/#" s3F/2G/-I/DA;/W

zL52 '34 8/Ì /;-I8#8/'MOnb # 9 >l# -I8/)3C3F8/G/ /7 +8/D #89²]R^F1?/#89*8# 8/#89[,FLXD/+3F>$3#8//8[F135$"#8=,'XD/3>8MEÃ_JIF1?/# 98# 8/#89N,XD/3F> AY# I)

Ω]#8/56D3;9 9)#,'XD/3>E jAY# ")

ω] A8# 8/ISs9)5>?;9 5#&3F/D;,7#892,8Y3Fy Æ-# 78/b52

MΩ2ρ+#T38-IF'pSA;-# 7/;52 P638/2G+/;/7,8Y3F

Mω2ρ]Æ#8/5=;

Ω [9 #87A# /

ω]"38/5/#ESA;9 9b2># +# M²0ST\']SA;Z#8/Z#8555=XD# )3Q-ð& /D)# 9d]E)y/a(+3Y9 ;-I;]-I#8È?;/7c;E+D B9;I$%52><-&3Y0 78*8# 8/#89U,XD/3F MÃ/C ST8È]8/9;>cST# È?9 >W#8/#8-I8/ 3* /D)# 9)Q# EF1#8-" /'5Æ]Æ# /5$"-# 78/b52&8

ñ

y# /+# -"/)3F;%>E J78\/56B?>E[52 -I/;/9 P# Y# -Iλ

MP /D)# 9STRSA 9 9°RSA 9 9°B+*SA;(/=#8

Vtrap =1

2Mω2ρ2

(

1 + λρ2

a2osc

)

.É ë M Ô'ú Ê

Q3F 9 9 # 89 /78c)A52FG/5c#8BF,M1º=º Éd¡X¤¢i¡¸´µÌ;/D#34;/C / >2%-E8y#-I 37D#8*)"((LdY# /785°]U# /+5^b" &+D B9;=(5>Z\8>$5;9 2>2%-R+#E3#8/<B3F/()52'5ËÊµ¾TÌÍÏÎPÐ638~6Ñ8MI!$83 9 >8]j8/"3# /$)52/Db,>Z#WXD# /Db%>cÈ?/SA/<#6~Íº6ÒÁ;68Ó?]

a]ÆSA 3YFy"9 /7 C3# 9 E,[ /DY#8-I)3E /Y#83F 8/°M*/°]Æb

n ["5/b%>]j>?(- # )5 CB+=ST#8ÈD9 >0;/DY#834;/7<;[c/?-&B8[#8( 39;'6'(/I /#c\89;-"ISA /D#34 8/+QSs9 5CB38-I;-I8#8/*)E\>Z(-#89;9d]°#E)E;

na3 1? ë @dM®0ST\\8/6SA;" J3F8/+52b;/",9bG+9;9 5Æ]ej3FP8A;/D#34;/3#8/EBT\>Q;-I8# /D']-I#8;/9;>Q78&B'# È? /7T578/Y#83>Q s+# -"/)33F 9 9 # 8T /D)# 9dM";/D#34;/$ R-I(['#8;9 >W# 5ZB?>W-I'# /R T# /ÔºÕ¾~eÑÖA×o638~6ºØ|Í]SA)3Y 78/'PR-# 9 9bL3# 9 y# /+5(8(L_ST#ª\9 /78IB+#ª\?;J8ÆQ3# /7M4y52 9;6# -")3E>2%-]#3F/D#83Fy /Y#83F 8/c [\8>W#833FY#ÉËEM 7+MÐAF%M? Ô @`Êe7

vint (r) =4πh2a

Mδ (r) = U0δ (r) ,

É ë M Ô ïÊSA

a <%L_ST#ª\c3# (;/7^9 /7 É,,8Ì9;SL_/7>H(>2(-Ia8/9 > (L_ST#ª\3# /7"SA;9 9N3F/D;BR"*;/D#34;/+ÊT# /5

M)TE#-I 3R-#8MÃ/[1? -"/D# 9+'# 9 ;:'#;/"8NÇTÄ ô /R# 9 È# 9 +7D#8]8;s#pB/a,8/+5+#yB Z( 78/$#8/5c-# 7/b52E8&3# (;/7a9;/7 °]+#8/5c/+3FEE6'34 \8 /Y#83F 8/Z(/78°]U3# /<B+"/5Z78Z L3# 9 9 5P4DB+#83YZ8/+# /3M)N)U8/T 2sB+#8 3J-"8 \# 8/+°,N (52>]SApT;/D#34;/&%/7 )8/Q *# Y# -IFYs# SA 9 9°B+*\#8 5Ì;/W*;/?\( 7#;/a8ÇTÄ ô M438/?\8/ /38]?;/a)T STR#-I#*X+# /D-V N/78># 9 8/7

zL_# 1?)]

∆Ez][ Ì-&+3Yl9 #87a#8/n0(/78l8*C;/DY#834;/°M #8-I2 8/<;-I9 ;'R# QI-I ;/<8JI39;5Z#89;/7Ì)*#12)[)R,8:/Z2M?]A9 \#8/p52 -I/ 8/9 # Y# -I)

σa]8SA

σ = N/Z)52/;%> E7#+y/;y9 /78$#89;/7"&>?-I-">Ì# 1?)]2#8È8/WB+*zL_# 12 M

σa3#8/aBQ9 # 5nU0

;U8/R387/;:pQË#34T# TQ#52)# 9jF1?/;/a U#8-I)3R39;5Ì;/WÉ,#8-I8/ 3R3F 9 9 # 84ÊJ788/5Ì%Y#* ∼ aoscMOCQG/5

σa =nU0

4ωh.

É ë M Ô ñ8Ê1º=ºÙ ÚÛ´¡¸¤B¡ ´µ /Q3F9;)# b%>a PÇsD(LdÄJ;/(;/c3F/52/# 's) A/?# 9Æ+/[8# L 8/N]2SA 3YcSA 9;9NB+E52)3F('5a /WR,9;9 SA;/7? Ô @_M

ö

E3F-I9;1Ì3F/52/# QS#ª\8[,/34;/Ψ

3#8/W#89;S#ª>2TB+*1?'5Ì#8Ψ (r, t) =

√

n (r, t)eiθ(r,t).É ë M Ô öDÊ

ì /?>*-I ;/&)'%)345*B?>QJË#34# U\89;23F;%>Q Us7#52 /DU8A3#89 #8#(θ

]v =

h

M∇θ. É ë M Ôþ Ê

)A -"9; #/9;'A*+#8*#8A#;/789)# b%>]DE\9 23F;%>G9)5W ;8# L 8/+# 9d]2²M 8M∇× v = 0.

É ë M ë8ÿ Ê48-ð6;/79;Ld\#89;'52/'A8"3F8/+52/#ES#ª\8Q,/3F 8/$;y,9;9 SA#[\# # 8/

∆θ8U*#(*# 8/5a#I3F9 5W3F/D8A-&(AB+E#"-&9b;9 *

2π7

∆θ =

∮

∇θ · dl = 2πm,É ë M ëÔ Ê

SAm

*#8/< /D78'M6?Q;Q,89 9;S[#*3F 39 # 8/Γ

#8/5<#W39;D('53F/D8T)78 \8/ÌB?>Γ =

∮

v · dl =h

M2πm =

h

Mm.

É ë M ë8ë ÊÃ_j#8A;/79)# b;';/"\89;23F;%>*G9 5°]

Γ3#8/IB+/8/2L_:+]SA 3Y38L+/5Q\8F1&(# 'MBÜP8)3F'P# 9)(Q# #8;/3F9)#8 3# 92)5P9 ;È8],PF1# -I9 8]?)3#8/]B2/R9 /Q;/D78Y# 9 °Q\89;23F;%>"G9)5W# 8/5a#639;D('5a3F\/+3F9 ;/7c#W\8(F1Z52?E/ *#ª\6cB+XD#8/;:5ÆM ì 33 #89P+;/D*)Q# *52/(;%>0#*Ì3F/D& #=\8(F1Z\#8/ 'McÃ/^8Eej3F \89 ><%Ss8L_52 -I/ 8/#898B9 -c]&# 12 y P8# 8/cSA;9 9U#89;S#ª>2B+&zL#12)]#89;/7SA)3Yc6527 U,52- ),:/Ì2MO<a#ª\8/C#*#8-I8/ 3"3F 9 9 # 8QS#ª\8",/3F 8/

Φm# I88L 8/+# 9P

ρmeimφe−ρ2/2 É, / #8-I 3I/bYÊFMPS 9;z = x + iy = ρeiφ M=/

STR+#ª\8Φm ∝ zme−|z|2/2 ]SA 3YW y#"38-I9 F1,/34;/Ì#+#8

m? /W3F-I9;19 #8/8MÃ_U#89;9+'([?8T# y( /79;]STy#ª\y# /Ì# #ª>6 °9 ?3#;/JSAEST#ª\R,/34;/°]j# /5W/3F*&52/b%>]\# /)]dM MT#8/$# #ª>8P\88)3F'MsÃ_

# /?>Ì? )[-&9; 9 8]by7;\A)(*a#I\8(F1WSA 3Y=)-69; 9;>=XD#8/;:5ÆMA\# # 8/C T#8# 8/+5<#c;/79;>CXD# /D :'5Z\81ZXD#89 2π

M ì 8/+5<#-&9b;9 >aXD#8/;:5a\881 (/7 m

]?; m2π

M Í IB?>9;?8È? /76# *(34L*8ES#ª\8R,/3F 8/$8#a(>2(-c]+SsE3# /c78y;/,8-I# 8/=# B8ybY\8F1(34ME Y# 9# /79 #8-I8-I/D-Ü p#a>2%-vXD# 9)A6-Ü P& /52 \? 5# 9-I8-I/DY# s"#8( 39;']ÆSA)3Y<#8" /7Y;-IRÅp9)# /+3YÈi/ Q3F/%Y# /Dh

]Lh =

∑

i hmiMOCASA;9 92SsÈ* /"/;P8

h]D# /56;/6s8#89?#8/789 #8-I-"/D- )

LMÃ/I8Y52&3F-I# A(>2(-8Æ5½j/DJ;:8]8STASA 9;9# 9)(Q9 DÈ6# Py#8/789 #8-I8-I/D-ò+s#8( 39;]

l = L/NMÃ_s/s82T+#J)uÝ|Te±¥|Ap+#ª\8A&B[# / /7T,8*\# 9 8

L+#-I;/;-I :Q*/7> *7D#8MSA 9;9B[#&3;)3#89,XD/3F>I#TSA 3Y#8/789 #8p-I8-I/D- )JG(p#8/%L,5"&R#8-M ì 9;STP,XD/3F ][788/5(# SA 9 9By/8/2L_8# /7

þ

0 0

Inte

ract

ion

(rep

ulsi

ve)

Rotational frequency

Singlyquantizedvortexarray

VortexMultiple quant.

Singly quantizedarray with hole

4;7 ë M Ô 7 Í 3Y-# 3652 #878Y# -v(SA /7"#8E,[#a9)( \89;>W;/DY#834;/7Éσa > 0

Ê"ÇTFL_Äp /( / 38/5/#Ì#8I#Z,/3F 8/ y /DY#83F 8/ (/7 # /5 # 8/#89,'X/3>8M

SA;$\# / /7a# /789)# A-I8-I/-cM4[#9;>W#8-I8/ 3R#8°]& /D25234L 8/Z J# /789)# [-"-I/D-ÜSA 9;99 #85=,-# 8/y J;/79;>=XD#8/;:5=\8)3F' /<#Ì9)( \89;>c /Y#83F /7a7D#8ME!=9; 9;><X+# /D :'5$\8( 3[# &/8QB\85°];/3E>=# &/78)3#89;9 >Ì/2Ë#ª\8#8B9 8M48Q#(#34;\E;/DY#834;/]"3;(L)3# 9,XD/3F>$,8*\8(F1=,-#;/<)*9)# 78y# /CI,'X/3>=,8*SA)3YC3F/52/# *SA 9;9X2>c# #8(']+# /+5c\88)3F'SA;9 9/\BE8B+(\5W#[#89;9dM?]# /2L#8-I8/ 3#8U78 \8J+NT88/b%>R['# 9 ;:s(# USA)3Y6#8p/ P3F8-I-I8/9;>/? îÍ@_MÐAF,/3F? Ô'ë @P#R(+52;'5$"#8I52 #878Y# -ð,8QB8/[SAb<9 \8& /2L#34;/P /a# /#8/# -I8/)3TY# ISA; /[ L3#89;9 5"-I#8/2L²G+9)5# ª12;-# 8/ÉË Í '3 ë M ÷Ê4MIÃ_Q/E82Q+#Q,8Q *Y# /7W+8/ #89²]U 78C/78C8# L 8/+# 9,'XD/3;'RSA;9 9P9;'#85ZI7D#8yW,8-#W-&9; 9 >ZXD# /D :'5$\8F1ÆM& )A#&#8( 39)# T,# [8NE# /+# -"/)3F;%>"8NQY# W# /5aQB#8È?;/7" U#8-I8/ 363F 9;9)#8Q527/#3F>M /<8R+# /5Æ]N,8RG125C# /789)# Q\89;23b%># /+5H(:3;/D9 >0(8/7$;/DL_# 8-I)3 /Y#83F 8/°]a7#ESA 9;9T,-.# / #8Y#ª>Z8ÉË /789 >ÌXD#8/;:5ÊT\8)3F'MpÃ/W y9; -IbyE /D#34 8/W/78>a52-I;/# 'A\8E/7>Ì#23F)#'5WSAb=6# /#8-I/ 3b%>MsÃ/=BF%ST/°]+E12 ([#%Y#SA;cB+8=;/79;>Ì# /5Ì-&9; 9 >ÌXD#8/;:5Ì\8( 3M"B9 -v s#8/# -I8/)3#89;9 >#85$# 8-[SAbZ#Y#83F \8E /DY#83F 8/#IB+/n%52 5 / ÐA%MÞ? ú @dMnÃ_/I82#;Qc /Y#83F 8/#8WSs'# È/87°]PW9;I# W\8>^( -I;9)# "<#I,"9 8/I;/3Ì; "=# /2L#8-I8/ 3b%>üÉ²# /5 Y#;/+Ê6 Q=B9;- #Ì58-I;/+#'M«4\8> (8/7 /Y#83F 8/]b" 6;/Db;\9 >H3F9 #8E+#&c3F8/+52/#SA 9;93F9;9)# a#8/50,8-3F9 (*E-I89 39;'Mc)Q/52/3F>C EZ8-I1D/D&B#89 #8/3F'5CBD>ZÌ:8L8 /[-"8 8/=8&# 8-]+#8/5c\/c,y+D('5c#8A&>2%-v y8#5ÆM"# 9)(ÌF12 (Q#8/ [,/5#8-I/D#89;9 >$52;/*(#68s3F/DL_ `L_-I#y8# L 8/N]JSAa=3F/52/# W8# "#8I#<SA89 8]JSAb82#Í'34 /7<c9)#;\b;/Ì8U*;/+52;\?)52#89°#-u? ö]+ñß@_M3Y# /78R /<\81=,8-I# 8/C#8R8/I7D'[,-#Ì9;>Z# -I8/)3E#8

Ôÿ

Inte

ract

ion

stre

ngth

Rotational frequency

Unstable Center of mass

Vortex

4;7 ë M ë 7 Í 3Y-#)352)# 78Y# - SA /7[A+#8s,p# /I#Y#834;\9 >*;/DY#834;/7Éσa < 0

Ê"ÇTFL_Äp /( / 38/5/#Ì#8I#Z,/3F 8/ y /DY#83F 8/ (/7 # /5 # 8/#89,'X/3>8M

^#09; 78D9 > #8/# -I8/)3Ì+8/D #893# / B=/52(?25H,8- $# c Rej3F \8Q /D)# 9V e = V −MΩ2ρ2/2,

É ë M ë ÷ÊSA)3Yl#Ì#y%-IF12)3#8/l#L_#8+ezZ;/üC9 # (c3#88]y-I#8È?;/7 bc/78F 3# 9 9;>/2Ë#ª\8Y# B9;y,8# /?>+# )3F9 TI 5Q# A* 78 /°MNQ#ª\8 5ESsÈ? /7ySA;6 -IFL52/5/DP\ª#8)# B9;']'>2%-)%52 56;/6 # /7Q,Y# -IA jY# NMy788/5I(# )P/,/5ISA//7>

εrot)-" / -I;:5Ì;/WQ>2%-Á/ '%,Y# -I8]D+#)TQ # /76,#8-"Q8UF,/+3F8Mì Ss9;9;L_ÈD/SA/&9bP7;\U/78>*8A8#;/7R>2%- #8

εrot = εlab−L ·Ω ]R,8A12#8-I9;O? þ @_MQOR3à ág]=_\Yâl¸ãÁ]=[enäabq5]=j\`båH_\Ynä]çæâ_\carn5We_ÃZ\j\Y_\[eWeèç_uabWejÃY"X+# /D-v-I3Y#8/)3# 9U8B9 -v J389)5$#8- /Z# Y# /7a#8/#8-I8/ 3Q L/D)# 9Æ3#8// TB+*89 \8'5F1#83F9 >",8A#6>2%-ò U-"[+# /%ST&+# )3F9 M# ST\"(\8Y# 9T# ª12 -I# 8/0-IF256# I3# /HBÌ+('5Æ]J# /+50W8B2L9 -V# # /5Ì#\?;(9 >B/=(5;'5Ì+( /7#"-"'# /2LdG9 5W# D#83Y°]D\# 9 5Ì,89)# 78A/?-&BYp8°# )3F9 MÃ_s)p `/a38-6B;/5SA;\#8)#;/# 923Y/ XD'J# /+5B#;/a>MÃ/"'(/p(5>8] # D#83Y6 #p81#834p52)# 78/#89; :# 8/°MNÃ_J(9)5BR8 /D'5Ì2# A*/+# -I*3F89 5ÌB+*-I)(9 #52 /7]2;/3RQ3Y/)XDQ F1#83F8/9;><;/0>MÌÃ/^Y#83F)3F]38-I2Y# 8/+# 9p9 ;-I;# 8/E,8Y3FE$( 3F*0 9 B+(P(+#83F]# /5Q,(-I8pSss#ª\8pR#52 3# 9 9;>R9 ;-I;Us/?-6B+U8# )3F9 # /+5*\# 9 8#8/789)# -I8-I/Y#I3F8/+()525°MÇT8/E#8+# )3F9 Q#&+#ª\8 /7& /°]P# /5C &S#ª\8",/3F 8/H)&#89bLS#ª>?[(>?-I-IF)3E/+52R+-62# 8/< P%STa#8( 39;'MRÃ_p6S#ª\8*,/34;/Z J#>?(- /8/Ld /DY#83F /7"B+D(/ SA;(/c;/Ì-A8NE( /789 FL_#8( 39;RST#ª\,/3F 8/]P#86#=(>?-I-IF :502523F] &/$9 ;-I;# 8/0Za/D-&B&8

Ô8Ô

#8( 39;'[233F?>?;/7Ì'#83Y0X+# /D-@-"'3Y# / 3# 9P( /79;Ld#8( 39;I(#8MO°φk (x)52/ 6È¥7 ^( /789 FL_#8( 39;"ST#ª\&,/34;/°]#8/5

Nk/?-6B+* # )3F9 233?>D /7(#

φkMTÃ_* #89°/?-&By P# )3F9 )

N]2&/8-# 9 ;:5WST#ª\,/3F 8/c U*/8/2L_;/DY#834;/7">2%-V)

ψ (r1, ..., rN ) =1√

N !√

∏∞k=1 Nk!

∑

P [φ1 (r1) ...φN (rN )] ,É ë M ë î?Ê

SAP

R-&2Y# 8/C8Y#8*# /+5Z-@)*\8*# 9 9P+D;B9;-&2L# 8/Q8A# )3F9 O? Ôÿ @_MaÃ/Cé423YÈ</ Y# 8/N]N8/"SA; |ψ〉 = |N1N2N3...〉SATNk

]D#8p#8B+\8] # Ty?33F+# 8/"/?-&BYP8+( /79;Ld#8( 39;A(# SA;c;/+52F1k? Ô8Ô @_Mp*;/79;Ld+# )3F9 [233# 8/W/D-&BA P#I3F#8;/W8B;#89)F \8'5B?>Y# È?;/76E3# 9)# 2523F N#BbY# 9°# /+5\423YÈa(# 87

〈0...0 1 0... | N1...Ni−1Ni Ni+1...〉 = Ni.É ë M ë8ú Ê

[5FG/; 8/8j[B+D(/I#8// 9 # 8/# /+53F#;/I8Y#)T#8,9;9 S67

ck |..., Nk, ..〉 ≡√

Nk |..., (Nk − 1) , ...〉c†k |..., Nk, ...〉 ≡

√

Nk + 1 |..., (Nk + 1) , ...〉 , É ë M ë ïÊ*#

c†k3F# J#E#8( 39;[;/a(#

φk]?#8/5

ck# // 9 # 'J#E+# )3F9 ,8-ò%Y#

φkMPÃ_NR /"#8( 39;R;/Ì A(# [# //; 9)#]DQ9b;/76S#ª\8,/34;/):Á? Ô'ÿ @_M

1ºÙ= êë¤1¹TÈmX¯§¡¸´¸Ã/[-I'# /2LdG9)5#8ª1? -#;/°] [+# )3F9 FL_# )3F9 A;/DY#834;/)J9)#835IB?>p;/DY#834;/Q ?8/p+# )3F9 PSAbQs-"'# /RG9 5Q25235QB?>[#89;9 °#8( 39;']SA)3Y$ J38Y&52/+5y/=6#8( 39;652/;%>8MRÃ/=68>W JÇTÄ ô ])[ # 9)"È?/SA/c#8T&ÎR%L_Åp;# \2(È? j#8ª12;-#;/°MÃ_y523F B+'E38/5/# /^-6 y# / \8Y# 9 9p52E#8#8-"] /%#5^ 1?9;)3F;9 >^%Y# /7Za Y# 9S#ª\8F,/34;/SA)3Ya)A52/5/DA8/Q9;23# 8/a U'#83YÌ;/5;\?)52# 9j#8( 39;MÃ/#a,9;9 >$3F/52/('5Z(# 8]°#89;9#8( 39;'ySA 9 9P?33F?>WI# -II( /79;Ld#8( 39;6%Y#φ

]# /+5*-# /?>DLdB252>S#ª\8[,/34;/=3# /WB*SA;(/c;/W-I8U)A#8

Φ (r1, r2, ..., rN ) =

N∏

i=1

φ (ri) .É ë M ë ñ8Ê

Q8Y52A# Y# -IF)T/c5FG/'5c#8Ψ (r) =

√Nφ (r) ,

É ë M ë öÊ# /+5É`;-IFL_ /52/+52/DYÊyÎRD(LdÅJbY# \2È?;jXD# 8/W'#85m? Ô @

− h2

2M5 2Ψ (r) + V (r) Ψ (r) + U0 |Ψ (r)|2 Ψ (r) = µΨ (r) .

É ë M ë8þ Ê8Q#y;y)//2L_9; /#8MsEXD# 8/=#8-IT&(>2(- aB+E /c*(L-I252>?/# -I)3[9 -"; 9)# 78

NM

Ô'ë

1ºÙ=º ì1X¢¥´µ±§íç¦¤µ¡î.¤b¡´µï¤±§.ð=¤¢i¡Ô§¤Bµ´µ²¤¯î.¤B¡¸´µ*XD# /D-V-I3Y+# /)3#89;7/?\# 9 Q8B9 - 9 ?8È2( -I9 E#y#6GY%A789)# /+3F87

H |Ψ〉 = E |Ψ〉 . É ë M ÷ ÿ ÊS8s3FY(EST&È?/S #*52+/52 /7Ì/$0[# -I 9b8/)# /$ /?\89;\5Æ]+)Q3#8/BE#I\8>a38-I9 3#'5WB9 -cMpÃ/WGY%yXD#8/;:'#;/°]2\0y# -I 9b8/)# /=38/ ( #I( /789 FL_#8( 39;u0y#8-" 9;8/ #8/

h (r)# /5W# /c /DY#83F 8/Ì#

vint (ri − rj)]

H =

N∑

i=1

h (ri) +

N∑

i<j

vint (ri − rj) .É ë M ÷ Ô Ê

Ã/c('3F8/+5ÌXD#8/;:'#;/Ì;y3#8/ÌB*SA;(/=#8u? Ôÿ @

H =

∞∑

k,l=1

〈k |h| l〉 c†k cl +1

2

∞∑

k,l,m,n=1

〈kl |vint|mn〉 c†k c†l cmcn,

É ë M ÷ ë Ês/ Y# 8/ |k〉 8N,

φk] /&8UST8Y5°( /79;Ld#8( 39;JST#ª\P,/3F 8/-I/D 8/5W# B\88M°6-IÌ(*#ISsa/S#ª\8IÌ5½j/Duy(9 \89 z AS#ª\8,/+34 8/+M4;Y(&Ssa#ª\8

φi]SA 3Y #8IÌ /789 FL_# )3F9 (9; 8/E=Ì;/789 FL_# )3F9 8B9 -cMPQ /52F1

i/DJ*9;\ª#8/DAXD#8/D- /?-6B+MO<Q/W#ª\ L3#89;9 5Ã423YÈE%Y#ψ

78 \8/6B?>*ÄJXMÉ ë M ë î?Ê4]SA)3Y6#8T?534YU8A( /789 FL#8( 39;y(# 'MPm423YÈ"(# 'P,8-#&38-I9 FyB#8 s /m0 9;BA(#3F]# /+5F,86I,9 9s89 2;/Ψ

T-# /?>DL_B+252>$>2%-@8s /DY#83F /7c# )3F9 3#8/ÌB*F12#8/52'5a /W- '(*(# A#|Ψ〉 =

∑

j

cj |ψj〉 .É ë M ÷8÷DÊ

NÌ(9;\*6 78/D\#89;*B9;-c]+/E&Ë#834[# yv4?3YÈc(#[# -Y9 >8878/# 9d]?# /5W-&9b;9 AÄJXMÉ ë M ÷ ÿ ÊJ,8-*9 F`SA; 〈ψi|I78

〈ψi |H |Ψ〉 = 〈ψi |E|Ψ〉 . É ë M ÷ îDÊÄP12#8/52 /7"*S#ª\8[,/3F 8/

Ψ /W\4?3YÈÌ%Y#'SsQG+/5

〈ψi |H |∑

k

ck ψk〉 = E 〈ψi|∑

k

ck ψk〉 ,É ë M ÷ ú Ê

# /+5a#8Y# /7;/7&# /5W1?9;b;/7"*878/# 9 b%>I8N\423YÈÌ(# 7;\∑

k

ck 〈ψi |H |ψk〉 = Eci.É ë M ÷8ïDÊ

Ã/H7/#89PÌSA;9 9TB+W(\8Y# 9J 78/,/34;/"# /50;7/?\# 9 E# "89 \8ÄJXMÉ ë M ÷ ÿ ÊFMNÃ_Æ8/T,9;9 SA23F'52A# B\8T,8p#89;9ψi

/A78P#Q-#;1& 78/?\ª#89;XD# 8/Hc = Ec,

É ë M ÷?ñ Ê

Ô ÷

SA

H =

〈ψ1 |H |ψ1〉〈ψ1 |H |ψ2〉 · · ·〈ψ2 |H |ψ1〉〈ψ2 |H |ψ2〉MMM M M M

.

É ë M ÷8öDÊ

N"/Sü89 \8[RB9;-c]?#89;9j/[+#8sI52I sG/+5Q-I# b19;-I/D# /+552)# 78/#89; :H

MESA 9;9U p3FQB+"(\8#89Æ 78/?\348Yc

]38/)%;/78233#;/Z/?-6B+R sâ423YÈ$%Y#']U3F8+/52 /7=\Y# 9\ª#89;Q s/78>E

MOnbZ#3F-I9;aÉË;/G/;'ÊB#8)A;/ψ

]+#8/5c;/G/;6-A /=ÄJXM*É ë M ÷ ë Ê4]/ÄJXM&É ë M ÷Dñ ÊASs9 5cB&1#834'M)]ST\8]Ss9 5c8p3F8EB&/?-I 3# 9 9;>Ì;-"L( B9 8]?# /5a8/[#8J6/3#M0Sn8/[52?J 52+/5T8/R# )3F9)# 8B9 -ð#A# /+5aLp /?\8'%;7D# 8/W F13F;# 8/T,AF1# -I9 Q'XD;B+#8 A(# +#Q#ª\8&#a 78[/78>M /"# 9)(Ì'%)34Y"/D-&BR P-R /3F9 52'5$ /E(- /cÄJXMÉ ë M ÷ ë Ê±? ÔÔ @dMñ ¾Ëâ2ã²¼+» ¼iAè¿=ÂH2ãÓò ¿?¾²¿DÂ ¿D»j'ÄJ#3YÌ9;-"/D;/W*-I# b1CÉ ë M ÷8öDÊp)

〈ψα |H |ψβ〉 = 〈ψα|

∞∑

k,l=1

〈k |h| l〉 c†k cl +1

2

∞∑

k,l,m,n=1

〈kl |vint|mn〉 c†k c†l cncm

|ψβ〉 ,É ë M ÷ þ Ê# /+5 |ψα〉 = |N1N2N3...〉

#8aÁ423YÈ^(#MCÌ3F# 8/H#8/5 #8// ;9)#;/8Y#8Y&3Y#8/78aW233#;/H/?-&BYNi

423YÈ^%Y# |ψβ〉]JZ#SA/m0y#8-" 9;8/ #8/$É ë M ÷ ë ÊP#83FJ8/;Jy9;J T#E3#89 #8P -IT# /8d423YÈ(#8MAE(878/#89;;%>8v423YÈÌ(#y/#R/9

H |ψβ〉 ∝ |ψα〉]ÄJXM<É ë M ÷ þ Êy'XD# 9)*:+MÃ_E)E3F9 #8Q# E-# b1C;/^ÄJXMCÉ ë M ÷8öDÊ[SA;9 9JB\8># Y(MsÉ²A3# /"By(-I38/2,+( 8/6*SA# P p3# 9 9;'5I#1y(-#;1&9 -I/zY]( /3FE# R9;-I/DA /W&-I# b1Ì ÄJXM[É ë M ÷öÊF]2B2[38-I-I8/9 >Q-ð A+('5IF,"EXD#8/D; 〈k |h| l〉 #8/5 〈kl |vint|mn〉

Ê4M9 #% /7#89 U;/610[# -I 9b8/)# /" ÄJXMTÉ ë M ÷ ë ÊU# 3# 9)# /D-&BYN#8/9;>6/'5"EBy3#89 39)#'5"8/3A,8p'#83YFJ8jXD#8/- /D-&BYM Í /3Fh

p;/79;Ld+# )3F9 ²0y# -I 9b8/)# /N]D#8/5k, l

#8ybYJ-62#89;9 >"8788/+# 92 78/2,/3F 8/]; A39;'# +#

〈k |h| l〉 = εlδk,l,É ë M î ÿ Ê

SAεl

#8Q( /79;Ld#8( 39;y 78//7;']D(*#T[3F/D;B 8/,8- -ð y52 #878/# 9dMJE /D78Y# 9) E /D#34 8/W/78>8]+(*ÄJXM ë M Ô ï]# E52"E529;#",/34;/Ì#8/(,8-"'5 /D〈kl |vint|mn〉 =

∫ ∫

Φ∗k (r) Φ∗

l (r′) δ (r − r′) Φm (r) Φn (r′) drdr′

=

∫

Φ∗k (r) Φ∗

l (r) Φm (r) Φn (r) dr.É ë M î Ô Ê

Ô î

ÐA9 #3F /7Φ∗

i (r)SA;*Æ%ST L52 -"/;/# 9',/3F 8/°,- Í '3 M ë M Ô ]

Rmi(ρ) eimiφ

]SARmi

(ρ)#868R>8R/È?/SA/°]ÆY#852)# 9# /+# -"/)3*ST#ª\F,/+34 8/+]6 /2L78Y# 9jB3F-I

∫

R∗mi

(ρ) e−imiφR∗mj

(ρ) e−imjφU0Rmk(ρ) eimkφRml

(ρ) eimlφρdρdφ =

= 2πU0δmi+mj ,mk+ml

∫

R∗mi

(ρ)R∗mj

(ρ)Rmk(ρ)Rml

(ρ) ρdρ .É ë M î ë Ê

QORó ô¥pI^Á[e]²pJ]=Yvab_uaBWej\YOCRS# /Ds%52>IQ/7>9;\89)s #">?(- NST#8È?9;> /DY#83F /7]?: L( /B8/#P:y-I+# 8](BK%34Q#R Y# /7+]Ss'# È?9;>E#8/#8-I8/ 3J+8/D #89²M4;Y(Ss=F P0y#8-" 9;8/ #8/ /n('3F/5 X+# /D :# 8/ #8338Y52 /7ZlÉ ë M ÷ ë Ê4MOC&E RB3# 8]j /Z3F-"2[87#8-I-" /7+]63F# 8/$#8/5Z#8// ;9)#;/8Y#8Y°# J-63YE'#8;°[# /+529;s+# /*s-69; 9;T /7#89 °# UST89)5*9b;/*SsR(QQGY%yXD#8/D :#;/0[# -I 9b8/)# /°MsE('3F/5ÌXD#8/D :#;/0y#8-I;9;/ #8/Z3F/D# /y" 78//78 8PI;/79;Ld+# )3F9 E8B9 -c]Æ# /5$ /7#89 〈kl |vint|mn〉

52\89 85H;/ W\?;"('34 8/ ;/?\89;\? /7=W /789 FL#8( 39;ÌST#ª\,/3F 8/" Ac# /#8-I/ 3a+8/ #89²M^'(W# a7;\/^B?>0Í 3Yx252 /78T'XD#;/a,#" /789 Q# )3F9 8][

− h2

2M52 +

1

2Mω2r2

(

1 + λr2

a2osc

)]

Ψ (ρ, φ) = εΨ (ρ, φ) .É ë M î÷DÊ

'X+# 8/0#R$B89 \85</?-I 3# 9 9;>]NSA)3Y0 6528/( /7Wé-I\-IF25Æ] ì /+52b1cÇRMT-I# b1* /EÄJXMÉ ë M ÷8öDÊÆ)UF*[ /38+# s6'34 J /DY#83F 8/8/W*B+#ª\?;A E(>2(-WM A/1DR%c AG/+5W#;#8B9;EF[ B#8 ,/3F 8/MÃ/Ì5p6-I / -" :R/+3#;/a8Y]b)s -I+(Y# /DT"(Q#6B#8 FESAD(9;-"/D&#8#E( -I;9)# 6#8*( B9 I=1?3F'5C89 2;/MÁ423YÈ(#ySA*ni

#8*233F# 8/=/?-6B+,86#8/#8-I8/ 3*;/789 FL_# )3F9 S#ª\8[,/34;/-ISAbÌE3F;)#M4['#83Y= #89#8/789 #8y-I8-I/D-L

]jST&#ª\*c([C#/S B#()ySA# 9 9j%Y#'p-#Y3Y T3F/52; 8/°M4(-I88]D8/9 >"(# 'JSA;

n = 0# /5

m ≥ 0SA 9;9BA /3F9 52'5ÆMPA-I \#;/6,)J3# /"B+[(/6,- 9 DÈ?;/7*# PA/78> E# -I8/)3Q3F 9 9 # 8EÉ,,-VSA)3YcESs'# È?9 >a# /+# -"/)3Q3F 9;9)#8A)A/ ?$52;j/DYÊe76 /^/8/Ld /DY#83F /7=3#(]N8#89J/78>Z)QÌ- /52 \D)52#89N#8( 39;*/78 ]E =

∑

i (|mi| + 2ni + 1) hω]# /5W*8#89U# /789)# -I8-I/D-ð)

L =∑

m hmNm]+SA

Nm AE/?-6B+y8#8( 39;'A /$#%Y#SA; #8/789 #8E-I8-I/D-

mM¬4"#Z78 \8/

L]W/78>C I3F9 #89 >C9;STE;[#89;9#8( 39;'J#ª\

n = 0M®4(-I88]?( /3F[[/78>52+/5T8/R-"2529;T86# /789)# [-"-I/D-c]Æ# 7D# /c,8Q#7;\/

L"/7>Ì)y9 STy;s#89;9# )3F9 +#y3F/D;BR

L#ª\R; \8É,A:DÊT#8/789)# -I8-I/-

mMp)A)9 Ss'%vU# /5# ^9 \89T#8ª12;-#;/°]NSAÌ#8# 8/^BF%ST/HW( /789 FL#8( 39;[/7>"9 \9)T T-&+3Ya78# p+# /ÌR;/DY#834;/a(/78°]nU0 hω

]8σa 1

M

Ô'ú

ì y+;/D]&5½:3F9b%>W &B9 - )y;-I9 >a78Y# -I-I;/7+M4A52)# 78/#89; :# 8/;9;%]# / ì ÐAÅ ìyô Ï 2 /Q;/7×U# /3:s-IF25Ì)T5ÆMp52)# 78/#89; :# 8/c Pv0[# -I;9;/)# /$-#;1c;/<ÄJXM6É ë M ÷öÊ78 \8[#/?-&BR J L9 2 8/]p# 9 9J /?\89;\? /7Z52;/D6233#;/^/?-&BY&8AW( /79;Ld#8( 39;aB#8 ,/3F 8/]?# /5ISA;52;j/DJ 78//78 M ì /S -I# b1"SA;9 9#ª\8T&B+R52 #8788L/#89; :'5*,8P'#83YL

#8/56(/7 6 ;/D#34;/°] # /5&\# >D /7['(#8#8-IFUSA 9;978 \8"QI88/b%>$$52Y#ªS52;5Z#(52)# 7#8-WM ô # 9)3F9 # 8/< -"F*#8/0E9 ;-I;*=SA# EÈ?;/5* (>2(-E3# /CB+(+52;'5Æ]# /5<'(/D(52>< E9; -Ib5<$>2%-Q 0N = 20

B8/*#8/5C#c-I# 12;-6-ó#8/789)# -I8-I/D- L = 41

MNE3# 9)3F9)#A/78>&;/I # /7R,#8-I8] ST(B2Y#83FLΩ

,8- Y# 9/78>8M")*52?Q/8Q+#ª\86cB+58/";/CI-I#8;/C87#8-vB2&3#8/<B528/#`ST#85]#8/5$52?[/ ['X;&#/S -IFL3F8/+(-I /787#8- /°My +#T /3FyR527 Æ,'528-ò /zL_52 34;/I J,:/82']D#89;9-I ;/ s /

ρφLd9 #8/8]#8/5WSsQ8/9;>/5Ì"ST8># B823# 9)# A#8/789)# -I8-I/Y#M

Ô ï

! õ

ö ÛÜÝN()÷JÙüÝ E ×)& &)+AÝN*ø(áÝnÝN+ÖØ×

*9;E#8I /F'5CBD><9;?ÈD /7$# */78><(3F-c];/789 FL_# )3F9 233#;/E/?-&BY ÄsX+MAÉ ë M ë8ú ÊN# /5&78Y# +N8+# /79)# -"-I/D-P+# )3F9 \8T # 8/#89,'X/3>8Mì %>?)3#89+/78>I+'34-c]D8# 9j/78>"\8P Y# 9j# /79 #8J-I8-I/-ò,8#"# )3F9 #8T>2%-c]2 y(SA/Ì;/Á4;7+Mp÷M Ô MP*(/78W NE# /# -I8/)3F;%>I

λ = 0.005]DSA 3YW TQ%>?)3# 9Æ\#89;*+('5Æ]2/9 A SA)(R F129 3b9;>a(# '5ÆMp /D#34 8/* /QpG7J N9)( \8]

σa = 1.5]ªB2U)U µi|A°#A%>D 3# 98\#89;]B2"78# 9 >ZF1#87878# '5$ /H8Y52EZ39;'# 9;><'(9;\6Ì(3F a+#852)# 7#8-WM Í /3Fv4U 7MA÷M Ô 5 9)#ª>?'(9;A,8[#a(-# 9 9U>2%-c]+#8/5WE-#12;-6-# /789)# p-I-"/D- 3# 9)3F9)#5)s38-I#8# \89 >6-# 9 9²]2#89;9+[ 78/D\#89;'p8°-#;1ÌÉ ë M ÷öÊ3#8/6B3#89 39)#'56,8P#3Y

LM ì 582;/7R-I /89 87>E /D+3F9 # ?>2()3] R9;ST((L_/7>&(#,s#E7;\/L

SA 9 9+BR3# 9 9;'5"vÎA39Ò=ÒAFMP\8(L# 9 929 Ss'%Ld/78>"%Y#y,8#E7;\/ # 8/#89,XD/3F>" 3# 9 9;'5"sÍ|Aù¸iÝÒÐÍÐMy/78 /G7#83# 9)3F9)#5I /I['%J,#8-"#8/5I788/5(#Q A/a"BRE(# R,8SA)3YL = 0

]2dMP8MQ/8/2L_ # /7I%Y#QSA8/9;>E;/79;Ld+# )3F9 R(# QSAbm = 0

)-I#3F3F 3# 9 9;>I233 5ÆMì pA#8I P Y#5Æ]8/78>& Æ#3Y"(# y3Y#8/78'MB48- Í 38M ë M ë M ÷] SsÈ?/SÞ# yEej3F[ PE Y# 8/c)#R#I-v88 8/#89Æ6#8/789 #8

-I8-I/D-L

# /5 Z Y# 8/+# 9,XD/+3F>Ω

]A+#8HB<B2Y#83F'5 ^#εrot = εlab −LΩ

ÉË;/Ì#-I 3A/bYÊFMp)p#Pyej3Fs8;9; /7&[SA89 78Y# /!4;7+MZ÷M Ô B?>C#Z3F#8;/H#8/789 a52+/52 /7Z/â\# 9 8Ω

]#8/5^;A)&-)9)# 78Q/78N]+#(#ESA;$#52;j/DA8# 9# /79)# A-I8-I/D-L

# /5=#I/S;/79;Ld+# )3F9 a3F/2G78# 8/^SA 9;9B3F-Ia9 Ss'%";/ /7>8M0a)6#C3F;)3# 9,XD/+3F>I,8ASA/W)TGY%A233Y]#8/5Ì# ),XD/+3F>IE(>2(-VSA;9 9°(b`#"/SÞ7/5a(# M4-I(y )8AEF13F;'5c%Y#',8y#83YL

SA 9;9U/ [B+&3# 9)3F9)#5Æ]c#ª\8I3# 9)3F9 # 8/$;-IM48Q-I8&3Y/)3#89'#88/]j/;*SA;9 9"/78 ,8T*%Y#'L = 0, 1

BQ3# 9)3F9)#5Æ]?BA) T/I9 ;-I;# 8/Ì /3FR>SA 9;9ÆB 9 \# /D#8A&>2%-v y(y;/D8#;/^ÉÈ-#;1Ì /=ÄJXMRÉ ë M ÷8öÊT) /c 3#(*#I3# 9)# ']2( /3[*)A8/9 >/QB#8 A(# * /W*1?+# /;/a /cÄJXMyÉ ë M ÷÷Ê4]# /+5 ì ÐAÅ ìyô Ï2 /Q5c52?/ y52)# 78/#89; :R#3# 9)# 4Ê4M

Ô ñ

0 5 10 15 2010

15

20

25

30

35N=5, σa= 1.5

total angular momentum L

tota

l ene

rgy

4;7<÷M Ô 7HZ/78>n(3F-Ø;/üC # /7^,Y# -I$,=# ÇTFL_Äp /% /3F/52/# ]λ = 0.005

]σa = 1

M ú ]# /5N = 5

Mc8#89s# /789)# *-I8-I/-L

/+R,- ÿ ëë M"I9;ST(Q(# ",8Q'#83YL

)R3# 9 9;'5<#8/ÔÎA9Ò\ÒA# /5$\8#89;9A9 Ss'%Ld/78>H(# $)a3# 9 9;'5 Í3|Íù¸iÝËÒÐÍÐM Ã/n8Y52H-# ÈW(34 c(3F-.-"Ì\? ;B9;]c%/7 H AW;/D#34;/^ /?# 9 9 >W9)# 78M*ÐA Y# 8/<#8y"6'34Q By% 9; /7Az&I78Y# N]j /3F6b* -"9; +#LΩ

AB2#34'5,8- */78>8U'#83Yc%Y#M

Ô ö

àâRTS ág]=_\Yâl¸ãÁ]=[enú`B]=Vbû5[¾aBVì T-I/D;/5Æ]?Q8B9 - # T#8/5+#8TB+/c%52 5ÌB+,88]?;/768T-I2L25I# / $F1#83Fa52)# 78/# 9 :#;/°Ml=9;I,9)( \8=#8/5 #(#34;\ /Y#83F 8/TSA 9;9°BE52)3('5Ì#8# 9;>MÙ1 ÚÛ¶m¦¯3´µ4p9);\T /D#34 8/+]D3F8-I#8)8/SA 9;9+-I#8;/9;>6B+-#52AE#&%52>&B?>Oü#3YÈDL8/°]PÏ*#ª\89)# È? &# /5ËN/52ý? Ôªë @d]PSAÌ-I#8/2LdG9)5^# D#83Y0 "5HSAb#=\# )# 8/+# 9PS#ª\8I,/3F 8/°]P#8/5CW# /+# -"/)3F;%><# /+50;/D#34;/E# #3Y# '5 +(B+# \89;>M $52a#8#8-IF)F12#8/52'5 ;/nZS#ª\8c,/34L 8/+68yc# -I8/)3aD3;9 9)#&,

m ≥ 0]n = 0

#8Ψ (ρ, φ) =

∑

m cmΦm]SA"3F?e:3F /DY

cm# "\ª#8)#;/# 9#8#8-IFQB2K%'34*c3F/52; 8/

∑∞m=0m |cm|2 = L/N

É²3F/(\# 8/ N#8/789 #8J-I8-I/D-Êp# /+5 ∑∞m=0 |cm|2 =

1É,/-# 9 :#;/+Ê4M Í ;/36B# 8/Z8>=8/9 >$#852-I;Q-I#89;9P38'34 8/+A$+8/ #89²]T a-IF?5 XD I$ /Y#83F 8/ # /5n#8/# -I8/)3F;%>^HB+ST# ÈjMnÃ/ '%;/7^=78Y# -.+#a3#89 39)#'I$# /#8-I/ 3c;/789 FL_# )3F9 S#ª\8T,/3F 8/]8;JS#8,/5"+#p,

λ ∼ 10−4 ] /?-I)3# 7-I/DPSAbI# /+# 9 > 3RB# \8RF12( 8/Ì;/cÄJXMÉ ë M Ô ÷ÊsS#8\8>7??5°MI#(I52)# 78Y# -v#*-I7[SA/<(/78< JI;/DY#834;/Z# /+5F1?/# 98#;/E# p\# ;'5Æ]#8U3# 9)3F9 # 5R,8-þ? Ô'ë @_])°# o4;7+MU÷M ë MN(9;)59 ;/'R'(/Dy#(6Y# /; 8/y# Q#8"52)38/D /?8[ /$6-I252>?/# -I)39 ;-I;]SA'#8sE52 5W9 ;/A-# È3F/D /D8T#8/(; 8/+MÃ_6/&2&#",8"ST# ÈC/87H;/DY#834;/°]W788/5^%Y#a)6#89;S#ª>2+#J8Æ#Q-69; 9;>XD#8/;:5"\8(F1I# /5"S#ª\8T,/3F 8/)P78 \8/B?>Ψ = Φm0

MÃ/ )3#8W;a)I$# /#8-I/ 3b%> R=8B9;- +#a52-" /# M ì I /Y#83F 8/0%/7 ^ * /3'#85Æ]USs\8]Æ/# 8s8B9 -.3Y#8/78'# /+5C9; /7=#(a E# & #=\8F1C# #ª>]NSAIa1?+# /;/0 8Y52N#8#8-"U /?\889 \8'°-I8p# /68/3F-I+//DMB48- ÐA%Md? Ô'ë @d]839;D(J[#(QB8/+5# >I*S#ª\8[,/3F 8/y# 9 ST#ª>2T;/?\89 \8[*;/79;Ld+# )3F9 R(# ]Ψ = cm1

Φm1+ cm0

Φm0+ cm2

Φm2.

ÉË÷M Ô Ê0] A;/+52 3s52/8y# /789)# P-I8-I/- 8j+# )3F9)# p;/789 FL_# )3F9 S#ª\8,/3F 8/°]# /5*STJ+#ª\8

m1 < m0 < m2# /5

m1+m2 = 2m0M)N+#8JSA 9;9B"3# 9 9;'5$#y(-Ib12'5z&#(]j /3F"b*3#8/$ /D\89 \8E\8)3F'#*# EB C-&9b;9 ># /+5Ì;/789 >ÌXD# /D;:5ÆMPÃ_

m1 = 0]2QSA;9 9°/9;>ÌB+E /789 >aXD# /D;:5W\8( 3T;/# #ª>M&Ã_

m1 > 0]°ISA;9 9pB#Ì-69; 9;>CXD# /D :'5Z\81Z#Q 78 /°]8/525aB?>a( /79;>aXD#8/D :5Ì\8( 3M4s\# 9 's8

σa9;STJ#8/aSAyQ52 '59 ;/'sG(T32TQ(9;)59 ;/T;/4;7+MJ÷M ë ]?E+#8[#8/b;/B+%Ss/W*-&9; 9 >WXD#8/;:5W%Y##8Q78 \8/B?>*F12( 8/CÉm

#-I5Ì+D(; \8ªÊΩm0

ω= 1 + λ (m0 + 1) − σa

(2m0 − 2)!

22m0−1 (m0 − 1)!m0!.

ÉË÷M ë Ê

Ôþ

1 1.005 1.01 1.015 1.020

0.05

0.1

0.15

0.2

Ω/ω

σa

m=0

m=1m=2

mixedm=(0,2,4)

mixedm=(0,3,6)

4;7"÷M ë 7yI#(I52)# 78Y# -Ü8s#a9 \89 >c;/DY#834;/7WÇTFL_Äp /( /<38/2L52/# ],- ÐAF%MN? Ô'ë @_MHc(9;)5^9 ;/"/D6+#8Y# /+(; 8/&+## 52)38/D /?8+Q;/CI-I252>?/# -I)3I9; -Ib']USA#8R528('5C9; /E'(/DY# /+(; 8/s#y#8*# 9 S#ª>?T3F/D /D8M

Ωm0

yRE9 ST(T,XD/3F>a,8ASA)3YW6%Y#Ψ = Φm0

(#8B9 8MJ'(Y# /+(; 8/#8*52)38/D /?8+ME#8/b;/y-#8È5WB?>ÌI52 5$9; /6#863F8/D;/?MR>=#8&3# 9)3F2L9)#'5=,8-#8/$/7>=3F8-I#8)8/$BF%ST/$6-69b;9;>ZXD#8/D :5Z\8(F1=%Y## /+5Ì#6)# 9N(# Q UQ,-ð;/cÄJXMÉË÷M Ô Ê4M48AF1#8-"9;]3F9 ["E38/D /?8B8/5# >SAm = 2

3F'#8sIB[*788/5a(# 8]?Q-Ib125W(#R# #8*9;ST(T/78>a Ψ = c0Φ0 + c2Φ2 + c4Φ4

MÙ1º ÿË¡¡o¤®¢i¡´µQ3#([ U# (Y#83F \8[;/D#34;/a#JB/W(5;'5aB?>Ï*#ª\89 #8È? ]üD#83YÈ28/# /+5Ç#ª>?-;/¬? ú @_]# /5Ì8A'(9;SA 9 9°#AGY%A#8/5WB+E38-I#8'5I YMÃ_+#8°7/#89;9 >yB/E,/5Q#UT# ,8;/D( /7#('° /*s+#852)# 7#8-.,8#Y#83F \8c;/DY#834;/M /=12+'34# ",ST# È?9 >^;/DY#834;/7#8( 39;']Ì9;"# a\>0 -" 9)# "SAF"Ì /DY#83F 8/H)"#(#34;\a89);\8]8;/+3FA/# j8B9 - s528-I /#5IB?>6y#8/#8-I8/ 3b%>E8QY# °MÃ/c AST8Y5]D/QF1234YQ\8( 3T -&9b;9 EXD# /D;:'# 8/NMÃ_T)# 9)(&;/Db;\9 >3F9 # J# s,T(8/7&/87a#Y#834;/°]R7#pSA 9;9Æ3F9bL9)# Py,8-ù39;+%Y°8U-I89 3F9;'M N/52/3F>Q)°y#y3# /*F1?/DUB#89 #8/3F'5B?>*:8Ld8 /D-I8 8/c/DA;/=#I#8-I8/ 3IÉ,#8-I8/ 3FLd9 (L_XD#8( 3'ÊP L/D)# 9d]#8/5=\/=,(RB# 9)# /+3F5cB?>W"3/Db,7# 9N,8Y3F*#Q# #8[#8y3F9 8+5a)A # '5ÆM

ë ÿ

1.01 1.02 1.03 1.04 1.05−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

Ω/ω

σa

c.o.m.

single+multiple

multiple

4;7÷M ÷7U#(52)# 7#8- 8Æ#8/#(#34;\9 >& /Y#83F /7EÇTFL_Äp;/+% /I38/2L52/# *#85c /$## -I8/)34L_9 XD#8( 3* /D)# 9USAbλ = 0.05

],-vÐA%M? ú @dMPU*y9 F`s8Æ[\8( 3# 99 ;/A[ J/E Y# 8/NM[# A#8J Æ-&9bL 9 >HXD# /D :'5^\8( 3E,8"\8>0Ss'# ÈC;/DY#834;/]p#$#(Ì y3/D(L_ `L_-#8 # 8/ ,8a(8/7I /Y#83F 8/]s#8/5 #0-I;125 #(c *;/79;> # /5 -&9b;9 >XD# /D;:5Ì\88)3F'MQ,8(W#8Q A#"#(*83/D(L_ `L_-#8s Y#;/°]252 3F+5ÌB?>aOn 9;È? /61Ís?ñT@ # /5c!=8(9 8/£? ö9@_MÃ/c)+#88]2E7#,8-#\y(B9;BzQ# y8# # 8/5Cc3F/D" yÌY# °M^Ì;/DY#834;/H/78>C "/8I#Í'34'5^B?>0 # 8/6 3F9 8+5Æ]8;/3T9)# \8Tb;/68A# )3F9 52?/ p3Y# /788M4;7"÷M ÷)Y# È8/c,- ÐAF%M? ú @dMQE8Z3\8E#yB/Z3#89 39)#'5c;/7&#8-IE\# # 8/#89Æ'3Y/)X&#8,y9 ;/]B?>W38-I#8 /7IE/7>a8P#-&9b;9 >^XD# /D :'5^\8(F1^%Y#aSA;H#I y#Z-Ib125 %Y#MCa)6#<3F9 #

52;/3FI,8-@9 \89bß7*\/C780 /3F#( /7c # 8/CSA;9 9p9 #5=33( \8(# Q8T-69; 9;>CXD# /D :'5C\88)3F']j# I/=52)38/D /?8Y# /+(; 8/TBF%ST/Ì-cMPÃ/('#85Æ]?QY# /+(; 8/W y# 9 ST#ª>2s\D)#I#I-Ib125W(#8M4P3/D(L_ `L_-#88#;/°] #*52;/Dp\ª#8)#;/# 9?S#ª\8T,/3F 8/I)P5ÆMOn 9;È? / eA²? ñ9@+# /+5!= 9)8/Á? Ô ÷9@++#ª\8A(SA/6#Py;/789 FL_# )3F9 8BbY# 9)] /ÌE(#Q 3/D(L_ `L_-#8T8#;/°]2(89)5aB*233 5c#833F8Y52 /76

Nm =lm

m!e−l,

ÉË÷M ÷ÊSA

Nm)*233F# /3> E(#

ΦmMy\)3#899; / /é4 7M÷M ÷E(SR3Fb 3# 92,XD/3F>6,8s;/+523F /7&3/D(L `L_-#8T8#;/a /WE39;5Æ]SA)3YW ,/5aB

Ωc/ω = 1 + 2λM3F;)3# 9p,'X/3>=,8E389 9 #8(I)E3#89 39)#'5Z /HÐA%MP? ú @J;/7c#=3/D(L `L_-#8yS#ª\8E,/34;/°]°BQ7D# Y52 /7"D3;9 9)#y9 /780#8R#a\#8)#;/# 9#L

ë2Ô

0 2 40

0.5

1N=5

m

occu

patio

n nu

mbe

r

0.1

0.8

0.087

0 2 40

0.5

1N=10

moc

cupa

tion

num

ber

0.052

0.9

0.049

0 2 40

0.5

1N=20

m

occu

patio

n nu

mbe

r

0.026

0.95

0.025

4;76÷M îo7 33#;/Z/?-6B+y,8*#ÌSs'# È?9;>=9 \86ÇTD(LdÄp /(;/C3F8/+52/2L#]λ = 0.005

#8/5σa = 0.04

]8, ú ] Ô'ÿ #8/5 ë8ÿ # -Is,8- 9 F`" 78DMÃ_3#8/B&(/W+#E233# /+3F A8U&(#m = 0, 2

52'3F#(*#81/N

MT>Ì#8F123F5&*52)# #8p3F8-I9 F9 >E /&-I252>?/#8-")3T9; -Ib']#8/5EA9b;/7(#Q #A8/*( /789 >aXD#8/;:5Ì\8(F1ÆM

Y# -IF'MWÃ_y#c\#8/ /7$D3;9 9 # E9;/7 ^78 \8'QÌ9;ST(E/78>8]UÌ3F9 8+50 /(#8B9 R#87# /(T389 9 #8(MB0ySs\8]8*3F9;9)# +([?33FYT#\ª#89;T σ |a| ;7+# /R8/'p,TSA 3Y*# ª12 -I# 8/p;/a)T'()T# y\# 9 5°]# /5#8[/ SA/Ì /Á4U 7Mp÷M ÷M

àâR¾Q ©]=Vbû5[ÒabVÙ1º= \X¢i¡£´µâ¡²,Èm±¡¸X²bË´â¡²²8¡¸X³!='# /2LdG9)5$3# 9)3F9)#;/y#86\ª#89;)5$ /$"9; -IbQ J9)# 78

N]ÆSA# /Z/?-I)3# 952)# 78/#89; :# 8/c3# 9)3F9 # 8/]2E/?-&B[ P# )3F9 #8BE\8>Ì9; -Ib5ÆMAÃ_3#8/$B"SA/=#[I528-I /# /DQ52\?)# 8/+,8-Ü-I#8/2L²G+9)5=;/C#G/b">2%-# RF12+'345ÌB* U*52

1/N? ñß@dMÃ/â4;7+M÷M î+]233F#;/"/?-6B+p j[( /789 FL_#8( 39;S#ª\8T,/3F 8/s# SA/,s*#8-I[#8#8-IFs\#89;'TF13F2,8TQ#8( 39;R/?-&BY ú ] Ô'ÿ # /+5

ë ÿ ,- 9 F`&;7QÉ,/8y# s[233F# 8//?-6B+T# /-#89; :'5Æ]DE#∑

m |cm|2 = 1Ê4MO<A#ª\

λ = 0.005] #R8# 8/#89,XD/3F>&

Ω/ω = 1.0045# /5ST# È?9 >&9 ;\ /Y#83F 8/

σa = 0.04]D# /5I,8-þ4;7+MP÷M ë SsyF12+'34s[788/5(#TQB+/A( /789 >EXD# /D;:56\881Æ]SA8/9 >Q /789 FL_# )3F9 JBbY# 9?,8SA)3Y

m = 1)-#83F3F)3# 9 9 >[233 5ÆMT78Y# +US^#,8-/?-I 3# 952)# 78/#89; :# 8/°]DSs*+#ª\8Q( 78/;G+3# /DA?33F+# 8/W *8BbY# 9)]

m = 0]m = 1# /+5

m = 2M±0ySs\8]2&233F# 8/=8P68BbY# 9)SA;

m 6= 1# &-I#89;9dMÃ/Ë#83F]' N233# /+3F °52'3F#(y9;'#852 /7A8Y52N#8

1/NMN N N\8>[# (Ë#83F>;/3Ib&SR+#E /Z-"252>?/#8-I 3"9 ;-I;]NSA69bYQ =4;7+M÷M ë# 1?3F'5I6B+R\# 9 5Æ]D8s9;J#878SA;D([ NÐA%M=? Ôªë @dMÃ/[G788]

1/N523'#8A T(SA/I,8J/9 >"/\# 9 [

σa]B2T;s89)5J-I7/#89;9 >8M

ë8ë

1 1.01 1.02 1.03 1.040

2

4

6

8

N=5 σa=0.04, l(Ω)

Ω/ω

l

4;7I÷M ú 7 ì /79 #8*-I8-I/D-@E#8( 39;I\Y(R Y# 8/+# 9,'XD/3>=,8E#ST# È?9 >^9)( \8c>?(-c]σa = 0.04

M =(#8"-# È^Ì#8/b;/ /78 BF%ST/c52;/D\881a%Y#]#8A3#89 39)#'5a /=ÐAF%M=? Ô'ë @dM

Ù1º=º ÚÛ¶m¦¯3µ,3d¡X¤¢i¡¸´µÐA'(9;#8R52 9)#ª>8'5I,8T%ST652;/DT\# 9 's8

σa]28/[# T)(-#89;9j/87-# Èy>2%-ò528-I /#5IB?>E[#8/#8-I8/ 3b%>]8# /5I/+#J)p9)# 78A/87R52)(9)#ª>[®y%-I;1?'5zp#(T#8°ST9 9?#8ÆJs-&9; 9 >EXD# /D :'5Q\8(F1Q#88M4;7R÷M ú SR#8/789)# J-"-I/D-òT# )3F9

lÉ, Æy7/5(# 'Ê#8#[,/3F 8/" +A Y# 8/+# 9,'X/3>

Ω/ω]ª,8p#[ST# È?9 >*9)( \8T /Y#83F 8/°]

σa = 0.04Ml

A39;'# 9;>XD#8/D :5W# /+5a/9;>IY# ÈT8/W;/D78A\# 9 MPQ?33F+#L 8/c/?-6B+A8E7/5W%Y#*,8y5½j/D # 8/#89j,'XD/3;'# ESA/ /Á4 7MT÷M ïM ì 52-I;(;/7E#87-I/D8 Í '3 MT÷M ë M Ô ]2;y y3F9 # #A&(# /DP-&9; 9 >6XD#8/;:5"\8( 3 /"A-I252>?/# -I)3s9 -";M0y/38]DF1#83F52)# 78/#89; :# 8/>? 9)5TE# -IQ#(*(34A#8AÐF%M=? Ô'ë @_MÌ(#8E;/!4 7MZ÷M ú -I#8È<Y# /+(; 8/0,XD/3F &78 \8/^B?>CÄJXM ÉË÷M ë Ê4MÃ_6)I3F9 #8E+#"c3F+/52/3F)&BF6,"-# 9 9;&\# 9 '&8l

#8/5Ω/ω

Mì [8 /5=2[ /<ÐF%M? Ôªë @d] [)[52Ea6Ë#83Fy#R"52/+52/368PB#;\s#8/# -I8/)3p/78>Q)PXD#852Y# 3J;/

mÉËTÄJXMTÉ ë M Ô ÷DÊ(ÊU# /5E,8

# /W8;/WÄJXMyÉ²÷M ë Ês;/+3F#8TSA;=# /789)# -I8-I/D-cM4[#9)# 78A\# 9 &8σa

&9;y#8652½j/D'M48-¼4;7+My÷M ë ]Ss&F12+'34 (Z# y(-";125zZ(# 8]#ST9 9[#$\8F1 %Y#MÃ_/Ì82a# a,81.004 ≤ Ω/ω ≤ 1.006

É Ôÿ # 8-YÊA# /+51.004 ≤ Ω/ω ≤ 1.0075

É ë8ÿ #8-YÊ4]+);/55"3Y6#Q(#8MA3F+/52 /7233# 8/"/D-&BY#8SA/& /4 7M÷Mñ?M233F#8/3F R8s%Y#m = 1

# /5m = 3

# (/<$52'3F#8I#81/N

MF,88]8'(Q# y38/ 55Iej3Fs ÆyG+/b/'s ÆQ(>2(-c]?# /59bYy#8*,/5cÌ#878*SA;ZÐA%MÃ? Ô'ë @_M[&#(&B+/5#8 [52a/8y-#Y3YF1#349 >8]B2 y / yST8>D /7+]? /3FE /=ÐAF%Mu? Ô'ë @_]*#8/(; 8/W)y3#89 39)#'5SA;W'(3FT# /£8~6%Y#Ψ = c0Φ0 + c2Φ2 + c4Φ4

]#"(# QSA)3YW5D'/ # +'# ;/WE(-#89;9°>2%-A3F/()52'5aM

ë ÷

0 2 4 6 80

0.5

1 Ω=1.025 l=4

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.018 l=3

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.015 l=2

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1N=10 Ω=1.005 l=1

m

occu

patio

n nu

mbe

r

4;7C÷M ïo7 33F+# 8/Þ/?-&BYW "^;/79;Ld+# )3F9 <S#ª\8Z,/34;/c,c#ST# È?9 >09)( \8W(>2(-c]σa = 0.04

M^ÄJ#3Y #8/9S&c788/5H(#W,8#E3# / Y# 8/+# 92,XD/3F>]D3YD(/I"(S [(34y Æy\81I(# +#y# +'# 'M

0 2 4 60

0.2

0.4

0.6

0.8

1N=10

m

occu

patio

n nu

mbe

r

0.130.1

0.65

0.0670.044

0 2 4 60

0.2

0.4

0.6

0.8

1N=20

m

occu

patio

n nu

mbe

r

0.13

0.059

0.74

0.0370.037

4;76÷Mñ7 33F+# 8/=/?-6B+[ JI( /79;Ld#8( 39;&ST#ª\*,/3F 8/[,8Q#FL9);\E(>2(-c]σa =

ÿ M ÿ ö]SAb Ôÿ ÉË9;`YÊ#8/5 ë ÿ ÉË 78D4ÊA#8-MÃ/$B cG78']l = 1.8

# /5Ω/ω = 1.0045

M&)Q QSA+#QSs3#89;9P#×y(-";125zI%Y#]°SA6# R-I#3F3F 3# 9233F# /3;' UE;/789 FL_# )3F9 [BbY# 9)M

ë î

1.01 1.02 1.03 1.040

2

4

6

8

N=5 σa=−0.001, l(Ω)

Ω/ω

l

1.01 1.02 1.03 1.040

2

4

6

8

N=5 σa=−0.002, l(Ω)

Ω/ω

l4;7Z÷M ö7 ì /789)# W-I8-I/D- W#8( 39;<\8Y # 8/#89,XD/3F> ,8%ST$Ss'# È?9 >0# (#34 \8Ì(>2(-]

σa = −0.001ÉË9;`YÊ"# /+5

σa = −0.002ÉË 78DYÊFMW9)#'# 6SAW-&9; 9 >HXD# /D :'5H\8)3F62336B38-Ic(-# 9 9 I#86 /Y#83F 8/Ì(/7 W /3F#('M

Ù1º=ºÙ ÿË¡¡o¤®¢i¡µýd¡¸¤¢i¡ ´d4A#(#34;\y /Y#83F 8/]DQ#(Q52)# 7#8- )F123F5a"BQXD;*52;j/DM0ST\']',8P#Q(-# 9 92\# 9

σa](3489)56B+A;-I 9 #8]8#F129)# /56;/Í '3 M ë M ë M ÷MB48

σa = −0.001# /+5 −0.002

][#8/789 #8J-I8-I/D-T#8( 39;)9 5&;/O4;7+M÷M ö[#8#y,/34;/&8s Y# 8/+# 9,'X/3>8MUT9;)( -I;9)# [+# 4 7MU÷M ú ]SAbE;-I8# /D52½j/+3Fs# BF%ST/EsT-&9b;9 >XD# /D;:5*\8F1*(#]ªT# s-# 9 9D7;/°8/8/Ld /D7# /79)# N-I-"/D#M)&6'34I 6 /3F#('50bW-I#878/;+52a8AW;/DY#834;/^ 6 /3F#('5Æ]p#8& / ,8- 3F-"+# 8/ Q$%ST^78Y# +M )ej3FW39;'# 9;> 3F8+/5"ZW#8(a8Ã4;7+M÷M ÷]ASA=,8c#^ST# È?9 > # (#34 \8$38/5/#=# y/I5 3F/;/?sY# /; 8/sB+%Ss/ÌQ-69b;9;>aXD#8/D :5Ì\8(F1#('Mì y;/c&3#8*SAb=9)( \8* /DY#83F 8/]#8/5Ì,8&# -I*#88/+]?&+#8B8/5# ;'° ÐAF%Mç? ú @25y/8UF1#349 >Q-#3YET8/'N /Ã4 7M÷M öMNT233#;//?-&BYp8j[788/5I(# ,J#*,Sn52½j/DT8#;/# 9,'X/3;'p# ySA/ /Á4 7MP÷M þ ,8/cB+D(/A# /+5

σa = −0.002M

ì 6-I#878/;+52E σa

y;/3'#85Ì,]+&# */9 8/7y#8/D>c(# s-&9; 9 >ZXD# /D :'5$\8)3FM*Ã/('#85Æ]j"#8"#('ySA"\Y# 9( /789 FL#8( 39;Q8B;#89 A#8Q-#83D38)3#89;9 >I?33F;'5ÆMp*>?#%(# T,-v#I(-I? 3F\"É²(\4U 7Mp÷M Ô'ÿ Ês("#'#83YW -V /Ì/WB38-IT*788/5a(# E#88# 8/#89,'X/3><8Ta>2%- * /3'#85ÆMc)E)&#=3F9 #8&52;/3F,8- R/78>(3F- ,8T9)( \8[;/DY#834;/]D#E%>D 3# 9F1# -I9 [8°SA)3Y)=4U 7Mp÷M Ô Mô # 9)3F9)#;/Q,652;/D&/?-&BY* A#8( 39;'ES #6#8/Z( 7 L/;G+3#8/DG/;L_;:Eej3Fy8/cE233F# /3;'A 6(# ]+#3#8/=B+"(/W,8-4;7+MR÷M ÔÔ MQF,&b*(-y# Q\Y# 9U(# '[#8E-#3F3F8 3# 9 9;>Ì?33F;'5É, /<)E3#(6%Y#'QSAb

m = 0, 1, 2# /+5<÷DÊ4]°SA)3Y0 *#89 WSA#*ST"12+'34

ë8ú

0 2 4 6 80

0.5

1 Ω=1.027 l=4

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.022 l=3

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.017 l=2

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1N=5 Ω=1.012 l=1

m

occu

patio

n nu

mbe

r

4;7C÷M þ 7 33F+# 8/Þ/?-&BYW "^;/79;Ld+# )3F9 <S#ª\8Z,/34;/c,c#ST# È?9 >^#(#34;\Ì>?(-c]

σa = −0.002M0ÄJ#3Y #8/9AS&c788/5H(#,8#I3# /W8#;/# 9j,XD/3F>]3Y/ÌSn&(34R8U*-&9b;9 >XD# /D;:5Ì\881a%Y#T+#y# +'# 'M

ë ï

5 10 15 205

10

15

20

25

N=5 σa=−0.004

total angular momentum L

tota

l ene

rgy

4;7E÷M Ôÿ 7T&>?Y#8(y%Y#8p#Ss'# È?9 >Ì#Y#83F \8EÇTFL_Äp /% /$3F/52/+# 8]σa = −0.004

MU*(#T,- #I-"?8a+# Y# B89 3FLd9 ;È[3\8]2# /+5a;N*(>2(-)8#5Æ]8#83YI(#A(33F';\9 >*B+'3F-"'9;ST(P;/I/78>MU89 5WÉË(Y# 78D4Ê9 ;/*)/DA8/9 >"-#8È8RE3\ª# Q U*>?Y#8(A(# '\D);B9 8M

,8-ÜÐAF%Ms? ú @_M²8*#RSs*#ª\l = 1

/= G78]+# /+5W,8)y\# 9 E8l

;/\? 8p3#8ÉË9)( \8ApST# È?9 >6# (Y#83F \8A /DY#83F 8/+ÊSTy#89;S#ª>2#85I\8F1,8-#;/°M4I#<-I8Ì/7# \8W\# 9 Ì8σa

]JSTÌSA 9 9(c#C52)% B2;/ 8yc /789 FL#8( 39;E233F#8/3F A# R3F9 9 >a-6B9 ASA# [STEF123Fy,8-ð"3F/DLd8`L-#8y Y#;/°]ÆSA)3YZS#8[/ Q"3#86 /$%Y#I52)3('5Z# B\8M\0ST\']"\#89;8σa

SA& E?33FYQ52?O¥|AR-I# 3YCSA#*)*F123F5Z,8- 4 7M÷M ÷MR T/"/5a"Ss>"#8B+2)MIÉË\# # 8/#89;9 >Êp3#89 39)#'5+#8B8/5# >,8y3/D(L_ `L_-#8J Y#;/W#(SA/Ì /W*G+78Q #34# 9 9;>Ì8/9 >a#8/sB+/5",p JB+/5#8>]#8J s# 9)(*-I/;/5I /ÐA%Mç? ú @_MP)p)pB3#8A/7>* 3/D(L_ `L_-#8NS#ª\8s,/3F 8/I#8B/3F8-I#8'5ERA/78> j#Q-&9; 9 >6XD#8/;:56\8(F1&(#8]D# /5&y'(/DP3# 9)3F9)#;/6S+#P )/8[#83F#89;9 >a&(# *SA)3YZ#"Y# /; 8/=233MÃ/cË#83F]+&Y# /b;/c BF%ST/#*3/D(L_ `L_-#8P%Y#y# /5#&%Y#y j,8- 52)3F('5";/\? 8#8#878Y# °]DSA)3YÌ)XD;E52;j/DMJ>?)3#892233F# /3;'U,8σa = −0.05

#8(SA/& /4 7M÷M Ô'ë ]SAsA3F 39;'52/ I12+'34'503F/DLd8`Ld-#8Q233# /+3F *#8&3# 9)3F9 # 5<,8-óÄJXM<ÉË÷M ÷DÊ4MQ52\?)#;/p,8- s,-69 #6# [SA/ /54 7MP÷M Ô ÷E,8 ú ] Ôÿ # /5 ë ÿ # -I+'34 \89;>M\4Q39 #8;%>8]j/9;>c%STW\ª#89;R Ω/ω

# & /3F9 52'5ÆM6"52\? # 8/# R'#8( /789 >(/Ì52'3F#8RSAbÌ*/?-&By U+# )3F9 MÃ/a*(-#89;9Æ>?(-J#AST[+#ª\8Q%52 5Æ]2;A -I+D;B9;[IF1#349 >9 ?3#y8 /TSAA#8/(; 8/BF%ST/\8(F1I# /53/D(L_ `L_-#8p%Y#p233FYM

ë ñ

0 2 40

0.5

1N=5

m

occu

patio

n nu

mbe

r

0.12

0.76

0.120.0025

0 2 40

0.5

1N=10

moc

cupa

tion

num

ber

0.15

0.7

0.140.0044

0 2 40

0.5

1N=20

m

occu

patio

n nu

mbe

r

0.17

0.67

0.160.0056

4;7I÷M ÔÔ 7 33F# 8/C/?-&BYR,8&#WST#8ÈD9 >$ /DY#83F /7=ÇTFL_Äp;/+% /038/2L52/# ]σa = −0.004

]Ω = 1.013

]# /+5l = 1

/=#3Y=78Y# °M=4-ð9 F`[;7D]N = 5, 10,

#8/520

MPQ78Y# T;/5 3#Q#* /6G+/bFL( :*6'34/a233#;/Ì/?-6B+T,)A\# 9 Q σa

M

0 2 4 6 80

0.5

1 Ω=1.026 l=3.7

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.022 l=2.8

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.018 l=2

moccu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.016 l=1.5

moccu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1 Ω=1.014 l=1.1

m

occu

patio

n nu

mbe

r

0 2 4 6 80

0.5

1N=10 Ω=1.012 l=0.6

m

occu

patio

n nu

mbe

r

4;7Q÷M Ôªë 7 ì /$#(#34;\9 >;/DY#834;/7IÇTD(LdÄp /(;/c3F/52/+# * /ÌE+#8 P3F/DL_ `L_-I#T Y# 8/NMs&3;Y3F9 A5/8*EF123F5W233#8/3F #3#89bL3F9 # 5a,8- ÄJXMyÉ²÷M ÷ÊFM

ë ö

0 2 40

0.005

0.01 Ω=1.016 l=1.6

0 2 40

0.005

0.01N=5 Ω=1.012 l=0.6

0 2 40

0.005

0.01 Ω=1.016 l=1.5

0 2 40

0.005

0.01N=10 Ω=1.012 l=0.6

0 2 40

0.005

0.01 Ω=1.016 l=1.5

0 2 40

0.005

0.01N=20 Ω=1.012 l=0.6

4;7y÷M Ô ÷o7UÑ[\? # 8/s j[/?-I)3#89+89 2 8/I,8- [3F/DL_ `L_-I#p?33F2L#8/3F A#A3# 9)3F9 # 5a,8- ÄJXMÉ²÷M ÷Ê4M

4σa = −0.01

]NÌ#89)# 78Ì52\?)# 8/+*,8- WF123F5 3F/DLd8`Ld-#8233# /+3F ]# /5a,8σa = −0.05

652\?)# 8/+T#8*-# 9 9JÉË#8/5c523'#8;/7"SA;/D-&B* #8( 39;'ÊF]°B2*,8- /DE5#Y#Ì)Q/=S#ª>c=9 ?3##c52)38/D /?8[#8/(; 8/NM"Ã/CË#34]°233#8/3F *- = /52)3# "# *Y# /+(; 8/c)y3F/D /D8MA [)/a3# Q,8R# 9)# -cM148[F1# -I9 8]ÐAF%M\? ú @#8SA/^+#"W\89)# ^BF%ST/HÌ-69b;9;>HXD#8/;:5 \8810(# c# /5H3F/DL_ `L_-I#J%Y#[\# /)(T;/[-I252>?/#8-")3[9; -IbA(&# s[Y# /b;/ST89)5aBE52)38/D /?8+]?B2/ y6,(-#89;9°>2%-MÙ1º= v³þ¡¸¤B¡ ´µ±´µ B¤¯3§ ¡ÍÜp# )# 8/+# 9E3#89 39)# 8/+c;/ H-I#8/2L²G+9)5# ª12;-# 8/k? ú @";-I9 >Þ# $B8/5# >^,8 /(#8B;9 ;%>H,8$/8/2L_8# /7C39;5 a#

σ |a| = −0.5MüU(/R(Sù#Q;/Z8Q3#(*)[9 -";Q9; [9;ST']j# 8/5

σa = −1MR)[89 5/8TB[ST8>D /7+];/3yT3# 9)3F9)#;/T#8(-Iy#

σa 1]2#638/5b;/# T)3F9 #89 >*/8U,9;G9 9;'5";/" 9 ;-I;MPU[ /?\8( 7# sB8/+5# >Q,8p(#8B;9 ;%>8]8(# ,8- ;71N#8/5#89;\89)sSs9 5I+#ª\8A&B[ /3F9 52'5 /yB#()M[;/(#8B 9bL;%>$3Y3YÈ=#yB/<,8-I5Z;/<-R p3Y-I)3#89 /D)# 9d]

µ = ∂E/∂NSA)3Y°]2SA/W/7# \8]D -I9 ;'# /W;/+%Y# B 9;;%>8Mì -I8612#34Q9; -IbE8sSAI*3# 9)3F9 # 8/Q# "\ª#89;)5< Q7;\/ZB?>Z#8/ '%/7ÌSsW#ª\8Ì,8/5H#I,"\#89;'"

σa8yL ÿ M Ô 8I7'#]=3Fb 3# 9,XD/+3F>W,8R;/+523F /7Ì8#;/$)

Ω/ω = 1.008]j#\# 9 E#Q)[/ [\>Ì-63Y52;/DE,- Ô M ÿÔ ]U#&3# 9)3F9)#5C;/N? ú @_McÃ/^86%52>8],E9;ST*\#89;'E8

σa]63F;)3# 9°,'X/3>a,y /523;/7Ì8# 8/=%Y# a3Y#8/78E#8/5W&7#y-I6B+R-I8#( 9 >IFT;/D&-I ;/°MO<[#8È8A #s#8/;/+52 3#;/# J[9 ST(N#8/5#8&9 \892# ª12;-# 8/E)/Q9;/78U\# 9 )5Æ]8# /+5&9 ;-I;P8P(52>E |σa| ≤ 0.1M

ë þ

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÷ ÿ

-# Èy A(+52>-I8R38-I9 FQST89)5BRI78"B>8/5Q9;ST(=N# /+5# Ì9;\89# ª12 -# 8/N]N/#8-"9;><Z;/39;+52Ì;/789 FL_# )3F9 %Y#ESA;^/7# \8a#8/789 #8-I8-I/DY#m < 0

#8/5Ì# 9 9 SüY#852)# 9Æ123bY#;/]n > 0

MJU52I ]2E2 /+#E(R<IB#()R;/<"78Y# -vSs9 5<#ª\EcB3Y# /75Æ]jB*I(ST89)5EBT;\?)# 9dMNÄP1?/52 /7RB#()Ss9)5E j3FJ7;\s)(sR9 #87-#)3FÌ52 #878/# 9 ;:8]B2[/&3F9)5W/3# E/9; È89;>c(#-# È*&3#89 39 # 8/Ë#8(MP)TSs9)5a# 9 9 Sns"786"9 #87T\ª#89;s σa

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nb-6B+ B+D(/M!$%A/8F13F5 ë ÿ M

ns-6B+& 9 Ss'%* 78//78 [$3# 9)3F9)#6,E#3YZ Y# 9J#8/789)# -I8-I/D-

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4U9)# 78#8/#8-I8/ 3p-]ª#833Y#83F>Q8+(B#;/E8>Q /8P/78°M,8s/s/5*89 \8s Í 3Yx252 /78'XD#;/&/?-I 3# 9 9;>MU#8/789 #8#8(p3#8/°]FK%+%s#;/IA#8-I8/ 33#(] By(# Y#56# /5I89 \85"# /+# 9 > 3# 9 9;>]8# /+58/R/5cSA;WQ,89 9;SA /7IXD# 8/Ì,8*#52 #89j# T7

∂2R

∂ρ2+

1

ρ

∂R

∂ρ− 1

a4osc

ρ2

(

1 + λρ2

a2osc

)

R+2Mε

h2R− m2

ρ2R = 0.

ÉËÇRM Ô Ê

/R3# /a78FT)58ÆyG(T5;\#;\y;N8/-# È8'p[B(; 8/R(ρ) = f(ρ)√

ρSA)3YÌ7;\

f ′′ +

(

− 2

ρ2− ρ2

a4osc

− λρ4

a6osc

+2Mε

h2− m2

ρ2

)

f = 0.ÉËÇRM ë Ê

4( -I9 3b%>]28/*8`/W# -")3R/;]ω = M = h = 1

]

f ′′ =

(

λρ4 + ρ2 − 2ε+m2 − 1/4

ρ2

)

f.É²Ç[M ÷Ê

NIG+/5W6(9;2;/]8/E3# /c*E(?8 /7-IF?5°];/7I*Ë#34#p\# 9 Æ8?pS#ª\8,/+34 8/6#ρ = 0

# /5ρ→ ∞ # PÈD/SA/°M Í /3FJbN)N3;)3#89+#&S#ª\8R,/3F 8/$)y#33# E# yB+8=E9 ;-I;]+/±y(D8 zR,8-

0

∞ #Ss9;9N#T& AS#ª>a# 8/+5Æ]# /5W-#Y3YWE89 2 8/+;/WE-I 559;MJNG/5ÌQbY# 8/c(+]/*3#8/ÌR,8A12#8-I9;R\-I\-IF25? Ô î] Ô'ú @d]SA)3Y)T;#8B9;y,8T/?-I 3R(9; 8/8N52½j/D #89'XD#;/P#A# y9 ;/# T /89 2;/"#8/5ESATTG(p52 \ª# \8T P#8B/DMUÃ_p #[,8"8Y52-I25Æ]

÷8÷

# /+5=bR [ej3F \8"( /3F68/9 >c8/&\# 9 #;/$ P& 78D[#8/5$ 5&)y/52'5= /#3YÌ(°M ì (-"E#8/Ì'XD#;/Ì8UQ%>?+d2y

dx2= g(x)y + s(x).

ÉËÇRM î?Êy(x)

3#8/B+R52\9 85I /a#*SsJ( J# B82#*/?-&Bs Æ+;/DYxn

]D+# Y#'5B?>Q;/D\#89 ∆x

M ì 552 /7QU#ª>?9 8ß/ °,8-&9)#A,yn+1

#8/5yn−1

78 \8'ÆT12' 8/

yn+1 = 2yn − yn−1 + y′′n(∆x)2 + y(4)n

(∆x)4

12+ O

(

(∆x)6)

.É²Ç[M ú Ê

wy /7=ÉËÇRM î?Ês,8E('3F/5Ì5;\#;\[7;\

yn+1 = 2yn − yn−1 + (∆x)2 (gnyn + sn) + y(4)n

(∆x)4

12+ O

(

(∆x)6)

.É²Ç[M ïÊ

R,8(c5;\#;\Q A#8ª1? -#5B?>

y(4)n =

d2yn

dx2

∣

∣

∣

∣

n

=d2(gy + s)

dx2≈ δn

2(gy + s)

(∆x)2=

=1

(∆x)2(gn+1yn+1 − 2gnyn + gn−1yn−1 + sn+1 − 2sn + sn−1) .

É²Ç[Mñ Ê)A3#8/ÌB*;/+((5W;/0ÉËÇRM ïÊ#8/5Ì89 \8'5,

yn+1]2 78/8;/76-A8U52

O(

(∆x)2) #8/5W 78]

yn+1 =

2yn

(

1 + 512 (∆x)2gn

)

− yn−1

(

1 − 112 (∆x)2gn−1

)

+ 112 (∆x)2(sn+1 + 10sn + sn−1)

1 − 112 (∆x)2gn−1

.É²Ç[M öÊ

ì 3F8/52 /76F12( 8/W3# /W#( 9 >B+Q,/5Ì,8yn−1

M»²8¡¸¤B¡¸«ªÔ¡

m = 03#(m = 0

3# 9 9 [,8E8-II(3 #89p#/D;/°M6Ã_*)RI-m2 − 1

4/ρ2 /ÄJXMEÉËÇRM ÷ÊT#R3# +('8B9;E/# E 78 /°MAU#ª\88)5Ì)]+/&3#8/=%Y# -"\Z-"25C9)#]# /5C# /01?+# /;/$$3# 9)3F9 # IIG(*,Sò(+M

ì 7D# /°]9;?ÈD /7H# Z=# -I8/)3c3F 9;9)#889 2 8/+I,8a$Y#85 #89ST#ª\,/3F 8/0/'# Q8;7;/N]N# /5<9 # 8/f (ρ) =

√ρR (ρ)

]U/)*9;'#85Z=*F12# /+( 8/f (ρ) =

√ρ

(

a0 + a2ρ2 + a4ρ

4 + a6ρ6 + ...

)

.É²Ç[M þ Ê

# È?;/7I38/5<52 \ª# \8I# /5Z;/+((;/7W /ÌÄsX+MWÉËÇRM ÷Ê[78 \8'[#Ì9 # 8/ BF%ST/ÌE3F?e:3F /DYMÃ/H( /7ZW78Y# -ØÉË#87# /(&Ì\D 8+(9 ><È?/SA/ # -I8/)3D3;9 9 # 9bYÊF];Ì/+Ì2a#c;/7H WF12#8/;/n,8ρ < 0, 1

78 \8'<-I(#833FY#[9;M÷ î

D + % ÷=+ÖLGI$ E # KÆM

? Ô @ ô Mü+MÅF)3YÈÌ# /+50&M Í -I;°]0/|Ae2143ço¾665|ÍiÝ866ÍºØ|Í 87\Ïù¸:9=A6Ò4]É ô #8-&B)5278sw/;\Y(;%>Åp] ë8ÿ8ÿë Ê4M? ë @ ì MüMoN778F]ÐA\jM!=25ÆMÅp?>2M ý ä ]÷ ÿ ñ?]ÆÉ ë8ÿ8ÿÔ Ê4M? ÷A@\pMjÅp;#8\2ÈD P# /5 Í M Í ;/7# d]0/±|962143çÒe;5|ÍiÝ866ÍºØ|Í É 1?,8Y5Pw/bL\8;%>Åp] ë8ÿ8ÿ ÷DÊ4M? îÍ@EÑ6M ì MÇT2#8/5cÑ&M Í MÐAÈD+#8]¸y# ÉØN8/52/+Ê ä< ý ]÷ ë ñaÉ Ô'þ8þþ Ê4M? ú @&Î"M!^MÏ*#ª\89)# È?)] ì MÑ&M®üD#83YÈ2(/°]# /5HÎR52/HÇ#ª>D-c]Åp?>2MÐA\jM ì ý ]

ÿ îD÷8ï ÿ ÷aÉ ë ÿÿ îDÊFM? ïA@ ì M+Ñ&MoüD#83YÈ28/°]2#8/5WÎ"M+!^MÏ*#ª\89)# È? ]DÅp?>?MÐA\M ì ý ] ÿë ÷8ï ÿÔ É ë ÿ8ÿ îDÊFM?ñß@\EM+ÏIMOn;9 È?;/N]oüM!HMo4TM+ÎR//°]# /5WÐ*M ì M Í -I;°]ÅJD>2MÐA\M°FM= ] ë8ë ï úÉ Ô'þ8þ öÊ4M? öA@EÇRM!=8(9 8/N]2Åp?>?MÐA\jM°FM= ä ] ë ï þDú É Ôþþ8þ ÊFM? þ @\pMPÑ6MBU# /5# °]p#8/5^ÄMP!^M®°;Ë(b:8]>5®;AÒØ~¾A@? ¾~oAØ~Ò4][ÉËÅ7# -I/°] 1DL,8Y5ÊFM? Ôÿ @QÄM2ÏIMwEM2ÎRD]0&MDÐ/78]2# /5 M0;///N]? ÍoÎA1CBoAÒÑØ~e;D¥¾|AÒÎ=É ì 5# -0 9 78e>ªÃ Å ÅpB9;)(;/7+] ÔþþÔ Ê4M? Ô8Ô @*!^MNB9 #5Æ]FE|ÍÒºØ~¾evA¥ÝG3=ÝTH±¾~|ÍÒÑÒùo~eÑØ|AP6I1ùoAºùKJL7v|ÍØÃANM\½ßO 8ù¸;ÍP?P|QJO6o? Ô'ë @ ì MÑ6M¸ü#3YÈ2(/°]?Î"M2!^MÏ*#ª\89)# È?)]# /5aÄM°/+52°]2Åp?>2M2ÐA\jM ìSR < ] ÿú ÷8ï ÔþÉ ë8ÿ8ÿ îDÊ4M? Ô ÷A@*ÇRM!=8(9 8/N]2 \#Q38-I-&/ 3#;/°M? Ô îÍ@QSASAS*M # °M / \?;M #3 M #Ò>Y¾>\8Ò>3 ÿÔ'ÿë >ª5?1>ª/?52ñ Ô M D-I9? Ô'ú @QSASAS*M -I#M -")(M #83 M Èo>ª7Bi>Aw ì ÷ þë >/ Ò>ª/?-6\UT3;NM j5?

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