Dynamic susceptibility of spin-1/2 Ising chain in transverse field
Dynamic susceptibility of the one-dimensional spin-1/2 transverse Ising model is obtained by using the numerical approach suggested earlier [Ferroelectrics 153, 55 (1996)]. The dependence of the susceptibility frequency shapes on the value of the transverse field at various temperatures is discussed...
Gespeichert in:
Datum: | 1997 |
---|---|
Hauptverfasser: | , |
Format: | Artikel |
Sprache: | English |
Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
1997
|
Schriftenreihe: | Физика низких температур |
Schlagworte: | |
Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/175711 |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
Zitieren: | Dynamic susceptibility of spin-1/2 Ising chain in transverse field / O. Derzhko and T. Krokhmalskii // Физика низких температур. — 1997. — Т. 23, № 7. — С. 721-726. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraineid |
irk-123456789-175711 |
---|---|
record_format |
dspace |
spelling |
irk-123456789-1757112021-02-03T01:27:46Z Dynamic susceptibility of spin-1/2 Ising chain in transverse field Derzhko, O. Krokhmalskii, T. Низкотемпеpатуpный магнетизм Dynamic susceptibility of the one-dimensional spin-1/2 transverse Ising model is obtained by using the numerical approach suggested earlier [Ferroelectrics 153, 55 (1996)]. The dependence of the susceptibility frequency shapes on the value of the transverse field at various temperatures is discussed. The way in which the frequency shape rebuilds as the transverse field increases is illustrated. 1997 Article Dynamic susceptibility of spin-1/2 Ising chain in transverse field / O. Derzhko and T. Krokhmalskii // Физика низких температур. — 1997. — Т. 23, № 7. — С. 721-726. — Бібліогр.: 15 назв. — англ. 0132-6414 PACS: 75.10.-b http://dspace.nbuv.gov.ua/handle/123456789/175711 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
collection |
DSpace DC |
language |
English |
topic |
Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
spellingShingle |
Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Derzhko, O. Krokhmalskii, T. Dynamic susceptibility of spin-1/2 Ising chain in transverse field Физика низких температур |
description |
Dynamic susceptibility of the one-dimensional spin-1/2 transverse Ising model is obtained by using the numerical approach suggested earlier [Ferroelectrics 153, 55 (1996)]. The dependence of the susceptibility frequency shapes on the value of the transverse field at various temperatures is discussed. The way in which the frequency shape rebuilds as the transverse field increases is illustrated. |
format |
Article |
author |
Derzhko, O. Krokhmalskii, T. |
author_facet |
Derzhko, O. Krokhmalskii, T. |
author_sort |
Derzhko, O. |
title |
Dynamic susceptibility of spin-1/2 Ising chain in transverse field |
title_short |
Dynamic susceptibility of spin-1/2 Ising chain in transverse field |
title_full |
Dynamic susceptibility of spin-1/2 Ising chain in transverse field |
title_fullStr |
Dynamic susceptibility of spin-1/2 Ising chain in transverse field |
title_full_unstemmed |
Dynamic susceptibility of spin-1/2 Ising chain in transverse field |
title_sort |
dynamic susceptibility of spin-1/2 ising chain in transverse field |
publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
publishDate |
1997 |
topic_facet |
Низкотемпеpатуpный магнетизм |
url |
http://dspace.nbuv.gov.ua/handle/123456789/175711 |
citation_txt |
Dynamic susceptibility of spin-1/2 Ising chain in transverse field / O. Derzhko and T. Krokhmalskii // Физика низких температур. — 1997. — Т. 23, № 7. — С. 721-726. — Бібліогр.: 15 назв. — англ. |
series |
Физика низких температур |
work_keys_str_mv |
AT derzhkoo dynamicsusceptibilityofspin12isingchainintransversefield AT krokhmalskiit dynamicsusceptibilityofspin12isingchainintransversefield |
first_indexed |
2025-07-15T13:06:09Z |
last_indexed |
2025-07-15T13:06:09Z |
_version_ |
1837718328350605312 |
fulltext |
��� �������
�� ��������
����������������������� "!�!�#$��%'&�(')*�� ,+-#$�.�*&/#'(* �0�#'(�1
243
57698;:=<?>�@A>-<CBEDGF":IH
:IJK:=F�3ML�NO>7D�:I5�P�Q,R"SUTV>W:K5�X;<7YW6E:I5Z:K5
F�[\6O57>^]7BO[\>GB_N�:`BEJ=a
bdc*e-f,gKh�ikj�lnm$o�prqscutsgKl$j�iuvdm$w x*j'yzy
{}|�~V�����z�������=�/�����/|����V|I~`�V�������=���V���'�I�\~K���`~��*�'�"�����V|.��~�����~��/�z���������������������K�����¡ /¢\£/£��/�=�*¤��/���\��|��¥�¦�§*¨�©�ª=«�¬�K®�¯`°�±/²�³´© µV§*¶�· ªz¸\© ¸¡· ¹�¨±�®=²�±�°\§�³´©�µK§*¶"· ª�¸\© ¸¡· ¹�¨
º�¹�»/§u©�¼�¼=K¬�½�VµKK§*»\K®'¾À¿�ÁÃÂ�Â�Ä
½ÆÅ.ÇÀ¨�§È©ÉµÈÊ`¹ÀÊ�µI�¶\¼K©�»�©zª=©�¼�Å˲\Ìͼ�°Àβ\Ç��¦�¬À©z§uVÇÀÊK©�²�ÇÀ¨}ª�Ê`¶�©�ÇÀ¦KÁ�Ï*д¼`®`¨/Ç\Ê`¸��®`Ê`ÒÑ�Ê�©ÉÇÀÓ´§u²�¬À�ª'©�Ê*²/»À¼`¨/©�ÇÀI¬Ë»�Å�¹ÀÊ�©ÉÇÀÓ
¼=°ÀÔÇÀ¹À§uI®�©Éµ�¨}ª^¨�¶À¶À®�²�¨�µ�°rÊ�¹ÀÓ�Ó/�Ê�¼��¬rV¨/®Kª�©��®ÔÕzÖ �}�K�`���/×z�V���Ø�K���Ã~ÔÙ�Ú�Û ¿�¾/¾ÝÜ�ÁVÂ�Â.Ä�Þ�ßV·�à�°ÀÔ¬ÀV¶ÀIÇÀ¬\�Ç�µ��²ÀÌs¼`°À
Ê�¹�Ê`µ�I¶À¼�©z».©�ª�©�¼�ÅáÌØ®��â�¹ÀVÇÀµKÅ�Ê�°À¨�¶ÀIÊÆ²�Ç˼`°�ȸ�¨/ª�¹À$²\Ì�¼`°Àu¼`®�¨�ÇÀÊ`¸\I®KÊ�ÈÌ�©�/ªz¬Ë¨�¼�¸\¨�®V©�²�¹ÀÊ�¼`V§u¶ÀI®�¨/¼�¹À®��ÊÆ©ÉÊ*¬�©�Ê`µV¹ÀÊ`Ê�I¬�·
à�°ÀÈã*¨IÅá©ÉÇsã*°.©zµV°�¼`°ÀÈÌ�®KIâÀ¹\�Ç�µVÅ�Ê�°À¨�¶Àu®KI»À¹.©�ªz¬ÀÊ�¨�Ê*¼`°\$¼�®`¨�ÇÀÊ�¸�V®KÊ`ÈÌ=©z�ª�¬s©zÇ�µ�®`V¨�Ê`�ÊÆ©�Êu©zª�ªØ¹\Ê`¼�®`¨/¼=�¬�·
ä�å*æ'º�«�ç Ú¡è�Ù�éÀè=ê�ë
ì*í�îÔï�ð�îÀñ`ò�ó�ô^î.ð�õVó�ï�ð�ö¡÷�õVø�ó�ð�ñ}ù�úsûrüVõIó=ð�ýÝô�ï�ò�î¡÷Ëó=ðnö
þ�ÿ öÀð�õ���î ÿ õIî��KóØî�÷=òEó�õËö¡ðEó�ô-ø�ï ÿKþ ö�ð þ õ���� �Vî�
þ ï � þ í�î\ï ÿ î þ ó�ñ
.ö ÷ õ þ ��ò�ó�î.õÎð�ï þ ï�ð�÷
����î�
Àö���õIî´ï��Æó þ õ���õIî����'÷=ð�î.õ�õuó�ðCõVï�÷Øó�ò
õ þ ö þ î�ø�í���õIó�
.õ���� � þ ö ÷ØõÃï���î�
.ö��'õKî7ô-öÀð��Ôï �´ó þ õsõ þ ö þ ó=õ þ ó�ñ
.ö ÷áô^î�
Àí�öÀð�ó�
Àõ
ø ÿ ï�ø�î ÿKþ ó�îÀõ�
Àö�ð���îOî���öÀô-ó=ð�îÀòÝî�� ö�
þ ÷��
� ù�� �"!�#%$sï'&sî���î ÿ � õ�ó=ð(
\î þ í�î´îÀö ÿ ÷�� ù*)�+�,�õ$ó þ &-öÀõ.-�ð�ï'&-ð
þ í�ö þ
Àö¡÷
��'÷=ö þ ózï�ð�ï �$õVï�ôsî þ ó�ô�î.ñ`ò�î.ø�îÀð�ò�î.ð þ õ�ø�ó=ð/
�ï ÿ�ÿ î.ñ
÷Øö þ ózï�ð0����ð�
þ ó�ï�ð�õ1�`ï ÿ þ í�ó�õ ôsï�ò�î¡÷dîÀð
�ï'��ð þ î ÿ ý ÿ î.ö þ
ò�ó2���Kó3
���÷ þ ózîÀõÈöÀð�ò7ó�ðWõ�ø�ó þ î�ï4�uôsö¡ð��Cø�öÀø�î ÿ õuò�î.ö¡÷=ó=ð¡ý5&Aó þ í
þ í�ó=õOø ÿ ï��'÷�îÀô ��6 �7+%!98\õIî\î4ö¡÷ØõIï ÿ î�
�îÀð þ ø�öÀø î ÿ õ �;: �<)%!
ò�îÀö¡÷Øó�ð¡ý=&Aó þ í õ��
Àí õ þ ��ò�ózîÀõ>�`ï ÿWþ í îEï�ð�îÀñ�ò�ó=ô�îÀð�õVó�ï�ð�ö¡÷
õ�ø�ó=ð�ñ}ù�úsû@?A? ô�ï�ò�î¡÷CB þ í�î ó�ð���î.õ þ ó�ý�ö þ ó�ï"ðrï��sò ��ð�öÀôAó3
ø ÿ ï ø î ÿÃþ ó�îÀõD
Àö�÷`÷ØõE�`ï ÿ ô^ï ÿ î î����`ï ÿÃþ õF#ÔüVð þ í�î ø ÿ îÀõIî.ð þ
ø�öÀø�î ÿ ï�� ÿ ý�ï�ö¡÷Aó=õ þ ï ø ÿ ï'��ó=ò�î öG� ÿ î.õIíH��ó�î�&Mï�ð þ í�î
ö�ð�ö¡÷���õVó�õOï �OõVø�ó�ð ò���ð�öÀôAóI
.õ�#KJ�ø�î�
Àó2�Kó3
.ö ÷�÷��L��&�î õ�í�ö¡÷�÷
î�� þ îÀð�ò þ í�îÝîÀö ÿ ÷=ó�î ÿ î�÷=ö���ï ÿ ö þ îÀò ð ��ô^î ÿ ó�
Àö¡÷
öÀø�ø ÿ ï�ö�
Àí
�`ï ÿ î�M���ó`÷=ó�� ÿ ó���ô õ þ ö þ ó=õ þ ó�
Àö¡÷ ô^î�
Àí�ö¡ð�ó�
ÀõD
.ö ÷
��'÷=ö þ ó�ï�ð�õ
�`ï ÿ õIø�ó=ð�ñ�ùÀúAû ?/N
.í�ö�ó�ð�õ � ù�,%�`ù�ù'! þ ï þ í�îÔöÀð�ö¡÷���õ�ó�õ-ï��
ò ��ð�öÀôAó3
ø ÿ ï�ø�î ÿKþ ó�îÀõ�ï � þ�ÿ öÀð�õ���î ÿ õIînüVõVó�ð�ý ô�ï�ò�î¡÷�öÀð�ò
&sîdõVí�ö¡÷�÷Aõ þ ��ò��L�Ëó�ð ø�ö ÿKþ ó�
��Í÷Øö ÿ �Ëó þ õ�ò�� ð�öÀô-óO
dõ�� õ�
\îÀø�ñ
þ ó���ó�÷Øó þ �L#
P îH
\ï�ð�õIó=ò�î ÿ õVø�ó=ð�ñ}ù.úAûQ
.í�öÀó=ð ò î.õR
ÿ ó���îÀòS� � þ í�î
$�öÀô-ó=÷ þ ï�ð�ó�öÀð
T
= Ω ∑U
=
Á
V
W U X + Y ∑U
=
Á
V
−
Á
W U Z W U
+
Á[ � 8�ù'B
&Aí�î ÿ î
Ω
ó=õ þ í�î þ�ÿ ö�ð�õR��î ÿ õÃî��Kó�î¡÷=òEö þ´þ í�îCõ�ó þ î �ÆöÀð�ò5\dó=õ
þ í�î�ó�ð þ î ÿ ö�
þ ózï�ð/��î þ &^î\îÀðÔð�î.ózý�í���ï ÿ ó�ð ý õIó þ îÀõF# ìL&sï
ò�ó=õVñ
þ ó�ð�
þ ó���îE
ÀöÀõIî.õ]
�ï ÿVÿ îÀõVø�ï"ð�ò�ó=ð¡ý þ ï ò�ó2���`î ÿ îÀð þ õ�óÉý ð�õ9ï4�
ó=ð þ î ÿ õIó þ îEó�ð þ î ÿ öF
þ ó�ï�ðG&Aó�÷`÷^��î�
�ï"ð�õVó=ò î ÿ îÀòL�ÒóR# î�#���\`_E,
8��`î ÿVÿ ï"ô-ö�ý ð�î þ ó�
a
\ï���ø ÷Øó=ð ýbBCöÀð�ò/\9cE,d8\öÀð þ ó��`î ÿ�ÿ ï�ô-ö�ý�ñ
ð�î þ óO
@
\ï���ø�÷=ó=ð¡ý�B�#.P î ö ÿ îÔó=ð þ î ÿ îÀõ þ îÀònó=ð þ í�î þ ó=ô^î.ñ`ò�îÀñ
ø�îÀð�ò î.ð þ � þ &Ëï�ñ`õVø�ó�ðE
�ï ÿVÿ î�÷=ö þ óØï"ðH��� ð�
þ óØï"ð�õ
〈W Uα( e )W U + |β 〉
�
&Aí�î ÿ î þ í�î öÀð ý'÷�î1� ÿ öF
�-�î þ õ ò�î.ð�ï þ î þ í�î ÿ ô^ï"ò���ð�ö�ôsóI
ö���î ÿ ö.ý î
〈( f�fgf )〉 ≡
ì ÿ ��h −βi ( f;fRf ) !�jÈì ÿ h −βk #�ì*í�îl
\ï ÿIÿ î¡÷=öÀñþ óØï"ðm����ð�
þ ó�ï�ðn��î þ &^î\îÀð þ í îEo�ñ2
�ï"ô-ø�ï�ð�îÀð þ õ9&sö�õrò�îÀñ
ÿ ó3��î.ò ó=ðqp$î��*#Eû #`P î õIí�ö¡÷`÷ ÿ îÀõ þ�ÿ ó�
þ4þ í�î öÀð�ö¡÷���õ�ó=õ
ô-öÀó=ð�÷�� þ ï þ í�îr
�ï ÿIÿ î�÷=ö þ ó�ï�ð0����ð�
þ óØï ðs��î þ &^î\îÀð þ í î
t ñu
\ï�ô-ø�ï"ð�îÀð þ õ ï � þ &sï õ�ø�ó=ð�õ��sð�ï þ ó=ð"ý þ í�ö þ ö ÷�÷-ï þ í�î ÿ
ð�ï"ð�v�î ÿ ï/
\ï ÿVÿ î¡÷Øö þ ó�ï"ð]���'ð�
þ ó�ï"ð�õw
.öÀð]��î��`ï'��ð�òEó�ðEø ÿ ó=ð�ñ
Àó=ø�÷�îx� �Ôò�óu���`î ÿ î.ð þ ó=ö þ ó�ï"ðzy
〈W U Z ( e )W U + |
{
〉 = −〈W U { ( e ) W U + |Z 〉 =
ù
Ω
|
| e 〈W U Z ( e )W U + |Z 〉
�
〈 W U � ( e )W U + |
{
〉 = −
ù
Ω }
| Ð
| e Ð 〈 W U Z ( e )W U + |Z 〉
#
8\û B
üVðWï ÿ ò�î ÿ$þ ï7î�� ö ÷
��ö þ î þ í�îxM(��öÀð þ ó þ �Wï4�$ó=ð þ î ÿ îÀõ þ ��ï ð î
õ�í�ï��'÷=ò ÿ î�& ÿ ó þ î þ í�în$AöÀô-ó`÷ þ ï�ð�ó=öÀð~8�ù'BÝó=ð þ î ÿ ô^õnï4�
�'î ÿ ôsó�ï�ø�î ÿ ö þ ï ÿ õ�&só þ í þ í�îsí�î¡÷=øCï4� þ í�î��"ï ÿ ò�öÀð�ñuP ózý�ð�î ÿ
þ�ÿ öÀð�õ��`ï ÿ ô-ö þ óØï"ð öÀð�ò þ í�îÀð ò�ó�ö�ý ï�ð�ö¡÷=ó�v/î þ í�î���ó`÷Øó�ð�î.ö ÿ
�`î ÿ ô^ó�ï�ð��`ï ÿ ôa#^�uö�õVóI
ÿ îÀõ��'÷ þ õCô-ö�����î õR��ô-ôsö ÿ ó�v/îÀòdö¡õ
�`ï�÷�÷�ï'&-õ�y´ìÆí î ÿ î¡÷Øö þ ó�ï"ð�õ5��î þ &^î\îÀðÝõVø�ó�ðnï�ø�î ÿ ö þ ï ÿ õ7öÀð�ò
�'î ÿ ôsókï"ø�î ÿ ö þ ï ÿ õ�ö ÿ î
� �������'�������������*�.���������'��������� ����� ���F ;¡�¡�¢
W U Z =
ù
û ϕ Á+ϕ Á−ϕ Ð+ϕÐ− f�f;f ϕ� − �+ ϕ� − �− ϕU+ �
W U � =
ù
û�� ϕ Á+ϕ Á−ϕ Ð+ϕ }
− f�fgf ϕU − �+ ϕU − �− ϕU− �
W U X = −
ù
û ϕU+ϕU− �
8��7B
&Aí�î ÿ î
ϕU+ = ∑� =
Á
V
���
η�+ + η�
�
ϕ�− = ∑� =
Á
V
Ψ� U
η�+ − η�
�
8 6 B
þ í�î þ�ÿ öÀð�õ;�`ï ÿ ô^î.ò $AöÀôAó`÷ þ ï�ð�ó�öÀð]8�ù�Bsí�öÀõ þ í�îx�`ï ÿ ô
T
= ∑�
=
Á
V
Λ �
η �+η � −
�
η � � η �+
= δ� ��
η �
η �
=
η �+ �
η �+
=
, �
8��%B
Λ � � Φ� U �ËöÀð�ò Ψ� U ö ÿ îdò�î þ î ÿ ôsó�ð�îÀò=� ÿ ï�ô þ í�îOî�M���öÀñ
þ ó�ï�ð�õ
∑U
=
Á
V
Ψ� U (����� + � U�| ) = Λ � Φ� | �
∑U
=
Á
V
Φ� U (� ��� − � U�| ) = Λ � Ψ� | �
∑U
=
Á
V
Φ � U Φ ��� = ∑U = Á
V
Ψ � U Ψ��U = δ� � �
∑� =
Á
V
Φ � � Φ � U = ∑� =
Á
V
Ψ��� Ψ��� = δ � U
8�� B
&Aó þ í
� � U ≡ Ωδ � U +
\
6 δU�� � + Á +
Y �
δU ¿ � − Á �
� � U ≡
\
6 δ� ¿�� + Á −
Y 6 δ� ¿ � − Á #
�'ï ÿ ��� ÿIþ í�î ÿ ò�î þ ö�ó�÷=õAõÃî\î@pÈî��Kõ�#,ù��`ù�,7�Æö¡ð�ò ù�ù�#uüÃð]��ó�î�&
ï �A8�� B���8��%B � þ í�îa
.ö¡÷�
��Í÷Øö þ ózï�ðOï4�
〈W U Z ( e )W U + |Z 〉
ÿ îÀò ��
\îÀõ þ ï
î���ø ÷�ï"ó þ ó�ð¡ý
ï�� þ í¡îxP ó3
�- � �È÷�ï�
.í��'ò�î! Ëï�ôsó�ð�ó3
\ó�õ þ í�î/ï ÿ î\ô
öÀð�ò þ í�î ÿ î.õR�'÷ þ
.öÀð���î î�� ø ÿ îÀõVõIîÀòE
�ï"ô-ø�ö�
þ ÷�� ó�ð þ í î
�`ï ÿ ô ï4� þ í�î#"��Kö����Kó�öÀð4ï(�Wû
(
û%$
+ & −
ù
)×
û
(
û%$
+ & −
ù
)öÀð þ ó=õ�� ô-ô^î þ�ÿ ó�
Eô-ö þ�ÿ ó��D
\ï�ð�õ þ�ÿ �%
þ î.òD� ÿ ï�ô î¡÷�îÀôsîÀð þ ö ÿ �
�ï ð þ�ÿ ö�
þ ó�ï"ð�õ
6
〈 W U Z ( e )W U + |Z 〉 = 〈ϕ Á+( e )ϕ Á−( e )ϕ Ð+( e )ϕ Ð−( e ) f;fRf ϕU − Á+ ( e )ϕU − �− ( e )ϕU+( e )ϕ Á+ϕ Á−ϕ Ð+ϕ Ð− f�fgf ϕU − Á+ ϕU − Á− ϕU+ϕU−ϕU + Á+ ϕU + Á− fgf�f ϕU + | − Á+ ϕU + � −
Á− ϕ� + |+ 〉 =
=
".�
'
−〈ϕ Á+ϕ Á−〉
⋅⋅⋅
−〈ϕ Á+( e )ϕU + |+ 〉
〈ϕ �+ϕ Á−〉'
⋅⋅⋅
−〈ϕ �−( e )ϕ� + �+ 〉
〈ϕ Á+ϕÐ+〉
〈ϕ Á−ϕÐ+〉
⋅⋅⋅
−〈ϕ Ð+( e )ϕU + |+ 〉
f�fgf
f�fgf
(�(%(
f�f;f
〈ϕ Á+( e )ϕ� + |+ 〉
〈ϕ Á−( ) )ϕU + �+ 〉
⋅⋅⋅,
� 8�+ B
&Aí�î ÿ î
〈ϕ�+( ) )ϕ *+ 〉 = ∑� = �
+
Φ� U Φ� *
�ï�õ�í � �
Λ� e − (βΛ ��, û )
!
�ï�õVí
(βΛ� j û )
〈ϕ�+( ) )ϕ *− 〉 = − ∑� = �
V
Φ� U Ψ�.-
õ�ó`ð�í0/21
Λ � e − (βΛ� jÈû )
!
�ï�õ�í
(βΛ � jÈû )
�
〈ϕ�−( ) )ϕ *+ 〉 = ∑� = �
V
Ψ� U Φ� *
õ�ó�ð�í � �
Λ� e − (βΛ� j,û )
!
\ï�õIí
(βΛ� jÈû )
�
〈ϕ�−( ) )ϕ *− 〉 = − ∑� = �
V
Ψ� U Ψ�3-
�ï�õ\í / 1
Λ�4e − (βΛ� jÒû )
!
¡ï�õ�í
(βΛ� jÒû )
#
8 : B
4�M���ö þ ózï�ð�õG8�� B��L8 : B��`ï ÿ ô þ í�îÝõ þ ö ÿKþ ó=ð¡ý ø�ï"ó=ð þ �`ï ÿ
��� ÿIþ í î ÿ ð���ôsî ÿ óI
.ö¡÷�
.ö ÷
��'÷=ö þ ó�ï�ð�õ�#�5$ï�ð�õ�ó�ò�î ÿ ó=ð ýEö�
.í�öÀó=ð
ï �76
=
û : ,Aõ�ø�ó=ð�õ &Aó þ í�\
= ±
ùËöÀð�ò7öA
�î ÿÃþ öÀó=ða��ö¡÷
��î�ï4�
þ í�î þ�ÿ öÀð�õR��î ÿ õÃî �Kó�î¡÷=ò ó�ð þ í�î ÿ öÀð ý�î
Ω =
,%#Kù��8�7��&^î
õIï�÷���îÀò96
×
6 õ þ öÀð�ò�ö ÿ òrø ÿ ï4�Í÷�îÀôq8:�%BCï � þ öÀó=ð�îÀò ó�ð þ í î
ÿ î.õ��'÷ þ
Λ� �
Φ� U � Ψ� U # J¡î þIþ ó=ð ý0$
=
��ûÔöÀð�ò]
\î ÿKþ ö�ó�ð &ó=ð þ í�î ÿ ö�ð¡ý�îr� ÿ ï"ô , þ ïm�(, ��&�î þ í�î.ð~
\ï�ôAø�� þ îÀò
î¡÷�îÀôsîÀð þ ö ÿ ��
\ï�ð þ�ÿ ö�
þ ózï�ð�õ@8 : BÔó=ð���ï�÷
��î.ò4ó�ð�8�+%B��`ï ÿ ö
ý�ó���îÀð þ îÀôAø�î ÿ ö þ � ÿ îAó=ð þ í�î ÿ öÀð�ý�î
β =
%' �7,7#`ùáöÀð�ò þ ó�ô^î
e ��ø þ ïnù/û4, öÀð�ò î���ö¡÷�� ö þ î.ò þ í�î;"��Kö'���Kó�ö�ð ð��'ô�î ÿ óI
.ö¡÷`÷
�
ï�� þ öÀó�ð�ó=ð ý ó�ð þ í î ÿ îÀõR�'÷ þ$þ í�î M ��öÀð þ ó þ �7ï �uó=ð þ î ÿ îÀõ þ � óR# î�#��
þ í�î/
\ï ÿ�ÿ î¡÷Øö þ ózï�ð���î þ &sî\îÀð þ í�î t ñu
�ï ô-ø�ï�ð�îÀð þ õ-ï �sõIø�ó=ð�õ
ö þ-þ í�î
õ�ó þ îÀõ<$ÔöÀð�ò;$
+ = þ ö'-�îÀðdö þ-þ ó=ôáî.õ e öÀð�ò e =
' �
ÿ î.õIø�î�
þ ó���î¡÷
�L#Æì*í�î ÿ î7ö ÿ î��`î�& ø ÿ ö�
þ ó�
Àö¡÷,÷Øó�ôAó þ ö þ ó�ï�ð�õ�ï�ð
þ í�ó�õAöÀø�ø ÿ ï"ö�
ÀíL�uóR# î�#g���Kó=ð�ó þ î�
Àí�öÀó=ðOõ�ó�v�î>69�uø ÿ î.õÃî.ð�
\îWï4�
��ï���ð�ò�ö ÿ ó�îÀõ-ù
≤
$%� ?
+ & ≤
69��öÀð�ò`�Kó�ð�ó þ î þ ó�ô^î e #Íì*í�î.õÃîî����`î�
þ õ�÷�îÀö�ò þ ïCò�î���ó=ö þ óØï"ð�� ÿ ï�ô ö þ ó=ôsîw��îÀí�ö���ózï ÿ ó=ð�í�î ÿ ñ
îÀð þuþ ï�ö¡ð7ó�ð%�Kó=ð�ó þ î^
Àí�öÀó=ð��(&-í�ó�
Àí7ó=õ þ í�îl
ÀöÀõKî�ï4�Îó=ð þ î ÿ î.õ þ
¢.@�@ A�� �;� �R�CB � �;��� ���b�����ED����3�%FHG����� C¡�¡�¢��JI��K@ML��%B ��¢
ó=ðÔõ þ ö þ ó=õ þ ó�
.ö¡÷uô�î�
.í�öÀð�óO
ÀõF# ì*í�î�� ö¡÷���îsï4� þ í�ó=õ´ò î���ó�ö þ ózï�ð
ò�îÀø�îÀð�ò�õkï ð þ í�î ��ö¡÷
��îÀõÎï4� þ í�î þ�ÿ öÀð�õ���î ÿ õIî^�Kó�î�÷=òCöÀð�ò þ í�î
þ îÀôAø�î ÿ ö þ � ÿ î�#��Òî���î ÿKþ í�î¡÷�îÀõVõF�Æõ�ó�ð
�î7õR�
Àí î����`î�
þ ó�õáî.ö¡õ��
þ ï ÿ î�
�ï�ý�ð�óIv/î �sï�ð�înôAö�� ò�î ÿ ó���înó=ð öD&-ó=ò�î ÿ ö¡ð¡ý�î ï��
ø�ö ÿ öÀô^î þ î ÿ õ þ í�î ÿ î.õ���÷ þ õG&Aí�ó3
.í ö ÿ î ð�ï þ õR� ���Vî�
þÝþ ï
þ í�îÀõKîdó=ð �I÷���îÀð
�îÀõ��ËóR#�î #u�^&Aí�ó�
Àí ÿ î��`î ÿWþ ï 6
→ ∞
#ËìÆí î
ò�ö þ öOø ÿ ï"ò��
�î.ò4ó�ð þ í�î]
.ö¡÷�
���÷=ö þ ó�ï�ð�õWò�î.õR
ÿ ó���î.ò4ö���ï���î
� ö�õ &sî¡÷`÷ÆöÀõ þ í�î ÿ î.õ��'÷ þ õ^�`ï ÿ
χZ*Z (κ
�
ω)
8�) B�ï4� þ öÀó=ð�î.òWï�ð
Ö ���.�����
χ ��� ( � ¿ ω)
¸�V®=Ê=¹�Ê
ω
¨I¼È¸/¨I®�© ²�¹�Êȼ�K§Æ¶�K®=¨�¼�¹\®�VÊV·�à�°��Ê=²/ª�© ¬7ª�© Ç�KÊÈ®=I¶/®`KÊ=KÇ\¼$¬�¨V¼=¨sÌz²�®ÒÌzV®�®=²�§Æ¨�Ó�Ç�K¼�© µs© Ç\¼�V®�Ê�© ¼=´µK²�¹/¶ ªz©�Ç/Ó �
= −
Á ¿�¼�°\
¬\¨IÊ=°\K¬�ª�© Ç�KÊ'¬/VÇ�²/¼�Ƭ�¨V¼�¨ÈÌz²�®'¨IÇ\¼=©zÌ�K®�®=²/§*¨VÓ/Ç�V¼=©�µ*©�Ç/¼=K®`Ê�© ¼=�µV²�¹�¶\ª�© Ç\Ó �
=
Á � � µV²�®=®=IÊ=¶/²/Ç/¬�Ê'¼�²
Ω = � · Ð
� µK²�®=®`KÊ=¶\²�Ç�¬/Ê'¼=² Ω =
Á/·
A�� �;� �;� B(� �;��� ���L�����<D����O�:FHG��g�� C¡�¡�¢4� I��K@ML4�:B ��¢ ¢.@ML
þ í�î.ó ÿ ��öÀõ�ó�õ�!Èø î ÿÃþ ö�ó�ð þ ï7ó�ð"�Kó�ð�ó þ îl
Àí�öÀó=ð�õ�#�ì*í�î ÿ î.õR�'÷ þ õ$ï��
ï4� ÿ ð��'ô�î ÿ ó�
.ö¡÷]
.ö¡÷�
���÷=ö þ ó�ï�ð�õ &sî ÿ în�`ï'��ð�ò þ ïm��î ó�ð
�%
\î¡÷`÷�îÀð þ ö�ý ÿ î\îÀôsîÀð þ &áó þ í þ í�îsî�� ö�
þ´ÿ îÀõR�'÷ þ õ,ï � þ ö¡ó�ð�îÀò
ö þ��
= ∞
8�pÈî��*# �%BsöÀð�ò �
=
, �
Ω =
\%j û@8�pÈî��*# +%BsöÀð�ò
&Aó þ í þ í�î´ð�ï þ ï ÿ ó�ï���õ*î��'ö�
þÈÿ î.õR�'÷ þ �`ï ÿ*þ í�î o�ol
\ï ÿ�ÿ î¡÷Øö þ ózï�ð
��� ð�
þ óØï"ð � û !�# ü þ õ�í�ï��'÷=òs��î î.ô-ø�í�öÀõ�óOv�î.ò þ í�ö þ4þ í�î
ò�îÀõ�
ÿ ó���îÀò öÀø�ø ÿ ï�ö�
ÀíÝö�÷`÷�ï'&AõWï"ð�î þ ïÝõ þ ��ò �E�Kó=ð�ó þ îÀñ�õ�ó�v�î
î����`î�
þ õ &Aí�óO
Àíz��í�ï'&^î���î ÿ �'ö ÿ î ��î���ï�ð�ò þ í�îáõR
�ï"ø�î�ï(� þ í�î
ø ÿ î.õIîÀð þ ø�öÀø�î ÿ #/��ó�ð�ö¡÷`÷
�L��&�î���î ÿ ó��Kó�îÀò ÿ î�÷=ö þ ózï ð�õE8\û%B
&Aí�ó3
Àía
\ï�ð�ð�î�
þ
〈 W U Z ( e ) W U + |Z 〉
&Aó þ íWï þ í�î ÿ
�ï ÿ�ÿ î¡÷=ö þ ó�ï�ð/����ð�ñ
þ ó�ï�ð�õÝö�� þ î ÿnþ í�î.õIî�
�ï ÿVÿ î�÷=ö þ óØï"ð ����ð�
þ ózï ð�õÝö ÿ î�
\ï�ôsñ
ø � þ î.ò ó=ðÔö-õ�ó�ôsó`÷=ö ÿ ô-öÀð�ð�î ÿ #
P î�õVí�ö¡÷�÷,ò�ó�õR
���õ�õ þ í�î7ò ��ð�öÀôAó3
Àõsï�� þ í�î þ�ÿ öÀð�õR�"î ÿ õKî
üÃõIó=ð ý ô^ï ò î�÷$÷�ï¡ï -�ó=ð¡ýEö þ´þ í�îCò���ð�öÀô-ó�
AõR��õ�
\îÀø þ ó3� ó�÷=ó þ �
χ Z�Z (κ
�
ω) ≡ ∑|
=
Á
V
î � κ | ∫
�
∞
� ) î � (ω+
�
ε)
� ù� 〈
��� U Z ( e ) � W U + |Z !
〉
�
ε → +
,@# 8�) B
ì*í�î � ÿ î�M ��î.ð�
�� õVí�öÀø�îÀõ ï � puî
χ[�[ ( '
ω)
öÀð�ò
üVô
χ [�[ ( '
ω)
ö þ � ö ÿ ó�ï ��õ þ�ÿ öÀð�õ;��î ÿ õKî��Kó�î¡÷Øò�õ�öÀð�ò þ îÀô-ñ
ø�î ÿ ö þ � ÿ îÀõ,ö ÿ îsõ�í�ï�&sð�ó=ð@�Íó�ý�õF#*ùËöÀð�ò�û #bJ"ô-ö¡÷�÷L&áó�ý"ý�÷�îÀõ
ó=ð þ í�îE
�� ÿ ��îÀõ]
\ï ÿVÿ îÀõVø�ï�ð�ò�ó=ð¡ý þ ï
β =
��&�î ÿ î4ó=ð þ�ÿ ï�ñ
ò��
\îÀò�� � þ í�î��Kó�ð�ó þ î þ ó=ô^î9
�� þ ï4���Wó=ðQ8�)"B/��î�
Àö�� õÃîOï4�
ÿ ö þ í�î ÿ õ�÷�ï�& ò î�
Àö��Wï(��
�ï ÿIÿ î�÷=ö þ ó�ï�ð�õ���î ÿ õ��'õ þ ó�ôsîa8�îÀõ�ø�îÀñ
Àó=ö¡÷�÷�� �`ï ÿ
Ω =
, #Éû �Où ö¡ð�ò �`î ÿVÿ ï"ô-ö�ý ð�î þ ó�
ó�ð þ î ÿ õVó þ î
�ï4��ø�÷Øó�ð¡ý�BF#ËìÆí�î9&Aózý�ý'÷�î.õ/
ÀöÀðH��î ÿ îÀô�ï��"îÀò4îÀó þ í�î ÿ � �
Ö ���.�� ·
χ ��� ( � ¿ ω)
¸/V®=Ê�¹\Ê
ω
Ìz²�®�¬\©zÌ�Ì�K®�KÇ\¼'¸/¨�ªz¹/VÊ'²/Ì�¼=®=¨�Ç/Ê=¸/I®=Ê=uÌ�©�VªÉ¬´¨V¼
β =
¾À·�à¡°\*Ê=²�ª�© ¬Ëªz©�Ç/IÊ'®=K¶�®=VÊ=KÇ�¼'¬/¨�¼�¨ÒÌz²�®
Ω = � · Ð�Ü=¼=°�KÊ=uµK¹�®=¸/KÊ'¨�®=¨}ªÉÊ=²-¶\ªz²�¼�¼�I¬7© Ç���© Ó�·*Á�Þ,¨VÇ�¬
Ω = � · Ä ��¼=°�˪ɲ/Ç/Ó/¦z¬\¨IÊ=°�K¬7ª�©�Ç�VÊ$¬\KÇ�²�¼=´¬�¨V¼=¨sÌz²/® Ω = � · � ��¼�°\�Ê=°\²�®`¼�¦�¬�¨VÊ=°�K¬7ª�© Ç\KÊ$µK²/®=®=KÊ�¶\²�Ç�¬C¼=²s¬�¨V¼=¨AÌz²�®
Ω = � · ��¼=°/*¬\²�¼=¼�K¬´ª�©�Ç�VÊ�®=V¶/®`KÊ=KÇ�¼�¬�¨V¼�¨ÈÌz²�® Ω = � · ¾À·
¢.@�
A�� �;� �R�CB � �;��� ���b�����ED����3�%FHG����� C¡�¡�¢��JI��K@ML��%B ��¢
ó=ð
ÿ îÀöÀõ�ó=ð¡ý þ í�î þ ó=ô�îD
�� þ ï ����ó=ð 8�) B � &sí�ó�
Àí ÿ î�M(� ó ÿ î.õ
\ï�ô-ø�� þ î ÿ´ÿ î.õIï � ÿ
\îÀõF��ï ÿ � �Ôó=ð�
ÿ îÀöÀõVó�ð�ý þ í�î ��ö¡÷
��îsï4�
ε
�
&Aí�ó3
Àí õ�÷=ózý�í þ ÷
� õIôsï¡ï þ í�õ þ í�î�� ÿ î�M(��î.ð�
�� õVí�öÀø�î.õ öÀð�ò
ò�î�
ÿ î.ö¡õKîÀõF�Íó=ð ø�ö ÿÃþ ó�
��'÷=ö ÿ � þ í�îCí�îÀóÉý í þ õ�ï�� þ í�î.ó ÿ ø�î�
���÷Øó�ñ
ö ÿ ó þ ó�îÀõ�#��Cõ���ö ÷�÷��L�"&sîCõKî þ
ε =
, #O,',Íù��7, #<, �7#
�'î þ �'õËð ï4& þ � ÿ ð þ ï þ í�îCò�ó=õ�
���õ�õVó�ï�ð�ï � þ í�î ÿ îÀõR�'÷ þ õF#
�Íó�ý(� ÿ î ùAõ�í�ï�&Aõ þ í î7ò�îÀø�îÀð�ò�îÀð�
�îCï �wpÈî
χ[*[ ( '
ω) − ωö�ð�ò?üVô
χ[�[ ( '
ω) − ω
�� ÿ ��îÀõ ï�ð
Ω
��ö ÿ óØï'�'õ þ îÀôsñ
ø�î ÿ ö þ � ÿ î.õF#�� ÿ ï"ô þ í�îÀõIî øÍ÷�ï þ õ�&^î îÀöÀõ�ó=÷�� õIî\î þ í�ö þ
üÃô
χ[�[ ( '
ω)
î�� í�ó���ó þ õ þ &sï ø�îÀö'-�õ9�`ï ÿ
Ω =
, #�û18 þ í�î
üÃõIó=ð ý�ñÃ÷=ó2-�î
Àö�õÃî4B�ö�ð�ò ï ð î�ø�îÀö�-��`ï ÿ
Ω =
ù�8 þ í�î�
.öÀõÃîAï��
ö�÷=ô�ï�õ þ ð�ï"ð�ó=ð þ î ÿ ö�
þ ó=ð¡ý7õ�ø�ó=ð�õ$ó=ðCöÀðGî�� þ î ÿ ð�ö¡÷7�Kó�î¡÷Øò7B #�� þ
÷�ï�& þ îÀô-ø�î ÿ ö þ � ÿ î
β =
�`�`ï ÿ
Ω =
, # ûrüKô
χ[�[ ( '
ω)
ÿ î.ñ
��î.ö¡÷=õÝö í�ózý�í õIí�ö ÿ ø ø�î.ö�- ó�ð þ í�î���ó�
Àó=ð�ó þ � ï �9v�î ÿ ï
� ÿ î�M���îÀð�
��
ω ≈
,7#O, �ÝöÀð�ò4ör÷�ï�&0� ÿ ï�öÀò ø î.ö -Ýö þ
ω ≈
≈
, # +��/�`ï ÿ �`î ÿ�ÿ ï�ôAö�ý�ð�î þ ó3
�ó=ð þ î ÿ õ�ó þ î�
�ï4��ø�÷Øó�ð�ýEöÀð�ò þ &sï
ø ÿ ï ð ï4��ð�
\îÀò ø�îÀö�-�õ ö þ � ÿ î�M���î.ð(
.ó�îÀõ
ω ≈
, # � �ZöÀð�ò
ω ≈
ù�#O,�û��`ï ÿ öÀð þ ó2�`î ÿIÿ ï"ô-ö�ý�ð�î þ ó�
�ó�ð þ î ÿ õ�ó þ î
\ï���ø ÷Øó=ð ý�#�üVð
þ í�î�
ÀöÀõVî
Ω =
ù üVô
χ[�[ ( '
ω)
ÿ î���îÀö�÷=õ�ï�ð�î í�óÉý í öÀð�ò
� ÿ ï öÀòOø î.ö�-Oö þ
ω ≈
,7# �Íù�öÀð�ò
ω ≈
ù�# ��,��`ï ÿ �`î ÿ�ÿ ï�ñ,öÀð�ò
ö�ð þ ó2�`î ÿIÿ ï"ô-ö�ý ð�î þ ó�
H
\ï ��øÍ÷Øó�ð ý�� ÿ îÀõ�ø�î�
þ ó���î�÷��s#��´õ þ í�î
þ îÀôAø�î ÿ ö þ � ÿ îsó=õ ÿ öÀó=õKîÀò þ ï
β =
ù�� þ í�îsõIó þ ��ö þ ózï�ð/M���ö¡÷=ó þ öÀñ
þ ó���î¡÷�� ÿ î.ô-öÀó=ð�õ þ í�î õIö�ô�î�#l� ï ÿ
Ω =
, # û üKô
χ[�[ ( '
ω)î��'í�ó���ó þ õ þ &Ëï ÷�ï &sî ÿ ö¡ð�òr� ÿ ï�öÀò�î ÿ ø�îÀö�-�õD
�ï"ô-ø�ö ÿ îÀò
&Aó þ í þ í�îsø ÿ î�
�î.ò�ó=ð¡ý>
ÀöÀõKî�� &Aí�óO
Àí�ö ÿ îáõIí�ó�� þ î.ò þ ïCí�ózý�í�î ÿ
� ÿ î�M���îÀð�
.ó�îÀõ
ω ≈
, #O,4) �
ω ≈
,7#�) 6 �`ï ÿ �`î ÿ�ÿ ï�ôAö�ý�ð�î þ ó3
Aó=ð�ñ
þ î ÿ õIó þ î ó�ð þ î ÿ öF
þ ó�ï�ð öÀð�ò þ ï ÷�ï'&sî ÿ � ÿ î�M���î.ð�
Àó�îÀõ
ω ≈
≈
, # ûÍù��
ω ≈
ù�#O,�,9�`ï ÿ öÀð þ ó2�`î ÿ�ÿ ï�ôsö�ý�ð�î þ óI
Eó�ð þ î ÿ õ�ó þ îÔó=ð�ñ
þ î ÿ ö�
þ óØï"ð � ó=ð þ í î7÷=ö þIþ î ÿ
Àö�õKî þ í�îW÷�ï &Añg� ÿ î�M���îÀð�
��Eø�îÀö�-��î�
\ï�ôsîÀõCí�ózý�í�î ÿ-þ í�ö�ð þ í�îEí�ózý�í�ñg� ÿ î�M(��îÀð�
�� ø�îÀö�-L# �'ï ÿ
Ω =
ù üVô
χ[�[ ( '
ω)
î���í�ó3� ó þ õ4ï"ðÍ÷
� ï�ð�î ÷�ï &sî ÿ öÀð�ò
� ÿ ï öÀò î ÿ ø�îÀö�-9
�ï"ô-ø�ö ÿ îÀò9&Aó þ í þ í�î�
ÀöÀõIî
β =
� ��&-í�ó�
Àí
ó=õuõ�í�ó2� þ î.ò þ ï�í�ó�ý�í�î ÿ � ÿ î�M ��î.ð�
��
ω ≈
, # �'+��`ï ÿ �`î ÿIÿ ï�ôAö�ý�ñ
ð�î þ ó�
5
�ï'�'ø�÷=ó=ð¡ýdöÀð�ò þ ï9÷�ï &sî ÿ � ÿ î�M ��îÀð�
��
ω ≈
ù4# 6 û/�`ï ÿ
ö�ð þ ó2�`î ÿIÿ ï"ô-ö�ý ð�î þ ó�
n
�ï'�'ø�÷Øó�ð¡ý�#�� þ í�ózý�í þ îÀô-ø�î ÿ ö þ � ÿ î
β =
, #�ù þ í�îE� ÿ î�M(��î.ð�
�� ø ÿ ï(�Kó�÷�î.õ ï � üVô
χ[�[ ( '
ω)
ö¡÷Øñ
ôsï�õ þ
\ï�ó=ð�
.ó�ò�î`�`ï ÿ �`î ÿVÿ ï"ô-ö�ý ð�î þ ó�
ö�ð�òÝöÀð þ óu�`î ÿIÿ ï�ôAö�ý�ñ
ð�î þ ó�
S
\ï ��ø'÷=ó�ð�ý�õF� ö¡÷ þ í�ï���ý�í ó þ ó�õ õ þ ó`÷�÷nø�ï�õVõ�ó��'÷�î þ ï
ÿ î�
�ïÀý ð�óIv/î þ &�ï ø î.ö�-�õ��`ï ÿ
Ω =
, #�û ö þ
ω ≈
, #�ù��L�
ω ≈
≈
, # :�: öÀð�ò
ω ≈
, #=ù��7�
ω ≈
ù'#O, ���`ï ÿ �`î ÿVÿ ï"ñ*öÀð�ò7ö�ð þ ó2�`î ÿ ñ
ÿ ï�ôAö�ý'ð�î þ ó�
�ï'��øÍ÷=ó�ð ý�� ÿ îÀõ�ø�î�
þ ó���î�÷��L� öÀð�òWï�ð�îsø�îÀö'-a�`ï ÿ
Ω =
ù�ö þ
ω ≈
ù�#O,�,ÔöÀð�ò
ω ≈
ù4#`ù�,@�`ï ÿ �`î ÿIÿ ï�ñ,ö�ð�òOöÀð þ ó�ñ
�`î ÿ�ÿ ï�ô-ö�ý�ð�î þ ó3
�
\ï � ø�÷=ó=ð¡ý�� ÿ îÀõ�ø�î�
þ óI�"î¡÷���#
üKð?ï ÿ ò�î ÿ þ ï0��ò�î ÿ õ þ ö�ð�ò_í�ï�& þ &sï�ñ`ø�îÀö�-Mõ�í�ö¡ø�î.õ
þ�ÿ öÀð�õR�`ï ÿ ô ó=ð þ ï ï�ð�îÀñ`ø�î.ö - õ�í�öÀø�îÀõOö�õ þ í�î þ�ÿ öÀð�õR�"î ÿ õKî
�Kó�î�÷=ò ó=ð�
ÿ î.öÀõÃî.õF��&sî ø î ÿ �`ï ÿ ôsîÀò öÀò�ò�ó þ ózï�ð�ö ÷�
.ö¡÷�
��Í÷ØöÀñ
þ ó�ï�ð�õ�ï4�
χZ�Z (
' �
ω)
ö þ
β =
�a�`ï ÿ
Ω =
, # � � , # 6 �b,7# �%�b, # �
8\õ�í ï4&sðOó�ðG��óÉý�#Íû%B #��´õw
.öÀð]��î7õÃî\îÀð]� ÿ ï�ô ��ózý�#Æû � þ í�î
í�ózý�í�ñ�� ÿ î�M(��î.ð�
��dø�î.ö�-9ï�ð þ í�î�
�� ÿ ��î ü�ô
χZ Z (
, �
ω) − ω�`ï ÿ
β = =
� �
Ω =
, #Éûrö þ
ω ≈
, #O+����`ï ÿ �`î ÿ�ÿ ï ô-ö�ý�ð�î þ ó3
\ï���ø�÷=ó=ð ýGöÀð�ò þ í îs÷�ï'&Añ�� ÿ î�M���î.ð�
��Cø�îÀö�-Cö þ
ω ≈
,%#�� �A�`ï ÿ
ö�ð þ ó2�`î ÿIÿ ï"ô-ö�ý ð�î þ ó�
H
\ï ��øÍ÷Øó�ð ý &só þ í ó=ð�
ÿ î.öÀõ�ó�ð ý þ�ÿ öÀð�õVñ
��î ÿ õIî��Kó�î�÷=òOô^ï'��î þ ï4&-ö ÿ ò þ í�î�÷�ï'&Añ�� ÿ î�M���î.ð�
�� ø�î.ö -Eö þ
ω ≈ ≈
, #<, �9�`ï ÿ �`î ÿ�ÿ ï�ôsö.ý�ð�î þ ó�
�
�ï���ø�÷=ó=ð ý9öÀð�ò þ ï'&Aö ÿ ò
í�ó�ý'í�ñg� ÿ î�M���îÀð
��Ýø�îÀö�-4ö þ
ω ≈
ù4#O,�û��`ï ÿ öÀð þ ó��`î ÿ�ÿ ï�ô-ö�ý�ñ
ð�î þ óO
a
\ï ��øÍ÷Øó�ð ý��b&-í�ó3
ÀíOó=ð þ � ÿ ð]��î�
\ï�ô�î5� ÿ ï�öÀò�î ÿ #ÆìÆí î
ø�ï�õVó þ ó�ï�ð�õ$ï � þ í�îAø î.ö�-�õ�ö ÿ îAöÀõ �`ï�÷`÷�ï &Aõ�y
ω ≈
, #<, �7�
ω ≈
, # ��+
�`ï ÿ
Ω =
, #��7�
ω ≈
, # ,�� �
ω ≈
, #Éû : �b,%#�� �5�`ï ÿ
Ω =
, # 6 �`ï ÿ
�`î ÿ�ÿ ï�ôsö�ý�ð�î þ ó�
Ýó=ð þ î ÿ õIó þ îÝó�ð þ î ÿ ö�
þ ózï�ð � ω ≈
,7# �3� �
ω ≈
≈
ù'#O, � �`ï ÿ
Ω =
, # �L�
ω ≈
, #<+�� �
ω ≈
ù�#<, ���`ï ÿ
Ω =
, # 6
�`ï ÿ öÀð þ ó��`î ÿ�ÿ ï�ôAö�ý�ð�î þ ó3
ó�ð þ î ÿ õVó þ î ó�ð þ î ÿ ö�
þ ó�ï"ðz# �'ï ÿ
Ω =
, # � þ &�ï ø�îÀö�-�õ í�ö���î ö¡÷ ÿ î.öÀò��
�ï"ö¡÷�î.õR
�î.ò�y
üVô
χ Z'Z (
, �
ω)
ÿ î���îÀö¡÷Øõ�ï�ð�î7ø�îÀö�-Oö þ
ω ≈
, #�ù�,EöÀð�ò
ω ≈
≈
ù'#O,�, �`ï ÿ �`î ÿ�ÿ ï�ñÔö�ð�ò ö�ð þ ó2�`î ÿVÿ ï"ô-ö�ý ð�î þ ó�
H
�ï'��øÍ÷Øó�ð ý��
ÿ î.õIø�î�
þ ó���î¡÷
�L#"�7� ÿKþ í�î ÿ ó�ð�
ÿ îÀöÀõKî�ï � þ í�î þ�ÿ öÀð�õR��î ÿ õÃî �Kó�î�÷=ò
÷�î.öÀò�õ þ ï þ í�îAõ�í�ó2� þ ï �uï�ð�î-ø�î.ö - þ ï
í�ózý�í�î ÿ � ÿ î�M���î.ð�
.óØî.õ��
óR#�î #u�
ω ≈
,%#`ù��dó2�
Ω =
,%# �dö¡ð�ò
ω ≈
,%# �Íù ó2�
Ω =
ùK�`ï ÿ
�`î ÿ�ÿ ï�ôsö�ý�ð�î þ ó�
�
�ï4��ø�÷Øó�ð¡ý öÀð�ò
ω ≈
ù�#�ù�,Wó2�
Ω =
,7# �
öÀð�ò
ω ≈
ù4# �4,Wó2�
Ω =
ù �`ï ÿ öÀð þ ó2�`î ÿ�ÿ ï�ôsö�ý�ð�î þ óI
x
�ï'��øÍ÷=ó�ð ý�#
üVð õ��'ôsôAö ÿ �L�b&�î5&sî ÿ î7ö��'÷�î þ ïÔð ��ô^î ÿ ó�
Àö¡÷�÷��Ôî���öÀô-ñ
ó=ð�î þ í îAò ��ð�öÀôAó3
Aø ÿ ï�ø�î ÿKþ ó�îÀõ,ï4�Îï�ð�îÀñ�ò�ó�ôsîÀð�õIó�ï"ð�ö¡÷Æõ�ø�ó=ð�ñ
ù�úsû üÃõIó=ð¡ý ô^ï"ò�î¡÷kó=ð ö þ�ÿ ö�ð�õR��î ÿ õÃî��KóØî�÷=òÔö�ð�ò þ ï7î���ö¡÷
��ñ
ö þ î þ í�î=� ÿ î�M���îÀð
���ñ�ò�îÀø�î.ð�ò�îÀð þ õ���õR
\îÀø þ óI��ó`÷=ó þ �L#�ì*í�î.õÃî
ÿ î.õ��'÷ þ õ$õKî�î.ô þ ï ��î�ó=ôsø�ï ÿIþ öÀð þ õ�ó�ð�
�î � þ ï þ í�î^��îÀõ þ ï4��ï'� ÿ
-�ð�ï'&C÷�îÀò�ý�î�� þ í�î��9
\ï�ð�õ þ ó þ � þ î þ í�î7ï�ð�÷��Oî�� ö�
þ ð���ôsî ÿ ó�ñ
Àö¡÷ ÿ îÀõ���÷ þ õ�ö���öÀó`÷=ö��'÷�î #uüVð öÀò�ò�ó þ ó�ï�ð��b&�î7í�ï"ø�î þ í�ö þ�þ í î
ô-öÀó=ð �`îÀö þ � ÿ îÀõnï � þ í�î õ��'õ�
�îÀø þ ó���ó`÷Øó þ � ó=ð þ í�î ý�ó���îÀð
÷=ó=ôAó þ ó=ð�ýE
Àö�õÃî.õWï �Cõ�ôsö�÷`÷áöÀð�ò ÷Øö ÿ ý�î þ�ÿ ö¡ð�õR�"î ÿ õKî9�Kó�î¡÷Øò�õ
&sï ��÷=ò���înï'�'õKî ÿ ��ö���÷�îÝó=ð þ í�îÝôsîÀöÀõ�� ÿ îÀôsîÀð þ õ ï4�Ôò���ñ
ð�öÀôAó3
ò�ózî�÷�î�
þ�ÿ óI
5
\ï�ð�õ þ öÀð þ ï �^M��'öÀõVó=ñ�ï�ð�îÀñ�ò�ó�ôsîÀð�õIó�ï"ð�ö¡÷R�
í�� ò ÿ ï�ý�îÀð�ñ2��ï ð�ò�îÀò �`î ÿ�ÿ ï�î�÷�î�
þ�ÿ ó�
Àõ ô-ö þ î ÿ ó�ö¡÷Øõ ÷Øó2- î
5Òõ � Ð "�
� "b�7$;"�
8�pÈî��*#Aù/ûnöÀð�ò ù��7B
(Ω
_�\�jÒû
)öÀð�ò ó=ð þ í�înö���õIï ÿ ø þ ó�ï"ð õ�ø�î�
þ�ÿ �'ô_ôsîÀöÀõ�� ÿ îÀô^î.ð þ õ��`ï ÿ
\ ñ`ö�ý�ý ÿ î�ý�ö þ îÀõ � ù 6 ��ù��%!
(Ω
c \%j û
)
#D$sï'&sî���î ÿ � þ í�î.õÃî
ø ÿ ï'��÷�î.ôsõröÀð�ò þ í�î ÿ î�
�ï ð�õ�ó�ò�î ÿ ö þ ó�ï"ð ï �ÔõIï"ô^îÝö�ø�ø ÿ ï�ñ
�'ó�ô-ö þ îdöÀø�ø ÿ ï ö�
Àí�îÀõa� õÃî.ò î.ö ÿ ÷=ó�î ÿ �`ï ÿ öÀð�ö�÷���õVó=õWï4�Gî���ñ
ø�î ÿ ó=ô�îÀð þ ö¡÷kò�ö þ ö ÿ î�M���ó ÿ îCöCõKîÀø�ö ÿ ö þ îCõ þ � ò��L#
P î &-ó�õ�í þ ï þ í�öÀð%- " ÿ ï4�`îÀõ�õKï ÿ � #��.#�
´ï�ð�
�Kö¡÷���î.õ,öÀð�ò
" ÿ ï4�`îÀõ�õKï ÿ �b#9pÈó�
Àí þ î ÿ �`ï ÿ õ þ ó�ôA�'÷=ö þ ó�ð ý ò�ó�õR
���õ�õVó�ï�ð�õ�#
/# /#Gó�õnó=ð�ò�î�� þ îÀò þ ï�� ÿ #���ï�õKîÀø�í��,ï�
�ï'&Aõ�-��r�`ï ÿ
�ï ð þ ó=ð���ï'��õ^�Kó=ð�öÀð�
Àó=ö¡÷kõ�� ø�ø�ï ÿKþ #
Á�·*¥'·���© K»�¿�à'·*º/µK°�¹Àª�¼=¯�¿�¨�Ç/¬Ô½,·���¨I¼=¼=©�ÊV¿�� |\|/�k� �I��~���Ù�� ¿ ���Ü�Á`Â\Ä�Á�Þ�·
ÐÀ·Àà¡°"·�� © �ØK§*V© �=V®I¿ � �I��~����K�$Û�� ¿ � � � Ü�Á`Â\Ä � Þ�·�À·�ä�·Àä�ÌzK¹\¼zÅ"¿ � |�|/�����I�/~ · Ú çÀ¿ � Â�Ü�Á` � � Þ�· À·"!Í·"�C·���µ�æ�²IÅ"¿$¥�·"!�¨V®=²�¹�µK°"¿Î¨VÇ�¬E½,·�!�·uå�»�®=¨�°/¨�§,¿ � ����~��
#Æ�V� ·�$�%�¿ÀÐ ���ÀÁÎÜ�Á` � Á�Þ�·¾À·'&À·(�¡¨��=¯`V®=²�ã*© µV¯ ¨VÇ�¬ ä�·Aä�ÌzV¹/¼�Å�¿�� �I��~��)#*�V��·'* Ù/Ù ¿ /¾/Ä �Ü�Á` � ¾ÀÞ�·Ä�·+&.·�,,·�,,·$ä�K®=±�¿-,�·�.E·�æ�¨�¶/�ª�¿0/*·�1�·2.E·43�¹À© Ê=¶\Vª`¿Î¨�Ç�¬
��·5. ·5� © �=°\²/Ì�Ì`¿ � �I��~��É�K� $ Ù�6/Û ¿�ÁÎÜ�Á`Â�7 ÀÞ�·
� ·�/*·8��¹.. ªzª�K®'¨IÇ\¬01*·À¥'·Àº�°�®=²�µK±"¿ � �I��~��9#Æ�V� ·:* 6 ; ¿�Ð�7 7�Ü�ÁK 7� �Þ�·
7À·Àå*·81*·�Ñ�¼�ÊV¿Àå*·�/u·�Ñ�¯`K®�Ó/©�Ç¡¿8<u·�¥Í·8='²�®=K¶À© Ç"¿�¨�Ç�¬��Æ·Àå*·�º�ªz¨V¸/Ç�²�¸�¿
� ����~��>#������������=� ·¡ç é ¿�Á � � ´Ü�Á�Â� ��Þ�·ÂÀ·�&À·Èº�¼=²/ª�¯K/¿*å*·?��².. ¶/¶�K®̀ ¼�¿Î̈VÇ�¬@/Æ·2��¹.. ªzª�K®I¿*� �I��~��+#����À·�* Ú 6 ¿
��ÀÁKẤ ÜVÁÃÂ�Â�¾\Þ�·
A�� �;� �;� B(� �;��� ���L�����<D����O�:FHG��g�� C¡�¡�¢4� I��K@ML4�:B ��¢ ¢.@5A
Á � ·��È·u½*K®=¯`°\±�²7¨IÇ\¬�à'·�='®�²�±�°�§Æ¨�ª�Ê=±À©�©�¿ Ö �������V��×��K�����`�z�`~ ÙVÚ�Û ¿�¾/¾
Ü�ÁKÂ�Â\Ä�Þ�·
Á/Á�·��È·"½kK®=¯K°/±�².¿�à�· ='®=²�±�°\§*¨�ª�Ê=±À©�©�¿�¨VÇ�¬�à�·�<'K®=±�°\²/ª Å\¨V±�¿ ���-�A�
ÙVÚ ç ê'ÙIÚ�� ¿ �Ð�ÁuÜVÁÃÂ�Â�Ä/Þ�·
ÁKÐÀ·�&.·�寷�ä�ªz¨VÊ�µK¨V±�¿�å*·�º�·�à�·�ä�© ®=IÊV¿�¨VÇ�¬ ��·�æ�·�º�¨′ !�¨V®=®`K¼=²À¿����/×��z�
�����À���$��� *7* �À| · % %�¿ � 7 � Ü�ÁK 7/Ð�Þ�·
Á �À·�º�·�.7¨I¼=¨V®�¨�©�¨VÇ�¬�à'· ��¨I¼=Ê=¹�»\¨I®=¨\¿ �¡��� ����~������I�V� � � |À· Ú/Û ¿ �\Ä �7Ü�Á`Â�7 ÀÞ�·
Á À·5&À·8='Ç/²�Ê=¼�V®V¿ �¡�����}� * �����I�/~�· ; ; ¿ 7 \Ä�Ä,Ü�Á�Â� ��Þ�·ÁK¾À·�,,·�º�¹/¯K¹/¹�®=¨À¿'à'·'à�²�±.© °À© ®=²�¿Í¨�Ç�¬��*·���°\¼�¨À¿ ���I�/~�� #���� ·�* % ; ¿
�� � ´ÜVÁÃÂ� \Þ�·
¢.@ � A�� �;� �R�CB � �;��� ���b�����ED����3�%FHG����� C¡�¡�¢��JI��K@ML��%B ��¢
|