� ��������� ����������� ������� ��� �!��"$#%��&���'(���)�%*,+.-����0/21�3%����%*547686:9;=<?>7@BADCDE�@F;HGI<?>KJML�NPORQ(O?STQ(OVUPWXUPOVYPQ(Z\[]XOR^`_�ZaUFb(NPWc_KOedS\^IfhgiZajkQlWcUF_mOonRb(WcZaUqp�rPS\UPstPu ODWXUI_vORnRb(WXZaUqw�x�y�ZajF^zjP_)bl_vZa] u O|{~}���YPQ(Z\[]XOo^���Wcb(NqSTbl]cO�S\_mbKb���Z���Q(Za^5O�S\n�NI_mOonRb(WcZaU�x� Z`O�S\n�N�YPQ(Z\[]XOR^�ZaU�_vO�YiSTQ�STb(O|_vNPORORb(_�Za��YiSTY�ORQVS\UPsIb(jkQ(U�WcUIZaUP]c�.b(NPZa_vO?YPQ(Z\[]XOo^�_���Zaj��S\U�blb(ZzNS u O��\Q�S\sPOos�x���Q(WHb(OD��ZajkQ�US\^`ODS\UPsqYPQ(Z\[]cOo^0U~jP^�[�O�Q%ZaUqOoS\neNFYiS\�aO\x�l���7�����M  ¡B¢�Jlp£NPWX�aNP]H��OoUPORQ(�aO�b(WXn|YNPZ\b(ZaU$¤ ��S u Oo]cOoUP�\b(N ¥�¦lnoZa]c]XWcsPOo_%�lWHb(N§S\U¨Oo]XOon�bvQ(ZaUSTbVQ(Oo_mboxzp���bvORQ?bvNPO©noZa]c]XWc_vWXZaU§b(NPO©S\UP�a]XOzª.[�ORb���OoOoU«bvNPO©WcUPnoWXsPOoU�b�YNPZ\bvZaU«sPWcQvOonRb(WcZaU«S\UPsb(NPO t US\]ksPWHQ(OonRbvWXZaU2WX_h^`OoS\_vjkQ(Oos�x�¬�S\]XnojP]XSTb(O:b(NPO�n�NS\UP�aO�WXU�b(NPO���S u Oo]cOoUP�\b(N�r~­�¥B®�¥¯(°2¥WXU�bvORQ(^`_?Za��ªkr±��NPORQ(O.¥i¯�WX_?b(NPO`_vn�STbmb(ORQ(Oos²YNPZ\b(ZaU���S u Oo]cOoUP�\b(N�x³�´NSTb�WX_�YiSTQmb(WXnojP]XSTQ(]c�Q(Oo^`STQvµ\ST[]cO�ST[�Zajkb�bvNPWX_�Q(Oo_vjP]Hbe¶�l���7�����M  ¡�·�J¸¬�ZaUP_vWXsPO�QIb(NPO«Q(OR]�STb(W u WX_)b(WXn§��S u O�Z\Y�ORQ(STb(Z\QqQ(OR_mbvQ(WcnRb(Oos¹b(ZºZaUPO«_mYiSTb(W�S\]sPWX^�OoUP_vWXZaUqS\UPsqb(Wc^`O\f

»©¼ ® ½ ¼½ ¤¿¾�Àv¦ ¼ ° ½ ¼½MÁ ¼�ÂghWXUPsFb(NPO?^�Za_mbK�aOoUPORQ�S\]ÃnoZ�Z\QvsPWXUSTb(OVbvQ�S\UP_v�=Z\Qv^³STb(WXZaU.Za�±bvNPO��=Z\Q(^

Á ¯ ®¹ÄiÅ(Å�ÆÇÄiÅ�È�¾eÀ ¾eÀ ¯ ®¹Ä~È�Å Á ÆÇÄÉÈ(È�¾eÀb(NSTb�]XO�S u Oo_ » ¼ WXU u STQ(WXS\UÉbox�l���7�����M  ¡�Ê�J|L���ZBQ(Oo]XSTb(W u WX_mbvWXn�YiSTQvb(Wcno]XOo_DZa�:^`S\_v_VË Å�S\UPs�Ë ¼ noZa]X]cWXsPOzNPO�S\s«ZaU¨�lWHb(N_mY�OoOosP_�Ì~Å�S\UPs`Ì ¼ S\_�_vOoOoU�[É��S?_mb�STbvWXZaUSTQv�2Z\[_vO�Q u ORQ�x�p���b(ORQ8noZa]c]XWXsPWcUP�DbvNPOR���=Z\Qv^,SD_vWXUP�a]cOYiSTQvb(Wcno]XO�Za��^³S\_m_%Í ^`Z u WcUP���lWHb(N§_mY�OROos�ÎÏQ(OR]�STb(W u O�b(Z³b(NPO�Z\[_vORQ u ORQ�xVghWXUPs§Í S\UPs«ÎWXU`bvORQ(^`_8Za�Ã˨Å�r�Ë ¼ r�Ì~Å�rkS\UPsBÌ ¼ xh}�_�Wcb8Y�Za_v_mWc[]XO��=Z\Q:b(NPO��=Z\Q(^`Oos`YiSTQmb(WXno]cOlb(Z�[�OVS�YNPZ\b(ZaUWX�±UPOoWcbvNPORQ%˨Å%Z\Q�Ë ¼ STQ(O?ÐRORQ(Zɶ�l���7�����M 5¡DÑ8J�¬�ZaUP_vWXsPO�Q�S�YiSTQmb(WXno]cO�Za�^³S\_v_7Ë,Za_vnoWc]X]�STbvWXUP��NSTQ(^`ZaUPWcn�S\]X]H�?[�ORb���OoOoU.°�ÄBÒÁ ÒÓÄMxhÔ�_mO�b(NPOl�´WX]c_vZaU�Õ�{�Za^�^`ORQ(�=Oo]cs�Ö~jS\UÉb(WcÐ�STb(WcZaU�Q(jP]XOKb(Z|_vNPZ×�Ób(NSTb�ØlÙ�®¸Ú�ÛPÜ©��NPORQ(OØ%Ù.WX_Kb(NPO?ORUPORQ(�\�FZa��b(NPO�Ú�bvNq_mb�STbvO\r�ÛqWc_�Ý8]�S\UPn(µ�Þ�_�nRZaUP_mb�S\U�b�r�S\UPs�ÜFWX_�b(NPO���QvOoÖ�jPOoUPn��IZa�b(NPO?Za_vnoWX]c]�STb(Z\Qox�l���7�����M  ß�¢�JK¬�ZaUP_vWcsPORQ�b(NPO|^�Z\b(WXZaUIZa��S©��Q(OoO�UPZaU�ÕàQvOo]�STb(W u Wc_mb(WXnDYiSTQvb(Wcno]XO|Za�h^`S\_v_%ËáWXUZaUPO�sPWc^`OoUP_vWcZaU�x�â±ORblWcb(_���S u O?�=jPUPn�b(WXZaU§STb�À�®¹ã2[�Oä ¤ Á�å ã�¦�®çæ´èé�ê ¼Éë�ìí:î\ïlðmñò ñ Â

� [Pb�S\WXU S\U²O�d�YPQ(Oo_v_vWcZaU �=Z\Q�b(NPO`YiSTQvb(Wcno]XO\Þ�_���S u O`�=jPUPnRb(WcZaU ä ¤ Á±å Àm¦D�=Z\QzS\]X]�À��,ãkx�{�NPZ×�b(NSTb��=Z\Q%S\]c]�À�r�� Á�� ®¹ã�S\UPsqbvNSTbKbvNPO�jPUPnoORQmb�S\WXU�b��qWcU Á WX_K�aW u ORUq[~�­ Á ¤ Àm¦�® ê ¼� Æ æ�Û�ÀË ê ë ¼ Â

�l���7�����M  ßK·hJ|L���ZF_mYWXU�Õ Å¼ YiSTQvb(Wcno]XOR_?�=Z\Q(^ SFnRZa^�Y�Za_vWHb(O2_m�k_)b(Oo^IxzÝ�STQvb(Wcno]XO��Wc_�WcU�S\UOoWc�aOoUP_mb�STb(OVZa� �7Å��%® Æ��� è S\UPsBYiSTQvb(Wcno]XO è WX_�WXUFS\U.OoWX�aOoUP_)b�STb(OVZa��� ¼�� ® Æ��� è xh�´NSTb�Wc_b(NPOVYPQ(Z\[iST[WX]cWcb��³b(NSTb�S©^�O�S\_vjkQ(Oo^�OoUÉbKZa�Ãb(NPODbvZ\b�S\]�_mYWcUq�lWc]X]��aW u O�S u S\]XjPODZa��ÐoORQvZɶqy:Zaj^³S��³�lWc_vNFbvZ�jkb(Wc]XWXÐRODb(NPO?Ý�S\jP]XW�_mYWcUI^³STbvQvWXnoOo_Rfê � ®�� ã ã�� å ê�� ®�� ã °��� ã�� å ê � ®�� ãã °��� Â

�l���7�����M  ßKÊhJKLKNPO�WXU�b(ORQ�S\nRbvWXZaU �VS\^`Wc]cb(ZaUPWXS\U"! S\^`ZaUP�zb(NkQ(OoO�_mb�STb(Oo_$#% � r&# è � r�S\UPs'# ( �WX_K�aW u ORUF[É�)! ®+* ã Ä ÄÄ ã ÄÄ Ä ã-,

ghWXUPsFb(NPO?ORWX�aOoU u S\]cjPOo_%S\UPsFb(NPO�noZ\QmQ(Oo_mY�ZaUPsPWXUP�`Z\QvbvNPZa�aZaUS\]ÃORWX�aOoUP�=jPUPnRbvWXZaUP_lZa� ! x�l���7�����M  ß�Ñ�JK¬�ZaUP_vWXsPORQ�S\UIWXU t UPWHb(O�_vÖ~jSTQ(O���OR]X]�Y�Z\b(OoU�b(WXS\]���WcbvN�S�_v]cWX�aN�blb�STY�ORQ�STb�b(NPO[�Z\bmb(Za^If Î!®�./ # Á # �=Z\Q$# Á #10 ÄÎ!®32 �=Z\Q4# Á #1�ÓÄ Â¬�S\]XnojP]XSTb(ODbvNPO?�\Q(ZajPUPsq_mb�STb(ODOoUPORQ(�\�FZa�±bvNPWX_�YiSTQmb(WXno]cODbvZ t Q(_)b%Z\Q(sPORQlWXU65ox�l���7�����M  ß87�J8p�YiSTQvb(Wcno]XODZa��^`S\_v_�Ë Wc_�WcUPnoWXsPOoU�b%��QvZa^0b(NPOD]XOR��blZaU�Sz_mb(ORYqY�Z\b(OoU�b(WXS\]¿fγ¤ Á ¦�® ã å Á 0ÓãÎ�9 åáÁ;: ã��NPORQ(O�Î<9=�Óãkx�}��Ãb(NPO�YiSTQvb(WXnR]XO�WX_��=ZajPUPs.b(Z�[�O�Q(O?>Oon�b(OosF��WcbvNqS|YPQ(Z\[iST[Wc]XWcb��³Za�-�@BAkr t UPsb(NPO?WXUPnoWcsPOoUÉb�µkWcUPORb(WXnVOoUPO�Q(�\�qWcUFb(O�Q(^`_KZa�hÎ<9\x

� ��������� ����������� ������� ��� �!��"$#%��&���'(���)�%*,+.-����0/��%��&�&³476�6�9;=<?>7@BADCDE�@F;HGI<?>KJML�NPORQ(O?STQ(OVUPWXUPOVYPQ(Z\[]XOR^`_�ZaUFb(NPWc_KOedS\^IfhgiZajkQlWcUF_mOonRb(WcZaUqp�rPS\UPstPu OKWXUz_vOonRb(WcZaU`w�xTy:Zaj©^2jP_mb�_vZa] u O%{~}�� YPQ(Z\[]XOo^�_�sPZaWcUP�|STb�]XOoS\_mb�b���Z���Q(Za^ O�S\n�N`_mOonRb(WcZaU�x� Z`O�S\n�N�YPQ(Z\[]XOR^�ZaU�_vO�YiSTQ�STb(O|_vNPORORb(_�Za��YiSTY�ORQVS\UPsIb(jkQ(U�WcUIZaUP]c�.b(NPZa_vO?YPQ(Z\[]XOo^�_���Zaj��S\U�blb(ZzNS u O��\Q�S\sPOos�x���Q(WHb(OD��ZajkQ�US\^`ODS\UPsqYPQ(Z\[]cOo^0U~jP^�[�O�Q%ZaUqOoS\neNFYiS\�aO\x�l���7�����M �¡B¢�Jhp´YiSTQvb(Wcno]XO���Wcb(N2Q(Oo_)b:^³S\_m_�Ë"9�NS\_8S u OR]XZ�nRWcb��.¤ Ì � å ã å ã�¦�WXU©bvNPO|¤ Á±å���å��kå Àm¦noZ~Z\Q(sPWXUSTb(OK��Q�S\^`OTx7��Q(WcbvO�sPZ×�lU©b(NPOlO�d�YPQ(Oo_v_mWXZaUP_:�=Z\Q�b(NPOKYiSTQvb(WXnR]XO\Þ�_8OoUPO�Q(�\�`S\UPs©^`Za^`OoU�Õb(jP^ WcU�b(NPWX_���Q�S\^�O\x%âÃORb�b(NPO�noZ~Z\Q(sPWXUSTb(O���Q(S\^`O\r�¤ Á ¯ å�� ¯ å�� ¯ å À ¯ ¦er�[�O|Q(OR]�STb(Oos�b(Z ¤ Á±å���å��kå Àm¦[É� S¨âÃZ\Q(OoU�b(ÐB[�Z�Za_)b©Za����®�Ì �\¾³S\]XZaUP��b(NPO Á ÕàsPWcQ(Oon�b(WXZaU�x�ghWcUPs bvNPO.YiSTQvbvWXno]cO\Þ _2OoUPORQv�\�S\UPsB^`Za^`OoU�b(jP^£WXU³b(NPO�¤ Á ¯ å�� ¯ å�� ¯ å À�¯X¦8��Q�S\^`O%S\UPs.OedkYPQ(Oo_m_�b(NPOo_vO%b���Z�Ö~jS\UÉb(WHb(WXOR_�_vZa]XOR]c�©WXUb(ORQv^`_�Za�±Ì å ¾VS\UPs.b(NPO�noZ\QmQ(Oo_mY�ZaUPsPWXUP�`Ö~jS\UÉb(WHb(WXOR_�WcUFb(NPO`¤ Á�å��Må��kå Àm¦:��Q�S\^`O\x�l���7�����M  ¡�·�J?p,YWXZaU«STb?QvOo_mb?sPOon�S��k_�WcUÉbvZISF^2jPZaU¨Y]XjP_?SFUPOojkbvQvWXUPZPx�ghWcUPs b(NPO2Q(Oo_)bmÕ��Q�S\^`OD_mY�OoOos§Za��b(NPO?^zjPZaUFWXUqbvORQ(^`_KZa�7b(NPO�^2jPZaUqS\UPsIYWXZaUI^³S\_m_vOo_�¤ Ë�FS\UPsIË�i¦KS\UPsb(NPO?_mY�OoORsIZa�7]XWc�aNÉb�¾ x�y:Zajqn�S\UIS\_v_vjP^`OVb(NSTbKb(NPO?UPORjkbvQ(WXUPZ�WX_K^³S\_v_v]cOo_v_ox�l���7�����M  ¡|Ê�J Ý�Q(Z u O¨b(NPO¨no]�S\_v_mWXn�S\]�Q(Oo]XSTb(WXZaU´�=Z\QFbvNPO�S u ORQ�S\�aO¨OoUPORQ(�\�´Za�?[]XS\n�µ~[�Z~sk�Q�S\sPWXSTb(WXZaU�f �Ø!® ����x7Ô�_mO:S��¨S dk��Oo]c]HÕ�w�Za]Hb(Ðo^³S\UPUVOoUPO�Q(�\�DsPWX_mbvQvWc[jkb(WcZaU��=jPUPnRb(WcZaU�f��h¤ Øz¦h®� î ï�������� x�l���7�����M  ¡DÑ8J.})U b(NPWX_2YPQ(Z\[]XOR^ ��Zaj$STQ(O.�aZaWXUP�§b(Z«sPORbvORQ(^`WcUPO.b(NPOBYNS\_vOqS\UPsÇ�\Q(ZajkYu Oo]cZ�noWHb���rÌ����BS\UPsIÌ���rQ(Oo_)Y�OonRbvW u Oo]H��riZa��SzQ(Oo]�STbvW u Wc_mb(WcnVYiSTQmb(WXno]cO\x�ghWcQv_mb�rP��ZajIUPOoOos�b(Z©_vNPZ×�b(NSTb

Ì����|® � ¤!��¦� ® Ø.¤#"�¦

" S\UPs Ì��?®%$&� ¤!��¦$ �

®%$ Ø.¤'"�¦$ "

L�NPO t Q(_mb%OoÖ�jS\]cWcb��BWXUq[�Z\b(NIZa��bvNPOo_vO|OoÖ~jSTb(WXZaUP_�n�S\Uq[�O|S\_v_vjP^�Oos�x��¸WHb(Nqb(NPOR_vO?Q(OR]�STb(WcZaU�Õ_vNPWHY_or�sPORbvORQ(^`WcUPO�Ì(�)�BS\UPs�Ì*�~x���Q(WHb(O?bvNPOo_vO|WcUqb(ORQ(^�_�Za�7b(NPO?Z\Q(sPWXUSTQm�FYiSTQvbvWXno]cO u Oo]cZ�noWHb��Ìix,+�Oon�S\]X]�b(NSTbKb(NPO?OoUPORQ(�\�.Za�hS2Q(Oo]�STbvW u Wc_mb(Wcn%YiSTQvb(WXnR]XODWX_�Q(Oo]�STbvOosqb(Z©Wcb(_�^`Za^`ORUÉb(jP^�[É�Ø ¼ ®-" ¼ ¾ ¼ Æ$Ë ¼ ¾ È

S\UPsqb(NSTbKb(NPOVQ(Oo]XSTb(W u WX_mb(WcnV^`Za^`OoU�b(jP^�WX_K�aW u ORUF[É�".® Ë.Ì. K°�Ì ¼ �\¾ ¼ Â

�l���7�����M  ß�¢�J:¬�ZaUP_vWcsPORQKS�UPZaU�ÕàQ(Oo]XSTb(W u WX_mbvWXnlYiSTQvbvWXno]cO%Za��^³S\_v_�Ë�jPUPsPO�QKbvNPO�WXU >jPOoUPnoODZa�b(NPODY�Z\b(OoU�b(WXS\]¿rγ¤ Á ¦�®!° �Û ¼Ë���� ¤ Á ¦�Æ � ¤ Á Æ $ ¦�� Â

{�NPZ×�Ïb(NSTb?b(NPORQ(OBSTQ(O©b���ZI[�ZajPUPs²_mb�STbvOo_|WX� $ � BS\UPs�b(NSTb|bvNPORQ(O³WX_?ZaUP]c�«ZaUPO�[�ZajPUPs_mb�STbvO?Wc� $ 03ax�l���7�����M  ß�·�J8Ý8Q(Z u O��8NkQ(OoUP�=Oo_)b�Þ _lbvNPOoZ\Q(Oo^0WXUFZaUPO?sPWc^`OoUP_vWcZaU�f$ �#" �$ À ® �)° $ Î

$ Á � Â+lORn�S\]X]�b(NSTb ".®�� ��� � x�l���7�����M  ßKÊ�JKp YiSTQvb(Wcno]XO�WXU¨S\U§WXU t UPWcbvO�_vÖ�jSTQ(O?��Oo]X]�Za����WXskbvN§ÄqNS\_%b(NPO�WXUPWHb(W�S\]���S u O�=jPUPnRb(WcZaU ä ¤ Á�å ã�¦�®� Á ¤¿Ä|° Á ¦ ÂS�¦���Z\Q(^³S\]XWcÐoO ä ¤ Á�å ã�¦ex[M¦�ghWXUPs ä ¤ Á±å Àv¦ex +�Oon�S\]c]�bvNSTblnoZa^©Y]XORb(ORUPOo_v_�bvOo]X]c_�jP_

ä ¤ Á±å ã�¦�®�� Ù ¾�Ù � Ù�¤ Á ¦S\UPs ¾eÙ`® � � Ù # ä � Â�l���7�����M  ß�Ñ8JK{~b�STQvb(WcUP���lWHb(Nqb(NPO�n�S\UPZaUPWXn�S\]±noZa^`^2jkb�STb(WcZaU.QvOo]�STb(WcZaUP_l�=Z\Q%Y�Za_vWcbvWXZaU�S\UPs^`Za^`ORUÉb(jP^ᤠ� Á±å " � �àr � �Må " � �àr � �kå "<���¿riO�b(n\xXxcx ¦�rk��Z\QvµBZajkblb(NPODnoZa^`^2jkb�STbvZ\Q(_

��� � å�� � ��� � å "���� å�=Z\Q � ® Á�å��Må�� x�Ô�_mODb(NPOR_vO?Q(OR_vjP]cb(_Kb(Z�Z\[Pb(S\WXU��� � å � ��� å��NPORQ(O?��Zaj.�lWc]X]�Q(ORn�S\]X] � � ® � "<�K° � " � r � � ® � " � ° Á "<�ariS\UPs � �l® Á " � ° � "<�ax

�l���7�����M  ß87�J?L���ZqWXsPORUÉb(Wcn�S\]��=ORQ(^`WcZaUP_DSTQ(OzY]XS\noOos�WcU«S\U WXU t UPWHb(O©_mÖ�jSTQ(Oz��Oo]X] x2LKNPO��WXU�b(ORQ(S\nRb���O�STµk]H�.��WcbvNqO�S\n�NIZ\b(NPO�Q�b(NkQvZajP�aNqb(NPOVY�Z\b(OoU�b(WXS\]ÎB¤ Á Å åvÁ ¼ ¦�®!°�ÄkÎ�9 � ¤ Á Å�° Á ¼ ¦ å��NPORQ(O�Î�9?Wc_lSznoZaUP_mb�S\U�bl��Wcb(NFsPWX^`ORUP_vWXZaUP_KZa�7OoUPORQv�\�qS\UPsIÄBWc_KbvNPO?��WXskb(NFZa�±b(NPOD��Oo]X] xS�¦%})�aUPZ\QvWXUP�qb(NPO©WXU�b(ORQ�S\n�b(WXZaU«[�ORb���OoOoU²b(NPOzYiSTQvb(Wcno]XOo_Rr t UPs�b(NPO©�\Q(ZajPUPs²_mb�STb(O©S\UPs���S u O�=jPUPnRb(WcZaUP_�S\UPsqORUPORQ(�\��x[M¦�Ô�_vO t Q(_mbmÕàZ\Q(sPORQKY�ORQvbvjkQv[iSTb(WXZaU.b(NPOoZ\Qv�³bvZ©Oo_)b(WX^³STbvO%b(NPO?O���OonRb�Za��b(NPODYiSTQmb(WXno]cO�ÕàYiSTQmb(WXno]cOWXU�b(ORQ(S\nRb(WXZaU�ZaUIbvNPO�OoUPORQ(�\�qZa�7b(NPO��\Q(ZajPUPs�_mb�STbvO\x ��dkY]�S\WcUINPZ×����Zaj�noZajP]cs§NS u O|�aZ\bmb(OoUb(NPWc_lS\UP_m��ORQl��Wcb(NPZajkb�jP_vWcUP��Y�ORQmb(jkQv[iSTb(WcZaUFb(NPOoZ\Qm��x

� ��������� ����������� ������� ��� �!��"$#%��&���'(���)�%*,+.-����0/��%��&�&³476�6��

;=<?>7@BADCDE�@F;HGI<?>KJML�NPORQ(O?STQ(OVUPWXUPOVYPQ(Z\[]XOR^`_�ZaUFb(NPWc_KOedS\^IfhgiZajkQlWcUF_mOonRb(WcZaUqp�rPS\UPstPu ODWXUI_vORnRb(WXZaUqw�x�y�ZajF^zjP_)bl_vZa] u O|{~}���YPQ(Z\[]XOo^���Wcb(NqSTbl]cO�S\_mbKb���Z���Q(Za^5O�S\n�NI_mOonRb(WcZaU�x� Z`O�S\n�N�YPQ(Z\[]XOR^�ZaU�_vO�YiSTQ�STb(O|_vNPORORb(_�Za��YiSTY�ORQVS\UPsIb(jkQ(U�WcUIZaUP]c�.b(NPZa_vO?YPQ(Z\[]XOo^�_���Zaj��S\U�blb(ZzNS u O��\Q�S\sPOos�x���Q(WHb(OD��ZajkQ�US\^`ODS\UPsqYPQ(Z\[]cOo^0U~jP^�[�O�Q%ZaUqOoS\neNFYiS\�aO\x�l���7�����M  ¡³¢�J�}�UIS�b���ZTÕà[�Z�sk�._vn�STbvbvORQ(WXUP�zO u OoU�b�r Æ���� � Æ���rPWcbKWX_KnoZaU u OoUPWXORUÉbKb(ZWXU�bvQ(Z~sPjPnoO �¨S\UPsPOo]c_mb�S\^ u STQvW�ST[]XOR_� ® ¤'"qÆ "�:¦ ¼¾ ¼ å À�® ¤#"�° "�:¦ ¼¾ ¼ å ® ¤#"¨° "��¦ ¼¾ ¼ å

��NPORQ(O�"�r "�Kr "� S\UPs "�,STQvO©b(NPO`�=ZajkQ|^`Za^`OoU�b�SFZa� �r��³r � rhS\UPs�� Q(Oo_mY�OonRb(W u Oo]c��xL�NPO³_vÖ�jSTQ(OR_�WXU�b(NPO`UÉjP^`ORQ(STb(Z\Q(_2STQ(O�bvNPO�Q(Oo]XSTb(W u WX_mb(Wcn�S\]X]H�¨WXU u STQ(WXS\UÉb|sPZ\b|YPQ(Z�sPjPn�b�xIâÃORbË��riË��lrË�� S\UPsqË���sPOoUPZ\b(ODb(NPOVQ(Oo_mbl^³S\_v_vOo_KZa� �r��³r � S\UPs���Q(OR_mY�Oon�b(W u Oo]H��xS�¦K{�NPZ � b(NSTb � Æ À�Æ ®¹Ë ¼ Æ$Ë ¼� Æ$Ë ¼� Æ$Ë ¼� x[M¦K{�NPZ � b(NSTbKb(NPO�nROoUÉbvORQ%Za�±^³S\_v_KOoUPORQ(�\�FZa� �WX_�������� ñ � ï � ñ �� ��ñ¼"! �

xnצK{�NPZ � b(NSTbKb(NPO?]�ST[IORUPORQ(�\�.Za�7p�¤ w¸STbKQ(Oo_mbe¦�WX_ �#� ï � ñ � ï � ñ �$�%��ñ¼ � � xsi¦K{�NPZ � b(NSTbKb(NPO?bvZ\b�S\]ì�ORUÉb(O�Q%Za���¨S\_v_�OoUPORQ(�\�FWX_'& � ¾ ¼ x�l���7�����M  ¡�·�J8Ô�_vO�b(NPO?w�ZaNkQ�Ö~jS\UÉbvWXÐ�STb(WcZaU³Q(jP]XOD�=Z\Q�S\UP�ajP]�STQK^`Za^`OoU�b(jP^ ¤ � ®¸Ë.Ì)(�®Ú �ÛP¦�bvZ©n�S\]cnojP]�STbvODb(NPODOoUPORQv�\�B]XO u Oo]c_�Za��SzNSTQv^`ZaUPWXnVZa_vnoWc]X]XSTb(Z\Q�x�¬�ZaUP_vWXsPORQlnoWcQvnojP]�STQKZ\Qv[WHb(_ZaUP]c��x �VSTQv^`ZaUPWXn`Za_mnoWX]c]�STb(Z\Q(_�NS u OBS�noOoU�bvQ�S\]��=Z\QvnoOB�aW u OoU²[É� ��Z�Z\µ�O\Þ�_�]�S���f+* ® ° �)(~x��Q(WHb(OV��ZajkQ%S\UP_m��ORQ%WcUqb(ORQ(^�_KZa�±b(NPO�S\UP�ajP]XSTQl��QvOoÖ�jPOoUPn�� � ® . � �T˨x�l���7�����M  ¡�Ê�J�pÇ_vZa]cWXs|nR��]XWcUPsPORQ���Wcb(NDQ�S\sPWXjP_-,©r sPOoU�Õ_vWHb��/.�S\UPs©ST[_vZ\QvYPb(WcZaU©noZ~O10`nRWXOoU�b32«¤ WcU2jPUPWcb(_�Za�MSTQ(O�SZ u ORQ�^³S\_v_(¦VWc_�[�OoWcUP�IWX^`S\�aOos«[É� �%Õ¿Q�S×��_ox.LKNPO�WXUPnoW ÕsPOoU�b�WXU�b(OoUP_mWcb��«Wc_54 9\x � O�Q(W u O³SF�=Z\Q(^2jP]�SF�=Z\Q?b(NPO`WcU�Õb(OoUP_mWcb��64�¤ Á ¦�WXUqbvNPO?Wc^³S\�aO�Y]�S\UPO\x RI

x

µ

I(x)

�l���7�����M  ¡DÑ8J?{�NPZ×� b(NSTb�bvNPOz_)Y�OoOos Za�:S\U¨Oo]cOonRbvQ(ZaU¨WXU§b(ORQv^`_�Za� �$n�S\U§[�O2O�dkYPQvOo_v_vOosS\_� ®¸_vWcUlSTQ(nRnoZa_�

7 �TË�8hÆ � å

��NPORQ(O 7 Wc_KbvNPO?µkWcUPORb(Wcn?OoUPO�Q(�\�FS\UPsIË�8%Wc_Kb(NPO?Oo]XORnRbvQ(ZaUI^`S\_v_ox�l���7�����M  ß?¢�J8p�YiSTQvb(Wcno]XODZa��^`S\_v_�Ë Wc_�WcUFb(NPO?_mb�STb(Oä ¤ Á±å Àv¦�®� î ï /�� � � � ñ � �� � � ��� å��NPORQ(O !S\UPs�ÄBSTQ(OVY�Za_vWHb(W u OVQ(O�S\]�noZaUP_mb(S\UÉb(_RxS�¦���Z\Q(^³S\]XWcÐoO%b(NPOD��S u OD�=jPUPnRb(WcZaU[M¦�gMZ\QÃ�lNSTbÃY�Z\bvOoUÉbvW�S\]�OoUPORQ(�\�D�=jPUPnRbvWXZaUzγ¤ Á ¦�sPZ~Oo_ ä _vSTb(WX_v���Vb(NPO�{�n�NkQ(Z�sPWcUP�aORQ7OoÖ~jSTb(WXZaUi¶nצhghWXUPszb(NPOljPUPnoO�Qvb�S\WXU�b(WcOo_:WcU©b(NPO�Y�Za_vWHb(WXZaU�S\UPs�^`Za^�OoUÉbvjP^Ir ê � S\UPs ê �rÉS\UPs`_mOoOlWc�Mb(NPOoWHQYPQ(Z~sPjPnRb%Wc_�noZaUP_mWX_mb(ORUÉb��lWHb(NFb(NPO?jPUPnoORQvb(S\WXU�b��.YPQ(WcUPnoWcY]cO\x�l���7�����M  ßK·�J�¬�ZaUP_mWXsPORQDS©[�O�S\s�Za�h^³S\_v_�ËábvNSTb%_v]cWXsPOo_�ZaU�S`��QvWXnRb(WcZaUP]XOo_m_l�lWHQ(O?QvWXUP�³Za�noWHQ(nojP^`�=ORQvOoUPnoODÄMx�L�NPWX_�Wc_�S\]X^`Za_)b�]XWHµ�O�S|��Q(OoO%YiSTQmb(WXno]cO%ZaUP]c�©b(NSTb��V¤ Á ¦�®��V¤ Á Æ Ä�¦�x�ghWXUPsb(NPO?_mb�STb(WcZaUSTQv�B_mb�STb(OR_lS\UPsqbvNPO?noZ\QmQ(Oo_mY�ZaUPsPWXUP�³OoUPORQv�aWXOo_ox � WX_vnojP_m_%S\U��FsPOR�aOoUPORQ�S\noWcOo_ox�l���7�����M  ßKÊhJ�¬�ZaUP_vWcsPORQ�S ( Õ � u ORnRb(Z\QD_mYiS\noO©_mYiS\UPUPOos�[É�«S\U Z\Qvb(NPZaUPZ\Q(^`S\]7[iS\_vWX_�#% � r# è � rS\UPs # ( � x�DORb4# � S\UPs # � � STQ(O��aW u OoUF[É�# � ® �?#% � ° �?# è � ° è # ( �S\UPs # � � ® �?#% � Æ è # è � ÂS�¦K¬�ZaUP_mbmQ(jPnRb��� #ÉS\UPs � �8#ÉWXU.b(ORQ(^`_KZa�±bvNPO�sPjPOo]�[iS\_vWX_ � # r � è # rS\UPs � ( # x[M¦�ghWXUPs ���# � � riS\UPs � �8# � riS\UPsInoZaU t Q(^ �� # � � ® � �8# ��l���7�����M  ß�Ñ8J�p�U�Oo]cOonRbvQvZaU�WX_�STb?Q(OR_mb|WcU²SqjPUPWX�=Z\Qv^ ^³S\�aUPO�b(WXn t Oo]Xs� � ® � � )�ÃÆ �=� )�¤ � � ��� �=�צ�x8LKNPO��VS\^`WX]Hb(ZaUPW�S\UBWX_! ® !498Æ ! ¯ ® î � �Ë � � Æ î � �Ë � � ÂghWXUPsFb(NPO t Qv_mbmÕ�Z\QvsPORQlY�ORQvb(jkQv[iSTbvWXZaUInoZ\QmQ(OonRb(WcZaUP_lZa�Ãb(NPO�Oo]cOonRbvQvZaUF��S u O��=jPUPnRbvWXZaUP_ox�l���7�����M  ß87�J�¬�ZaUP_vWcsPORQlbvNPO?ZaUPO?sPWX^`OoUP_mWXZaUS\]�Y�Z\b(ORUÉb(WXS\]

γ¤ Á ¦�®�� ã �=Z\Q Á 0ÓãÎ�9 �=Z\Q�ã�0 Á 0ÓÄã �=Z\Q Á �ÓÄ å��Wcb(NIÎ<9=�´ãkx�p�_v_mjP^`WXUP��S\UqWXUPnRWXsPOoU�b%_mbmQ(O�S\^0Za�±YiSTQvb(Wcno]XOR_l��Q(Za^�b(NPO?� STQl]cOo��b�¤ Á � ° 2º¦��Wcb(N?SK^�ZaUPZ�n�NkQ(Za^³STb(Wcn7OoUPORQv�\�?Za�ÉØ!® �Û � S\UPs?S�nojkQvQ(OoU�b7Za�)4z® ar Z\[Pb�S\WXU?S\U?OedkYPQ(Oo_m_vWXZaU�=Z\Qhb(NPO:QvO?>OonRb(ORs©nojkQmQ(OoU�b3, S\UPs�b(NPO�bvQ(S\UP_v^`Wcbmb(Oos|nojkQmQ(OoU�b���xhp�]c_vZ�Z\[Pb�S\WXU2S\U�OedkYPQ(Oo_m_vWXZaU�=Z\Q%b(NPO�nojkQvQ(OoU�bVWXUIbvNPO�Q(Oo�aWcZaU§ã 0 Á 0¹Äix���Z\QvµFbvNPOo_vO�Zajkb��=Z\Q%bvNPO�b���Z`n�S\_vOR_or�Ø �¸Î<9S\UPsIØ 0 Î<9\x � WX_vnojP_m_�b(NPODYNÉ��_vWXn�S\]�WcUÉb(O�QvYPQ(ORb�STbvWXZaUqZa�±��ZajkQ�Q(OR_vjP]cb(_Rx

� ��������� ����������� ������� ��� �!��"$#%��&���'(���)�%*,+.-����0/��%��&�&³476�6��;=<?>7@BADCDE�@F;HGI<?>KJML�NPORQ(O?STQ(OVUPWXUPOVYPQ(Z\[]XOR^`_�ZaUFb(NPWc_KOedS\^IfhgiZajkQlWcUF_mOonRb(WcZaUqp�rPS\UPstPu ODWXUI_vORnRb(WXZaUqw�x�y�ZajF^zjP_)bl_vZa] u O|{~}���YPQ(Z\[]XOo^���Wcb(NqSTbl]cO�S\_mbKb���Z���Q(Za^5O�S\n�NI_mOonRb(WcZaU�x� Z`O�S\n�N�YPQ(Z\[]XOR^�ZaU�_vO�YiSTQ�STb(O|_vNPORORb(_�Za��YiSTY�ORQVS\UPsIb(jkQ(U�WcUIZaUP]c�.b(NPZa_vO?YPQ(Z\[]XOo^�_���Zaj��S\U�blb(ZzNS u O��\Q�S\sPOos�x���Q(WHb(OD��ZajkQ�US\^`ODS\UPsqYPQ(Z\[]cOo^0U~jP^�[�O�Q%ZaUqOoS\neNFYiS\�aO\x�l���7�����M  ¡B¢�J2L�NPO�OoUPORQv�\��Za��SFYNPZ\bvZaU�WX_?�aW u OoU«[É� Ø�®0ÛPÜ�S\UPs²Wcbv_?^`Za^�OoUÉbvjP^áWc_�aW u ORU`[~� ".® ÛPÜ �\¾|¤ Z\Q�"B®¸Ø _vORbmb(WXUP�2¾K® ¦�r��lNPO�Q(ODÜ`WX_8bvNPO���Q(OoÖ�jPORUPnR�³Za��b(NPO%OR^`WcbvbvOos]XWc�aNÉboxS�¦ÃÔ�_mO�b(NPO�âÃZ\Q(OoU�b(Ð�bvQ�S\UP_v�=Z\Qv^³STb(WXZaU�Za�kb(NPO�OoUPORQv�\��b(Z t UPs�b(NPO���QvOoÖ�jPOoUPn���rÉÜ�¯àr\Za�kb(NPO�]cWX�aN�bZ\[_vORQ u Oos [É�²SqQvOonoOoW u ORQ2^`Z u WXUP��������� ����������������zb(NPO³_vZajkQvnoO\xq�´NSTb2WX_�bvNPO³��Q(OoÖ�jPORUPnR���NPOoUqb(NPODQ(OonoOoW u ORQ%Wc_�^�Z u WXUP�����������������������D��Q(Za^�b(NPO?_vZajkQ(noOTx[M¦ ��Z×�¸noZaUP_mWXsPORQ�b(NPO�n�S\_mOV��NPORQ(O%bvNPO�]XWX�aN�b�WX_:Oo^`Wcbmb(OosBSTbKS\U.S\UP�a]XO%Za�ê�WXU`bvNPO���Q�S\^`O�Za�b(NPO?_vZajkQ(noO�S\UPsIZ\[_vORQ u ORs�STb%S\UqS\UP�a]XODZa�hª\¯�STbKb(NPO?Q(ORnoOoW u O�Q�x

K

K’

β θ

θ’

source

receiverS�¦�ghWXUPsFbvNPO � Z\YPY]XORQl_vNPWX��bvOosI��Q(OoÖ~jPOoUPnR��x[M¦K{�NPZ � b(NSTbb�S\UKª ¯ ® _vWXU�ª

� ¤ noZa_iªD° ��¦ Â�l���7�����M  ¡|·�J páYNPZ\b(ZaU¹Za�|OoUPO�Q(�\� Ø _vn�STbmb(ORQ(_FZ ��Za��S$��Q(OoO Oo]XORnRbvQ(ZaU£¤ ^³S\_v_BË�8�¦b(NkQ(ZajP�aNIS\UIS\UP�a]XODZa�hªkx8{�jkYPYPQ(Oo_m_vWXUP�³� S\nRbvZ\Q(_lZa�h¾ rS�¦�_vNPZ � b(NSTb�b(NPO?OoUPORQ(�\�FZa�±bvNPO�_vn�STbmb(ORQ(OosFYNPZ\b(ZaUqWc_��aW u OoUq[É�

Ø ¯ ® Ø�Ë�8Ë�8�Æ Ø.¤ K°²nRZa_Mª�¦ Â

[M¦K{�NPZ � b(NSTbKb(NPO?µ�WXUPORbvWXnVOoUPORQ(�\�FZa�±bvNPO�Oo]cOonRbvQ(ZaUIS\��b(ORQKb(NPO�noZa]c]XWc_vWXZaU.WX_K�aW u OoU.[~�� 8�® Ø ¼ ¤ �°²noZa_iª�¦Ë�8�Æ Ø.¤ �°²noZa_iª�¦ Â

�l���7�����M  ¡�Ê�J {�NPZ×� b(NSTb.S\U WX_vZa]XSTb(OosºYNPZ\b(ZaU n�S\UPUPZ\bB[�O§noZaU u ORQvb(ORs´WcUÉb(Z S\UÓOo]XORn�ÕbvQ(ZaU�@ Y�Za_vWcbmQ(ZaU�YiS\WcQ�f � � î ï Æ î � x � Z�UPZ\b�S\_v_vjP^`O�bvNPO³S\UP�a]XOo_�Za��bvNPO`Oo]XORnRbvQ(ZaU S\UPsY�Za_mWcbvQ(ZaUqSTQ(ODOoÖ�jS\] xbefore

e

e

+

−

after

θ

θ−

+

�l���7�����M  ¡DÑ8J�}�U§b(NPO2_mb�S\UPsSTQ(s sPORQ(W u STb(WcZaU Za��b(NPOzw�ZaNkQD�=Z\Q(^2jP]�Skr�b(NPO2UÉjPno]XORjP_�WX_V_mb(S Õb(WcZaUSTQv��S\UPs�r±b(NPORQ(Oo�=Z\QvO\r7b(NPO`Wc^�Y]XWcnoWcb�S\_v_vjP^�YPbvWXZaU�Wc_�b(NSTb�bvNPO`^³S\_v_?Za��b(NPO`UÉjPno]XOojP_�Wc_WXU t UPWHb(O\x ��Z ��O u ORQ�r�b(NPOIOo]cOonRbvQvZaU´S\n�b(jS\]X]H� Z\Qm[Wcb(_³STQvZajPUPsÓbvNPOInoOoU�b(ORQ.Za��^³S\_v_©Za��b(NPOOo]cOonRbvQ(ZaU�ÕàU~jPnR]XOojP_B_m��_mb(Oo^IxÇLKNPOqYPQvZ\Y�ORQ³sPO�Q(W u STb(WXZaU ��ZajP]XsÇQ(O�Y]�S\noOqbvNPOIOo]XORnRbvQ(ZaUº^³S\_m_��Wcb(N.b(NPO ��� � ������ ������Za�±b(NPO?_m�k_)b(Oo^5�aW u OoU.[~�2q® Ë�8eÍË�8hÆ$Í å

��NPORQ(OBÍ WX_�b(NPOBUÉjPno]XOoSTQ©^`S\_v_ox§Ô�_vO62ÓbvZ t UPs b(NPOBnoZ\QvQvOonRbzO�dkYPQvOo_v_vWcZaU$�=Z\Q�b(NPO.w�ZaNkQOoUPORQv�\��Ø%Ù�x�{�NPZ ��b(NSTb%b(NPO�QvOo]�STb(W u O|ORQvQvZ\QVWXUIb(NPO��\Q(ZajPUPs�Õà_mb�STb(O�ORUPORQ(�\�IZa��N��kskQ(Za�aORU¨Wc_�aW u ORUF[É�­zØØ ® Ë�8Ë � å

��NPORQ(Oz­zØÏWX_lb(NPO�sPW ��ORQvOoUPnoO|[�O�b���OoOoU§b(NPO|OoUPORQv�\��S\_v_vjP^�WXUP�BS`_mb�STbvWXZaUSTQv�FU~jPnR]XOojP_VS\UPsb(NPO?OoUPORQ(�\�.�=Z\Q t UPWHb(O�Õà^³S\_v_KU~jPno]cOojP_ox

�l���7�����M  ß?¢�JVÝ8QvZ u O2b(NSTbD�=Z\Q�UPZ\Qv^³S\]XWcÐ�ST[]XO�_vZa]cjkb(WXZaUP_�b(ZBb(NPO�{�n�NkQ(Z�sPWcUP�aORQ|OoÖ~jSTb(WXZaU�rb(NPOD_vORYiSTQ�STbvWXZaUInoZaUP_)b�S\UÉb�Ø�^zjP_)b�[�ODQ(OoS\]¿x-��WXU�b�f��«Q(Wcb(OVØÏS\_�Ø 9�Æ����¸¤ �lWHb(NFØ9?S\UPs��Q(O�S\]�¦�rÃS\UPs«_vNPZ×�£b(NSTbDWX��b(NPO�UPZ\Q(^`S\]XWXÐoSTb(WXZaU�noZaUPsPWHb(WXZaU WX_Vb(Z.NPZa]Xs«�=Z\Q�S\]X]7Àer��º^2jP_mb?[�OÐoORQvZPx�l���7�����M  ß�·�J8p�YiSTQvb(Wcno]XODZa��^`S\_v_�ËáSTYPYPQ(Z�S\n�NPOo_%S\UIST[PQ(jkYPb�Y�Z\b(OoU�b(WXS\]�skQ(Z\Y�fγ¤ Á ¦�® � ã å �=Z\Q Á 0Óã å°lÎ<9 å �=Z\Qlã©Ò Á Â

�´NSTb%WX_�b(NPOVQ(O?>Oon�b(WXZaUInRZ�O10�noWXOoU�b%Wc�7Ø�® Î<9 �B(�l���7�����M  ß�Ê�J8p�YiSTQvb(Wcno]XODWX_KWXUqS©nojk[Wcn�S\]���Oo]X] fÎB¤ Á ¦�® � ã å �=Z\Q Á±å���å�� [�O�b���OoOoU§ã�S\UPsIÄ å2 å OR]X_vOR��NPORQ(O Â

ghWXUPsFb(NPO?_)b�STb(WXZaUSTQm�B_mb�STb(Oo_lS\UPsFb(NPO?noZ\QvQ(OR_mY�ZaUPsPWcUP�`OoUPORQ(�aWcOo_ox�l���7�����M  ß�Ñ8J�ghWXUPs S\]X]Ãb(NPO�OoUPORQv�\��]XO u OR]X_�Za��S³Ö�jS\U�b(jP^�YiSTQvb(Wcno]XO�Za��^³S\_m_%Ë ^`Z u WcUP�jPUPsPORQ�bvNPO�O ��OonRb:Za��SDb���Z|sPWc^`OoUP_vWcZaUS\]kY�Z\b(ORUÉb(WXS\]Za��b(NPOK�=Z\Q(^ ż Ë � ¼ ¤ Á ¼ Æ è � ¼ Æ & ( Á � ¦�xp%Q(OVb(NPORQ(O�S\UÉ�.sPOo�aORUPORQ�S\noWcOo_�¶�l���7�����M  ß�7�Jh¬�ZaUP_mWXsPORQ7SKYiSTQvb(Wcno]XO�Za�k^³S\_m_±Ë£WXU�SKb(NkQvOoO�Õ�sPWc^`OoUP_vWcZaUS\]~nROoUÉbmQ�S\]ÉY�Z\b(OoU�b(W�S\]Za�±bvNPO��=Z\Q(^γ¤ (a¦�®�� ã å WX�KÄ`Ò (�Ò�� å2 å Oo]c_vOR��NPORQ(O Â

L�NPWX_�noZ\QvQ(OR_mY�ZaUPsP_Kb(ZzS2YiSTQvbvWXno]cOVbmQ�STYPY�OosB[�O�b���OoOoUFb���ZzNSTQ(sF_mYNPORQ(Wcn�S\]�_vNPOo]c]X_ox�ghWXUPs.b(NPO� ®�ãzbvZ\b�S\]���S u O?�=jPUPnRbvWXZaU�S\UPsIORUPORQ(�\��x

� ��������� ����������� ������� ��� �!��"$#%��&���'(���)�%*,+.-����0/��%��&�&³476�6��;=<?>7@BADCDE�@F;HGI<?>KJ�L�NPORQ(O|STQvO�UPWXUPODYPQ(Z\[]XOR^`_�ZaUFb(NPWX_KO�dS\^qf8ghW u O?WcUI_vOonRbvWXZaUIp�rS\UPs�=ZajkQ�WXU�_vOonRb(WcZaUzwVx�y:Zaj|^2jP_mbh_vZa] u O�Slb(Z\b�S\]�Za�i{~}��ºYPQ(Z\[]XOR^`_���Wcb(N2STb�]XO�S\_mb�b���Z���Q(Za^!O�S\n�N_vOon�b(WXZaU�xhy�Zaj³^2jP_mb�S\]X_vZ�_vZa] u O%b(NPO%b���Z|YPQ(Z\[]cOo^`_�^³STQvµ�ORs��lWHb(N��Éx � Z�OoS\neNBYPQ(Z\[]cOo^ ZaU_vORYiSTQ(STb(OV_vNPOoORb(_KZa��YiSTY�ORQ�S\UPsBb(jkQ(U.WXUBZaUP]c�`bvNPZa_vO�YPQ(Z\[]XOo^�_���Zaj.��S\U�b�b(Z2NS u OV�\Q�S\sPORs�x��Q(WHb(OV��ZajkQlUS\^`O?S\UPsqYPQ(Z\[]cOo^5U~jP^�[�ORQ%ZaUIOoS\neNFYiS\�aO\x�l���7�����M  ¡³¢�J${�NPZ � bvNSTb.bvNPO _mY�OoOos´Za�?b(NPO§Oo]cOonRbvQ(ZaU´WXUÓb(NPO¨Ú�b(N¸w�ZaNkQ.Z\Qv[WHbFZa�?SN��kskQ(Za�aOoUFSTb(Za^ Wc_��aW u OoU³[É�BÌ2®�¾�� Ù �lNPORQvO ® � î ¼ � �Û¾lWc_�b(NPO t UPOD_)bvQ(jPnRb(jkQvO?noZaUP_)b�S\UÉbox�l���7�����M  ¡�·�J:¬�ZaUP_vWXsPORQ�S?��S u OlYiS\n(µ�ORb:b(NSTbWX_K�=Z\Q(^`ORsF[É�Fb(NPOD^`Za^`ORUÉb(jP^�sPWc_mbvQ(WH[jkb(WXZaU�f

� ¤!��¦�® � ã � 0�°lÄ� °lÄ`Ò ��ÒÓÄã Ä 0 �

L�NPOKQ(OR_vjP]cb(WcUP�D��S u O��=jPUPnRb(WcZaU�r �h¤ Á ¦�]cZ�Z\µk_h]cWcµ�Ob(NPO t �ajkQ(O?b(Z2b(NPODQ(WX�aN�b�xS�¦�ghWXUPsFbvNPO�noZ~Z\Q(sPWXUSTb(OeÕ�_mYiS\noOD��S u O��=jPUPnRb(WcZaU �h¤ Á ¦�x[M¦�ghWXUPs � _vjPn�NIb(NSTbKb(NPO?��S u OD�=jPUPnRb(WcZaUIWX_�UPZ\Q(^³S\]XWcÐoOos�xnצ8{�NPZ � b(NSTb:S?Q(O�S\_vZaUST[]cO%sPO t UPWHb(WXZaU`Za�í Á �=Z\Q:��ZajkQ�S\UP_)��ORQ�b(Z �h¤ Á ¦7�aW u Oo_�­ ��­ Á �3WXUPsPO�Y�OoUPsPOoU�bVZa� � x�l���7�����M �¡|Ê��±Jhp�U�OR]XOonRbmQ(ZaUz��WcbvN�jPUkµkUPZ×��U�OoUPORQv�\�2_vn�STbvb(O�Q(_8OR]�S\_mb(Wcn�S\]X]H����Q(Za^�S�YPQ(Z\b(ZaUb�STQ(�aO�b�x�L�NPO|S\UP�a]cOo_%S\UPsI^�Za^`OoU�b�SzZa�±b(NPO�_vnoSTbvb(ORQ(ORsqYiSTQvb(Wcno]XOo_lSTQ(O|S\]c]�^`O�S\_vjkQ(ORs�x�{�NPZ×�b(NSTbKb(NPODWXUPnoWcsPOoUÉb%OoUPORQ(�\��rPØ 8×rn�S\UF[�O�sPORb(O�Q(^`WXUPORsFb���Z���S��k_RfS�¦�Ø 8�®-" 8×noZ\bMª 8hÆ " ��noZ\biª �S\UPs[M¦�Ø 8�®¸Ë ���¿noZ\b� �¼ noZ\biª �?° �� å��NPORQ(O�" 8�S\UPs " ��STQ(O�b(NPOK^`Za^`OoU�b�S%Za�b(NPOK_vn�STbvb(O�Q(Oos�OR]XOonRbmQ(ZaU�S\UPszYPQ(Z\b(ZaU�raQ(Oo_)Y�OonRbvW u Oo]H�S\UPs§ª 8?S\UPs§ª �.STQvO?b(NPOoWHQ%Q(Oo_)Y�OonRbvW u O�_vn�STbmb(ORQ(WcUP�BS\UP�a]XOo_oxKp�_v_vjP^�O?b(NSTb%bvNPO�b(NPO|WcUPnoWXsPOoU�bOoUPORQv�\�qWc_�^2jPn�Nq�\Q(OoSTb(ORQlbvNS\UFb(NPO?Oo]XOon�bvQ(ZaUq^`S\_v_|¤ Ø 8�� Ë�8R¦ex�y:Zajq^³S��.S\]X_mZ©UPOoORsnoZ\b è ® _vWcU K°²noZa_ Â

�l���7�����M  ¡?Ñ8J páYNPZ\b(ZaU¸b(Z\QvY�OosPZ$Wc_ t QvOos�NPZ\Q(WXÐRZaUÉb�S\]c]c�Ó��Q(Za^ b(NPO _mb�STQ(_mNPWcY�g7}�ÔD{P{�¨S\sPWXÖ~jPO2STbVS§¤ u ORQv�F]XZaUP�ɦ�Q(Z�n(µ�ORb�_mNPWcY§b(NSTb�Wc_�^`Z u WXUP� u ORQvb(Wcn�S\]X]H�F��Wcb(N§S`_)Y�OoOos¨Za� È � ¾ x� UPO�_vOonRZaUPs�]�STb(ORQorkb(NPO �¨S\sPWXÖ~jPO t QvOo_�S©_mOonoZaUPsqb(Z\QmY�OosPZ³STb�b(NPODQ(Z�n(µ�ORb�_vNPWHY�xS�¦%})U�bvNPO©Q(Oo�=O�Q(OoUPnoO³��Q�S\^�O©Za�:bvNPO �¨S\sPWXÖ�jPOTr�NPZ×�,� STQ|STYiSTQvb�STQvO©b(NPO©WX^�YiS\nRbDY�ZaWcUÉbv_�Za�b(NPODb���Z©b(Z\QvY�OosPZ�OR_�¶[M¦�})UFb(NPODQvOo�=ORQ(OoUPnRO���Q�S\^`ODZa�±b(NPODQ(Z�n(µ�ORb�_vNPWcY�rNPZ×� � STQ%STYiSTQvblSTQ(ODb(NPO?Wc^�YiS\nRb�Y�ZaWcUÉbv_�¶�l���7�����M  ß?¢�J7pºUPZaU�Õ¿Q(Oo]XSTb(W u WX_mb(Wcn8YiSTQvbvWXno]cO�WX_7WcU2S%ZaUPO�Õ�sPWc^`OoUP_vWcZaUS\]�WcU t UPWcb(O�_mÖ�jSTQ(O���OR]X]¿x{�NPZ×��bvNSTbDbvNPO©��Q(ORÖ�jPOoUPnR� Za�8Q(S\sPW�STb(WcZaU Oo^�Wcbvb(ORs �=Z\QDb(NPO2bvQ�S\UP_vWHb(WXZaU Ú b(ZFÚ�° ©OoÖ�jS\]c_b(NPO���Q(ORÖ�jPOoUPnR�IZa��^�Z\b(WXZaUq��NPOoU§Ú � ax��´NSTb%YPQ(WXUPnRWcY]XO�sPZ~Oo_%b(NPWc_�_(STb(WX_m����S\UPs��lN��IWc_Wcb�WX^©Y�Z\Qvb�S\U�be¶�l���7�����M  ßK·hJ8L���Z|YiSTQmb(WXno]cOo_�Za��^³S\_v_8Ë STQ(O%STbvb�S\n�NPOos³b(Z|b(NPO%ORUPsP_KZa��S|^`S\_v_v]XOR_v_:Q(Wc�aWXsQ(Z~s«Za��]cOoUP�\b(N²Äix�L�NPO�_m��_mb(Oo^áWX_?��Q(OoOzb(ZFQ(Z\b(STb(O©WcU«b(NkQ(ORO�sPWX^�OoUP_vWXZaUP_?ST[�Zajkb?b(NPO t dPOosnoOoU�b(ORQlZa��b(NPOVQ(Z�s�xS�¦K{�NPZ � b(NSTbKb(NPO?S\]X]XZ×��OosFOoUPORQ(�aWcOo_%Za�Ãb(NPWX_�Q(WX�aWcsFQ(Z\bvZ\Q%STQ(O

Ø%Ù©® �Û ¼ Ú�¤ Ú�Æ ¦ËqÄ ¼ å �=Z\Q�Ú�®¹ã å å è å ÂcÂXÂ[M¦Ã�´NSTb�STQ(O:b(NPO�UPZ\Qv^³S\]XWcÐoOos�OoWc�aOoUP�=jPUPnRb(WcZaUP_��=Z\Q�b(NPWX_h_m��_mb(Oo^³¶ � WX_vnRjP_v_�S\UÉ�|sPOo�aOoUPORQ�S\n���x�l���7�����M  ßKÊ��±J.ghWXUPs$b(NPO.[�ZajPUPsÇ_)b�STb(O.OoUPORQ(�\�$S\UPs ��S u Oq�=jPUPn�b(WXZaUÇZa�%S¨YiSTQvb(Wcno]XO.Za�^³S\_v_KË _vjk[��)OonRblb(Z`S\UqSTbvbvQ�S\nRbvW u O � Õà�=jPUPnRb(WXZaUIY�Z\b(OoU�b(W�S\]γ¤ Á ¦�®!°%Î�9 Ä � ¤ Á ¦ å��NPORQ(O|ijWc_lSz]XOoUP�\b(NFYiSTQ�S\^`ORb(O�QlS\UPs�Î<9=�Óãkx�l���7�����M  ß�Ñ8JKp Ö~jS\UÉb(jP^�YiSTQvb(WXnR]XO|Za��^³S\_v_lË r�ORUPORQ(�\�IØ�^�Z u WXUP�`WcU�ZaUPO�sPWX^`OoUP_mWXZaU_mbvQvWcµ�Oo_�SzQ(ORnRb�S\UP�ajP]�STQ�[iSTQvQ(WcORQlZa��NPOoWX�aN�b�Î�9%®��\Ø �BAzS\UPsqb(NPWcn�µ�UPOo_v_ $ xS�¦ � ORbvORQ(^`WcUPO�b(NPO�bvQ(S\UP_v^`WX_m_vWXZaU²noZ~O10`nRWXOoU�b2S\nRQ(Za_m_|b(NPO�[iSTQvQ(WcORQ�xq¬�Za^©Y]XORb(OR]c�¨_vWX^©Y]XWX�����ZajkQKQ(Oo_vjP]Hb�x[M¦|�¸NSTb�STQ(OBb(NPOq^³S dkWX^2jP^ S\UPs$^`WXUPWc^zjP^ bvQ�S\UP_v^`Wc_v_vWcZaU noZ~O10`nRWXOoU�b(_`S\UPs$�=Z\Q©��NSTbu S\]XjPOo_�Za� $ sPZ�b(NPO��FZ~nonojkQe¶�l���7�����M  ß�7�J�p�YiSTQvb(Wcno]XO�Za��^³S\_m_�Ë S\UPsFn�NSTQ(�aO��|Wc_�WXU.S�ZaUPO�Õ�sPWc^`OoUP_vWcZaUS\]�NSTQ(^`ZaUPWcnZa_vnoWc]X]XSTb(Z\Q?Y�Z\b(OoU�b(W�S\]�^`Z u WcUP�F��Wcb(N ��Q(OoÖ~jPOoUPnR� � x�}�U S\sPsPWHb(WXZaU�rÃWcb�WX_�_mjk[��)ORnRb|b(Z�S ������Oo]cOonRbvQ(Wcn t OR]Xs�8xS�¦�ghWXUPsFbvNPO�O�dPS\nRblOedkYPQ(Oo_m_vWXZaUI�=Z\QKb(NPO?OoUPORQ(�\��x[M¦V¬KS\]cnojP]�STbvO|b(NPO©ORUPORQ(�\�§jkY¨b(Z.bvNPO t Qv_mb�UPZaU�ÕàÐoORQ(ZFnoZ\QvQ(ORnRb(WXZaU¨WXU¨UPZaU�Õ�sPOR�aOoUPORQ�STb(O2Y�O�QmÕb(jkQv[iSTbvWXZaUFbvNPOoZ\Qv��x8¬�Za^©YiSTQ(OD��ZajkQKQ(Oo_mjP]cblbvZz��NSTbl��ZajF�=ZajPUPsIWXU.YiSTQvb�Skx

FLORIDA INTERNATIONAL UNIVERSITYDEPARTMENT OF PHYSICS

UNIVERSITY PARK, MIAMI, FL33199

Ph.D. Qualifying Exam, Fall 2011 – MODERN PHYSICS

THURSDAY, AUGUST 18, 2011 1PM - 5PM

INSTRUCTIONS:

1. There are nine problems on this exam with

• five in SPECIAL RELATIVITY/MODERN PHYSICS;

• four in QUANTUM MECHANICS.

2. You must solve a total of SIX problems with

• at least two from SPECIAL RELATIVITY/MODERN PHYSICS;

• at least two from QUANTUM MECHANICS.

3. You MUST solve the problems marked REQUIRED.

4. Do each problem on separate sheets of paper and turn in only those problemsyou want to have graded.

5. Write your name and problem number on each page.

1

RELATIVITY / MODERN PHYSICS

1. Xenofo and Ydex are fighting a duel with identical paintball guns. Xenofo is at theorigin of an Earth reference frame and Ydex is at x = L. An earth observer notesthat they fire their guns simultaneously. An observer on a space-craft moving athigh speed in the positive x direction notes that Ydex fired before Xenofo by anamount T . How fast is the space-craft moving relative to the Earth?

2. In the lab reference frame, an electron and a positron emerge from a nuclearreaction with equal speeds of βc. The electron is observed to be moving at anangle θ above the positive x axis, and the positron is moving at an angle θ belowthe positive x axis.

(a) What is the velocity of a moving observer (S’) who sees the two particlesmove outward along the positive and negative y′ axis respectively.

(b) What is the speed of each particle in S’?

3. A ball of mass m and speed βc collides completely inelastically with a ball ofmass 2m moving in the opposite direction with speed v.

(a) If the two balls are at rest after collision, determine v.

(b) Thermal energy, Q, is generated by the collision, and is dissipated to thesurroundings as the balls cool down to the temperature they had before thecollision. Determine Q.

4. REQUIRED In a Stern-Gerlach experiment, silver atoms of mass m, whose spinis due to a single electron, are obtained from an oven at temperature T . Youranswers to all parts should be in terms of given symbols and any other standardphysical constants.

(a) What is their root mean square (RMS) speed?

(b) The atoms pass perpendicularly through a magnetic field of length L thathas a gradient, dB

dz , perpendicular to the beam. For atoms in the two beamshaving the same RMS speed, what force do they experience?

(c) How far apart will they be as they emerge from the field?

2

5. A radioactive nucleus decays via the α-emission reaction

A→ B + α (1)

with a half-life of t1/2.

(a) Calculate the kinetic energy of the α-particle released in the rest frame ofA.

(b) At time t = 0 a pure sample of A is prepared. At time t = T , the activityis measured to be R. What is the original mass of the sample?

3

QUANTUM MECHANICS

6. Consider two particles of mass m interacting with a potential in one dimensionof the form

V (x1, x2) =1

2k1x

21 +

1

2k2x

22 + αx1x2, (2)

where x1 and x2 are the locations of the two particles. Assume that k1, k2, α > 0and k1k2 > α2.

Write down the time independent Schrodinger equation and find all the boundstate energy eigenvalues.

7. Let H be a quantum mechanical Hamiltonian with eigenfunctions |φi〉 and cor-responding eigenvalues Ei with i = 0, 1, 2, · · ·∞. Further assume that

E0 < E1 < E2 < · · · < E∞. (3)

(a) Let |ψ〉 be a normalized wavefunction (not necessarily an eigenfunction ofH). Prove that

〈ψ|H|ψ〉 ≥ E0. (4)

Hint: Expand |ψ〉 in the eigenbasis of H.

(b) Prove that every attractive potential in one dimension, i.e., V (x) < 0 forall x, has at least one bound state. Hint: Consider the normalized wavefunction,

ψα(x) =(

α

π

) 14

e−12αx2

; α > 0 (5)

and calculate,

E(α) = 〈ψα|H|ψα〉; H = − h2

2m

d2

dx2+ V (x). (6)

Show that E(α) can be made negative by a suitable choice of α.

8. Consider a one-dimensional harmonic oscillator in the ground state |0〉 at t =−∞. Let a perturbation

H1(t) = −Xe−t2

τ2 (7)

with X being the position operator be applied between t = −∞ and t = +∞.What is the probability that the oscillator is in the state |n〉 at t = ∞?

4

9. REQUIRED Consider the one dimensional harmonic oscillator.

(a) Prove that[a, a†] = 1. (8)

(b) Prove that

H = hω(a†a +

1

2

)(9)

(c) Prove that

H|n〉 = hω(n +

1

2

)|n〉. (10)

(d) Prove that〈n|T |n〉 = 〈n|V |n〉 (11)

for any eigenstate |n〉.

5

Some useful information

• Some useful information about quantum harmonic oscillators in one dimension:Using standard notation, let ω be the classical frequency of the oscillator andlet m be the mass of the particle. Let |n〉, n = 0, 1, · · · ,∞ label all the boundeigenstates. The coordinate and momentum operators are given by

X =

√h

2mω(a + a†); P = i

√mωh

2(a† − a); (12)

with a† and a being the step up and step down operators obeying

a|n〉 =√

n|n− 1〉; a†|n〉 =√

n + 1|n + 1〉. (13)

• √α

π

∫ ∞

−∞e−αx2

dx = 1. (14)

for all α > 0.

6

FLORIDA INTERNATIONAL UNIVERSITYDEPARTMENT OF PHYSICS

UNIVERSITY PARK, MIAMI, FL33199

Ph.D. Qualifying Exam, Fall 2012 – MODERN PHYSICS

THURSDAY, AUGUST 16, 2012 1PM - 5PM

INSTRUCTIONS:

1. There are nine problems on this exam with

• five in SPECIAL RELATIVITY/MODERN PHYSICS;

• four in QUANTUM MECHANICS.

2. You must solve a total of SIX problems with

• at least two from SPECIAL RELATIVITY/MODERN PHYSICS;

• at least two from QUANTUM MECHANICS.

3. You MUST solve the problems marked REQUIRED.

4. Do each problem on separate sheets of paper and turn in only those problemsyou want to have graded.

5. Write your name and problem number on each page.

1

RELATIVITY / MODERN PHYSICS

1. REQUIRED In the lab system S, a mass m1 = 9 is moving towads a targetmass m2 = 16, which is at rest. The energy of m1 is E1 = 27.75. Assumetime-space dimension is 1 + 1.

(a) Calculate the space momentum p1 of m1. What is its velocity v1?

(b) Find the Lorentz transformation Λ : S → S ′,

Λ =(

cosh ξ − sinh ξ− sinh ξ cosh ξ

), (1)

to the center-of-mass system S ′, i.e. find the numbers for cosh ξ and sinh ξ.What is the boost velocity v?

(c) What are the energies E ′1 and E ′

2 in the center-of-mass system?

(d) Supposes m1 and m2 collide and stick together, forming one combined massM . What is M? Compare M to m1 + m2.

Hints: We are using units such that c = 1 and all masses/energies are given insome unit mu. (For example, choosing mu = 931.5MeV, the target mass couldbe Oxygen 16

8O and the projectile would be Beryllium 94Be).

2. The proper length for one spaceship is twice that of the other. To an observeron earth, both ships have the same length. If the faster spaceship is moving withβ = v

c = 0.95

(a) Find the speed of the slower ship.

(b) If both ships are moving in the same direction, find the speed of the slowership in the rest frame of the faster ship.

Hint: Make sure that the sign of the velocity you find makes sense.

3. An observer in a rocket moves toward a mirror at speed v relative to the mirror.A light emitted by the rocket moves toward the mirror and is reflected back tothe rocket. If the rocket is a distance d away from the mirror when it emits thelight pulse as measured in the mirror’s reference frame, what is the total traveltime of the pulse (from the rocket and back) as measured in

(a) the rest frace of the mirror and

2

(b) the rest frame of the rocket?

4. The activity (decays per second), R, of a radioactive sample decreases to R32 in

time T .

(a) What is the half-life and decay constant?

(b) What fraction of the original number of nuclei will have decayed after T100

and T10 .

5. The radial part of the wavefunction for the Hydrogen atom in the 2p state isgiven by

R2p(r) = Are−r

2a0 , (2)

where A is a constant and a0 is the Bohr radius. Use this to calculate the averagevalues of r, r2 and the standard error in r.

QUANTUM MECHANICS

6. REQUIRED Eight electrons are placed in a three-dimensional infinite cubicwell with sides of length L.

(a) Determine the ground state energy.

(b) Determine the energy of the first excited state.

7. This problem involves the case of a harmonic oscillator in the ground state (n =1). In some cases you may use physics arguments instead of direct calculation tofind the answers.

(a) Find 〈x〉 and 〈x2〉.(b) Find 〈p〉 and 〈p2〉.(c) Use your results from the previous two parts to determine ∆x∆p.

(d) What is special about this case and why?

3

8. Consider a quantum mechanical particle of mass m constrained to move in aone dimension. Let x label the space variable and let x ∈ [0, L]. Also assumeperiodic boundary conditions: x = 0 is identified with x = L. As such, you canalso visualize space as a circle with perimeter L. Let this particle interact witha potential V (x) which satifies the property

V (x + a) = V (x) (3)

for all x. In addition a satisfies the relation Na = L with N being an inte-ger. Assuming no degeneracies in the spectrum of bound states, prove that alleigenfunctions satisfy the condition

ψ(x + a) = Cψ(x) (4)

with C being one of the N ’th root of unity, i.e; CN = 1.

9. Consider Hermitian matrices γi, i = 1, 2, 3, 4 that obey

γiγj + γjγi = 2δijI; i, j, = 1, 2, 3, 4. (5)

The Kronecker delta is defined as

δij ={

1 if i = j0 if i "= j

. (6)

(a) Show that the eigenvalues of γi are ±1 for all i. Hint: Go to the eigenbasisof γi, and use the equation for i = j.

(b) By considering the relation (5) for i "= j and noting that trace is cyclic,namely; Tr(ACB) = Tr(CBA), show that γi is traceless for all i.

(c) Show that γi cannot be odd dimensional matrices.

4

Some useful information

• The ground state eigenfunction of a harmonic oscillator is

ψ(x) =(

mω

πh

) 14

e−mωx2

2h . (7)

• √α

π

∫ ∞

−∞e−αx2

dx = 1. (8)

for all α > 0.

5

FLORIDA INTERNATIONAL UNIVERSITY

DEPARTMENT OF PHYSICS

UNIVERSITY PARK, MIAMI, FL 33199

INSTRUCTIONS

There are nine problems on this exam with

• four in SPECIAL RELATIVITY/MODERN PHYSICS;

• five in QUANTUM MECHANICS.

You must solve a total of SIX problems with

• at least two from SPECIAL RELATIVITY/MODERN PHYSICS;

• at least two from QUANTUM MECHANICS.

You MUST solve the problems marked as REQUIRED.

Do each problem on separate sheets of paper and turn in only those problems you want

to have graded.

Write your assigned number (DO NOT WRITE YOUR NAME) and problem number on

each page.

Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013

1. (Special Relativity/Modern Physics)

Derive the relativistic explanation of superluminal motion.

Superluminal motion is the apparent faster-than-light motion seen in some radiogalaxies

and other sources. Consider an active galactic nucleus (A) that emits a relativistic jet

from its center at time t1. This jet has a speed v in a direction that makes an angle ! with

respect to the line joining the observer on earth (O) and the origin of the relativistic jet at

time t1 (A). Light emitted from this jet is seen by this observer as a streak in the sky.

(a) Obtain an expression for the apparent transverse speed of this streak as a

function of v, ! and the speed of light c in the limit where the distance OA is very

very large.

(b) Find the maximum value of the apparent transverse speed for a fixed v.

(c) Show that this maximum value can be larger than c even though v has to be less

than or equal to c.

2. (Special Relativity/Modern Physics)

An Imperial Star Destroyer passes a rebel observation post at a relativistic speed. The

observation post"s sensors measure the length of the Star Destroyer to be 800m. (You

can assume that this is an instantaneous measurement of the length).

(a) Assuming that the length of the Star Destroyer is 1,600 m at rest, how fast is the

ship traveling relative to the observation post? Leave your answer in terms of c,

the speed of light.

(b) In response to the approaching Star Destroyer, the observation post sends an

alert signal to the rebels on a nearby planet. If the signal travels at the speed of

light, how fast does this signal travel according to the observer on the Star

Destroyer? Explain your answer.

(c) As the Star Destroyer passes and moves away from the post, the post"s sensors

monitor Darth Vader in his private gymnasium on the Star Destroyer practicing

his light saber skills. If the sensors measure infrared photons from the light saber

with # = 2500 nm, determine the wavelength of light that Darth Vader observes.

Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013

3. (Special Relativity/Modern Physics) REQUIRED

When a muon decays at rest,

why can the electron only have a maximum energy of 53 MeV? The mass of the muon

is 106 MeV/c2 and you can assume that the neutrinos are massless. You can also

ignore the mass of the electron.

4. (Special Relativity/Modern Physics)

A large number of hydrogen atoms are in the n = 3 state.

(a) Considering all possible transitions that the atoms can make before reaching

their ground state, what wavelengths would you observe in the emission

spectrum?

(b) If one such atom, initially at rest, undergoes a transition directly from the n = 3

state to the ground state,

i. what is the momentum of the emitted photon, and

ii. with what kinetic energy will the atom recoil?

Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013

5. (Quantum Mechanics)

Consider three spin 1/2 particles, each bound in the 1-dimensional harmonic oscillator

potential. The single particle state vectors are given by |n,m>1, |n,m>2 and |n,m>3, where

n is the energy state number (n=0,1,2,3,...) and m is the z-component of the spin (m =

±1/2).

(a) Assume that the particles have equal mass, but are distinguishable.

i. Determine the ground state energy of this system of distinguishable particles.

ii. Determine the ground state vector(s) as a product of the single particle state

vectors. What is the degeneracy of the ground state?

(b) Now assume that the particles are identical fermions.

i. Determine the ground state energy of this system of identical particles.

ii. Determine the ground state vector(s) as a product of the single particle state

vectors. What is the degeneracy of the ground state?

6. (Quantum Mechanics)

A hydrogen atom is in the ground state. A pulsed electric field E(t) = E0!(t) is applied to

the atom. Calculate the transition probability to the excited state n with the energy En

and the eigenfunction !nlm (n,l, and m, are quantum numbers) and calculate the

probability of the atom remaining in the ground state.

7. (Quantum Mechanics) REQUIRED

A particle with mass m is in the n!th bound state (with the energy En and the

wavefunction "n(x)) in a 1-D square potential well ( the potential depth is V0 and the

width is w). Find

(a) the probability of the particle outside the potential well and

(b) the expectation values of V(x) and V2(x) under the condition that V0 >> En.

8. (Quantum Mechanics)

An operator A, representing observable A, has two normalized eigenstates #1 and #2

with eigenvalues a1 and a2, respectively. Operator B, representing observable B, has

Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013

two normalized eigenstates !1 and !2 with eigenvalues b1 and b2, respectively. The

eigenstates are related by "1 = (#3 !1 - !2)/2 and "2 = (!1 + #3 !2)/2.

(a) Observable A is measured, and the value a2 is obtained. What is the state of the

system (immediately) after this measurement?

(b) If B is now measured, what are the possible results and what are their

probabilities?

(c) Right after the measurement of B, A is measured again. What is the probability of

getting a2? [ Note that the answer would be quite different if I had told you the

outcome of the B measurement.]

9. (Quantum Mechanics)

Find the differential cross section for elastic scattering of a particle initially traveling

along the z-axis from a non-spherical, double delta potential

where is a unit vector along the z-axis.

(a) Since the potential is not spherically symmetric, obtain the scattering amplitude

from the general equation

where is the momentum transfer.

(b) Since the incident particle is initially traveling along the z-axis, and it scatters

elastically from the potential, the magnitudes of the momenta are the same

before and after the collision. Let $ be the angle of the outgoing momentum with

respect to the z-axis. Find the differential cross section d%/d& = | f($) |2 in terms

of constants and the magnitude of the momentum, k.

Ph.D Qualifying Exam Modern Physics Thursday, August 22, 2013

Modern  Physics      Ph.D.  Qualifying  Exam   Fall  2014  Florida  International  University  Department  of  Physics    Instructions:  There  are  nine  problems  on  this  exam.    Five  on  quantum  mechanics  (Section  A),  and  four  on  general  modern  physics  (Section  B).    You  must  solve  a  total  of  six  problems  with  at  least  two  from  each  section.      You  must  also  do  the  problems  marked  Required.      Do  each  problem  on  its  own  sheet  (or  sheets)  of  paper.    Turn  in  only  those  problems  you  want  graded.    Write  your  student  ID  number  on  each  page  but  not  your  name.    You  may  use  a  calculator  and  the  math  handbook  as  needed.    Section  A  1.  A  particle  is  moving  inside  a  1-­‐D  potential  𝑉 𝑥 = ∞  for  x<0;  𝑉 𝑥 = !!!!!

!  for  

x>0.    Find  the  particle  energy.    2.  A  particle  of  mass  m  moves  between  two  infinite  potential  walls  separated  by  a  distance  d.    

a) Find  the  ground  state  energy.    b) One  of  the  potential  walls  is  suddenly  and  instantaneously  moved  to  a  

distance  2d.  What  is  the  probability  of  finding  the  particle  in  the  new  ground  state?  

 3.  An  electron  with  magnetic  moment  µ0  and  spin  up  at  t=0  enters  a  region  of  a  static  magnetic  field  𝐻 = −𝐻!𝑧.    There  is  also  a  AC  magnetic  field  given  by  𝐻! = 𝐻! cos 𝜔𝑡 𝑥 + 𝐻! cos 𝜔𝑡 𝑦.  What  is  the  probability  of  finding  the  electron  in  the  spin  up  state  at  time  t?    4.  A  particle  with  positive  energy  (E  >  0)  encounters  a  Dirac  delta-­‐function  potential  barrier  

𝑉 𝑥 = 𝛼𝛿 𝑥  where  α  is  positive  real  constant  with  the  units  of  energy×length.    

a) Solve  the  Schrödinger  equation  for  the  regions  x  <  0  and  x  >  0.  If  you  make  any  assumptions,  you  must  verify  your  answer  to  receive  full  credit.  

b) State  and  apply  the  appropriate  boundary  conditions  for  the  potential  interface  at  x  =  0.      Obtain  expressions  relating  the  different  coefficients.  

c) Using  the  results  from  part  b),  determine  the  transmission  coefficient  T  for  a  particle  incident  and  transmitted  through  this  potential  barrier.  Discuss  the  limiting  cases  for  T  as  the  energy  E  approaches  zero  (𝐸 → 0)  and  E  approaches  infinity  (𝐸 → ∞).  

     

5.  Required    Consider  an  unperturbed  harmonic  oscillator  with  Hamiltonian  𝐻! =

!!

!!+ !

!𝑚𝜔!𝑥! ,    

with  eigenstates  of  the  form  |𝑛 ,  and  energy  eigenvalues  𝐸!! = 𝑛 + !

!ℏ𝜔.    

A  perturbation  of  the  form             𝐻! = !

!𝛽𝑚𝜔!𝑥!  

is  added  to  the  system,  with  β  being  a  small  parameter.  a) Using  the  definition  for  the  harmonic  oscillator  raising  and  lowering  

operators,  show  that  the  expectation  value  of  x2  can  be  written  as  

𝑛 𝑥! 𝑛′ =ℏ

2𝑚𝜔𝑛! + 1 𝑛! + 2 𝛿!,!!!! + 𝑛′ 𝑛! − 1 𝛿!,!!!! + 2𝑛! + 1 𝛿!,!!

𝑛!  

 b) Using  non-­‐degenerate  perturbation  theory,  obtain  the   first-­‐order  correction  

to  the  energy.  c) Using   non-­‐degenerate   perturbation   theory,   obtain   the   second-­‐order  

correction   to   the  energy.  You  may  only  consider   the  n  =  0  and  n  =  1   states  when  doing  the  expansion.  

Hint:    Recall  the  following  relationships    𝑎± =

!!ℏ!"

∓𝑖𝑝 +𝑚𝜔𝑥                          𝑎!|𝑛 = 𝑛 + 1|𝑛 + 1  

𝑥 = ℏ!!"

𝑎! + 𝑎!                                              𝑎!|𝑛 = 𝑛|𝑛 − 1  

           

Section  B  6.  Light  of  wavelength  300  nm  strikes  a  metal  plate,  producing  photoelectrons  that  move  with  a  speed  of  0.002c.  

a) Determine  the  work  function  of  the  metal.  b) What   is   the   critical   wavelength   for   this   metal,   so   that   photoelectrons   are  

produced?  c) What  is  the  physical  significance  of  the  critical  wavelength?  

 7.    Provide  a  brief  qualitative  description  for  each  physical  phenomenon:  

a) Compton  Effect.  b) Difference  between  bosons  and  fermions.  c) Michelson  Morley  Experiment.  d) Rutherford  scattering.  

 8.  Required    A  pion,  𝜋!,  has  a  rest  mass  of  139.57  MeV/c  and  is  moving  with  a  speed  𝛽 = 0.5𝑐.    It  then  decays  into  a  muon/neutrino  pair,  𝜇!𝜈! ,  with  the  muon  traveling  off  at  an  angle  of  100  relative  to  the  original  direction  of  the  pion.    What  is  the  energy  and  momentum  of  the  muon  and  what  is  the  angle,  energy,  and  momentum  of  the  neutrino?    The  muon  has  a  rest  mass  of  105.65  MeV/c  and  you  may  assume  the  neutrino  is  massless      9.  Police  radar  is  used  to  measure  the  speed  of  oncoming  cars.    It  works  by  broadcasting  microwaves  of  a  precisely  known  frequency,  f.  The  moving  car  reflects  these  waves  that  are  then  picked  up  by  the  radar  gun.    The  gun  computes  the  speed  of  the  car  by  measuring  the  beat  frequency  (which  is  small  compared  to  the  broadcast  frequency)  between  the  broadcast  frequency  and  the  detected  frequency.    

a) Find  the  frequency  of  the  reflected  wave  in  terms  of  f,  v,  and  c.  b) Show  that  the  speed  of  the  car  is  given  by  𝑣 =≅ !

!!!"#$!𝑐  for  a  car  moving  

much  less  than  the  speed  of  light.                      

Modern  Physics      Ph.D.  Qualifying  Exam   Fall  2015  Florida  International  University  Department  of  Physics    Instructions:  There  are  nine  problems  on  this  exam.    Six  on  quantum  mechanics  (Section  A),  and  three  on  general  modern  physics  (Section  B).    You  must  solve  a  total  of  six  problems  with  at  least  two  from  each  section.      You  must  also  do  the  problems  marked  Required.      Do  each  problem  on  its  own  sheet  (or  sheets)  of  paper.    Turn  in  only  those  problems  you  want  graded.    Write  your  student  ID  number  on  each  page  but  not  your  name.    You  may  use  a  calculator  and  the  math  handbook  as  needed.    Section  A  1. An  electron  (spin  ½)  at  rest  is  in  a  uniform  magnetic  field  𝐵 = 𝐵!𝚥 + 𝐵!𝑘  

(By>>Bz).  The  Hamiltonian  is   zz

yy S

meBS

meB

HHH +=+= '0  (m  is  the  electron  

mass  and  e  is  the  electron  charge).  Find  the  first-­‐order  perturbation  corrections  of  the  electron  spin  wave  functions.  

 2. (Required) A particle of mass m is confined to move freely in a ring of radius R. (a) Find the energies and the eigenfunctions of the particle. (b) A perturbation term  

𝐻! =𝑉!,−𝛼 < 𝜑 < 0𝑉!, 0 < 𝜑 < 𝛼  0,              elsewhere

   

 is  added  to  the  particle,  where  φ  is  the  azimuthal  angle  and  α  is  an  arbitrary  fixed  value.  Find the first order corrections of the energies for the three lowest energy states.    3.  An  electron  in  a  hydrogen  atom  jumps  from  n  =  4  to  n  =  1  state.    (a)   Is   energy   absorbed   or   emitted   in   this   process?     If   energy   is   emitted,   which  spectral  line  series  is  this  transition?    Explain.  (b)  Determine  the  energy  transfer  in  eV  and  photon  wavelength  in  nm  for  this  process.  (c)  Explain  why  classical  physics  fails  to  correctly  describe  the  emission  spectrum  for  the  hydrogen  atom.  (d)  Discuss  Bohr’s  model  of  the  hydrogen  atom.    In  your  discussion,  be  sure  to  include  his  three  revolutionary  postulates  that  led  to  this  model.      

4.  A  1-­‐D  particle  in  an  infinite  square  well   (0 ≤ x ≤ a)has  the  initial  wavefunctionΨ 𝑥, 0 = 𝐴𝑥(𝑎 − 𝑥)  inside  the  well  and  Ψ 𝑥, 0 = 0    outside  the  well.  (a)  Find  the  normalization  constant  A.  (b)  Find  the  expectation  value  for  the  position  x.  (c)    Find  the  expectation  value  for  x2.  (d)  Use  the  results  of  (a)  and  (b),  find  the  standard  deviation  σx.  

(e)  If  σ p = 10a ,  is  the  Heisenberg  Uncertainty  Principle  satisfied?  

 5.  An  operator  𝐴,  representing  observable  A,  has  two  normalized  eigenstates  Ψ!and  Ψ!  with  eigenvalues  a1  and  a2,  respectively.  Operator  𝐵,  representing  observable  B,  has  two  normalized  eigenstates  Φ!and  Φ!  with  eigenvalues   b1  and  b2 ,  respectively.  

The  eigenstates  are  related  by  Ψ1 =3Φ1 −Φ2

2  ,  and  Ψ2 =

Φ1 + 3Φ2

2.  

(a)  Observable  A  is  measured,  and  the  value  of  a2 is  obtained.  What  is  the  state  of  the  system  immediately  after  this  measurement?  (b)  If  B  is  now  measured,  what  are  the  possible  results  and  what  are  their  probabilities?  (c)  Right  after  the  measurement  of  B,  A  is  measured  again.  What  is  the  probability  of  getting  a2 ?    6.  There  are  three  particles  and  four  distinct  single-­‐particle  states  Φ1 ,Φ2 ,Φ3 ,  and  Φ4 .  How  many  different  three-­‐particle  states  can  be  formed,  (a)  if  the  three  particles  are  distinguishable,  (b)  if  the  three  particles  are  identical  fermions,  and  (d)  the  three  particles  are  identical  bosons?      Section  B  7. In a nuclear physics experiment, a photon beam with the energy of Eγ is incident on a fixed proton target (liquid hydrogen). Suppose you are interested in production of the strange particle Λ, via the reaction of γ+p K+ +Λ, what is the minimum Eγ you need to produce the Λ? The masses of the proton, K+, and Λ are 0.938, 0.494 and 1.116 GeV, respectively.  8. (Required) Consider the process of Compton scattering. A photon of wavelength λ is scattered off a free electron initially at rest. Let λ’ be the wavelength of the photon scattered in a direction θ relative to the photon incident direction. (a) Find λ’ in terms of λ and θ and universal constants. (b) Find the kinetic energy of the recoiled electron.  9.  (a)  A  laser  emits  a  pulse  of  light  which  travels  at  a  speed  of  c  (in  vacuum)  relative  to  the  laser.  Does  this  mean  that  the  speed  of  the  laser  relative  to  the  light  pulse  is  also  c?  Give  your  reasoning  for  your  answer.  

 (b)  A  laser  emits  a  pulse  of  light  in  the  positive  y  direction  at  the  same  time  that  a  space-­‐ship  passes  the  laser  at  a  speed  of  0.9c  in  the  positive  x  direction.  What  is  the  speed  and  direction  of  the  light  pulse  relative  to  the  space-­‐ship?