problemset10

The second from the k terms is bk w i w ik v

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Unformatted text preview: to �xy + ik, where �xy is just the x and y derivatives and k = k z , so that ˆ � �� � � � b [�xy + ik] · b�xy · bk· ˆk = D = +i . a [�xy + ik] a�xy ak ´ � 1 ˆ ¯ + a v · v� , we obtain two terms from �w, Dk w� �, ˆ ˆ from the two terms above. The first, from the �xy terms, works exactly like D (since we analyzed D in any number of dimensions, and this is just the 2d case), and so is anti-Hermitian as before. The second, from the k terms, is: � � ˆ bk· ¯ �w, i w� � = (¯ik · v + v · iku) u ak Ωxy ˆ � � =− iku · v + ik · vu Plugging this into the 2d inner product �w, w� � = ¯� b uu �1 Ωxy Ωxy � = �−i ˆ ˆ∗ and so this part is anti-Hermitian too. Hence Dk = −Dk . 1 bk· ak � w, w� �, (c) Here, we just apply the ordinary quotient/product rules of differentiation: � � ˆ D ˆ ˆw ˆ ˆ � ∂ wk , iDk wk � +...
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