相干态是超完备的一个相干态|α)可以用其它相干态|α)展开qnα12Z.[α) = e2nVini利用『α）《αdα=（证明见后）有[α')<α'lα)dα"[α)[ α")exp(-la}2 + α*α' -}lα'l120展开系数17
17 l 相干态是超完备的 一个相干态|�⟩可以用其它相干态|�′⟩展开 |�⟩ = �) 4 7 # I6 �6 �! |�⟩ 利用∫ |�⟩⟨�|�#� = �（证明见后）有 |�⟩ = 1 � L |� ⟩ : �: � �#�: = L |� ⟩ : 1 � exp − 1 2 � # + �∗�: − 1 2 �: # 展开系数 �#�:
下面证明『α）《αdα=（思路）Ed2αlaa]d’α=e-nnle12Vn!n(α*)n.α"e-la|2dα[n'XnlVn'!.n!n'n2元α=laleio[α|n+n'+1 . -lal’ d]al |(αt)n.an'e-lal°d2αei(n'-n)e de = Tn! Sn,n'd2α=α.dα .de00[α)(α]d2α=π[n)n| = πn18
18 l 下面证明∫ |�⟩⟨�|�6� = � (思路) 9 |�⟩⟨�|d6� = 9 �% / $ 6 ⋅ < 9% �9% �: ! |�: ⟩⟨�|�% / $ 6 ⋅ < 9 (�∗)9 �! d6� = 9 �% / $ ⋅ < 9% < 9 (�∗)9 ⋅ �9% �: ! ⋅ �! d6� |�: ⟩⟨�| 9 �∗ 9 ⋅ �9% �% / $ d6� � = � �&; d6� = � A d|�| A d� 9 < = � 9(9%(# ⋅ �% / $ d|�| 9 < 6> �& 9%%9 ;d� = ��! �9,9% 9 |�⟩⟨�|d6� = �< 9 |�⟩⟨�| = �
(细节)下面证明「α）<αld?α=元a2（a*)nlα)(α|dα =『mZdα福Vn!2(a")n.an-1a/2d2αn'n]Vn'!-n!nn|n'n-(α")n .αn'e-lal’d2αVn'!.n!n'nα=lαlei8[(a")n .a"'e-la’ d2αei(n'-n)e deα|n+n'+1.e-a’ d]α]d?α=.d .de1[ la|n+n'+1 . e-lal’ d]a]=2元0n,n00olα|2n+1.e-la1’d]al=2元8n,n伽马函数的积分定义= Tn! On.n'[α)(αdα = ,[n)(n| = πn19
19 l 下面证明∫ |�⟩⟨�|�&� = � (细节) 6 |�⟩⟨�|d&� = 6 �' ( ! & ⋅ : )" �)" �*! |�* ⟩⟨�|�' ( ! & ⋅ : ) (�∗)) �! d&� = 6 �' ( ! ⋅ : )" : ) (�∗)) ⋅ �)" �* ! ⋅ �! d&� |�* ⟩⟨�| = : )" : ) |�* ⟩⟨�| �*! ⋅ �! 6 �∗ ) ⋅ �)" �' ( ! d&� 6 �∗ ) ⋅ �)" �' ( ! d&� � = � �+, d&� = � + d|�| + d� 6 - . � )/)"/0 ⋅ �' ( ! d|�| 6 - &1 �+ )"') ,d� = 2��),)" 6 - . � )/)"/0 ⋅ �' ( ! d|�| = 2��),)" 6 - . � &)/0 ⋅ �' ( ! d|�| = ��! �).)" 6 |�⟩⟨�|d&� = �: ) |�⟩⟨�| = � 伽马函数的积分 定义 ←
量子态或算符的相干表象展开基于完备性，任意量子态f）可以在相干表象下展开If) = dα[α)(αlf)注意：《αlf)涉及到复数的运算，由于超完备性，其形式可能不唯一任意算符↑可以在相干表象下展开基于完备性，T=I.T.I:d2αd2β /α)(α|TIβ)βlT2d2αd2β(α|TIβ)/α)<βl同样的：<αTβ>涉及到复数的运算，由于超完备性，其形式可能不唯一20
20 量子态或算符的相干表象展开 基于完备性，任意量子态|�⟩可以在相干表象下展开 |�⟩ = � � ∫ ��� |�⟩⟨�|�⟩ 注意： � � 涉及到复数的运算，由于超完备性，其形式可能不唯一 基于完备性，任意算符 �a 可以在相干表象下展开 �a = � · �a · � = � �� L ��� ��� |�⟩⟨�|�a|�⟩⟨�| = � �� L ��� ��� ⟨�|�a|�⟩|�⟩⟨�| 同样的：< � �> � > 涉及到复数的运算，由于超完备性，其形式可能不唯一
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