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49:: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
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300:{\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.}
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of that
Hilbert space, and then it is also called a
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689:Unitary transformations in quantum mechanics
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27:Endomorphism preserving the inner product
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420:A closely related notion is that of
470:{\displaystyle U:H_{1}\to H_{2}\,}
132:between two inner product spaces,
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427:, which is a bijective function
122:{\displaystyle U:H_{1}\to H_{2}}
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314:, as one can see by setting
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416:Antiunitary transformation
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182:{\displaystyle H_{2},}
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397:{\displaystyle H_{2}}
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152:{\displaystyle H_{1}}
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73:). In other words, a
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336:{\displaystyle x=y.}
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67:inner product spaces
30:For other uses, see
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45:that preserves the
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79:bijective function
57:More precisely, a
43:linear isomorphism
37:In mathematics, a
642:complex conjugate
606:{\displaystyle y}
586:{\displaystyle x}
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350:In the case when
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53:Formal definition
18:Unitary transform
16:(Redirected from
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684:Wigner's theorem
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654:Antiunitary
423:antiunitary
189:such that
698:Categories
558:⟩
546:⟨
538:¯
534:⟩
522:⟨
513:⟩
495:⟨
454:→
282:∈
251:⟩
238:⟨
219:⟩
200:⟨
107:→
69:(such as
648:See also
573:for all
310:It is a
482:complex
61:is an
77:is a
41:is a
593:and
377:and
159:and
644:.
613:in
700::
412:.
626:1
622:H
601:y
581:x
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543:=
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516:=
510:y
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498:U
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458:H
449:1
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441::
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390:2
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235:=
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224:H
215:y
212:U
209:,
206:x
203:U
177:,
172:2
168:H
145:1
141:H
115:2
111:H
102:1
98:H
94::
91:U
34:.
20:)
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