HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  df-kb Structured version   Visualization version   GIF version

Definition df-kb 29050
Description: Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, 𝐴 𝐵 is an operator known as the outer product of 𝐴 and 𝐵, which we represent by (𝐴 ketbra 𝐵). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with definition df-bra 29049, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-kb ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-kb
StepHypRef Expression
1 ck 28154 . 2 class ketbra
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 chil 28116 . . 3 class
5 vz . . . 4 setvar 𝑧
65cv 1630 . . . . . 6 class 𝑧
73cv 1630 . . . . . 6 class 𝑦
8 csp 28119 . . . . . 6 class ·ih
96, 7, 8co 6793 . . . . 5 class (𝑧 ·ih 𝑦)
102cv 1630 . . . . 5 class 𝑥
11 csm 28118 . . . . 5 class ·
129, 10, 11co 6793 . . . 4 class ((𝑧 ·ih 𝑦) · 𝑥)
135, 4, 12cmpt 4863 . . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥))
142, 3, 4, 4, 13cmpt2 6795 . 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
151, 14wceq 1631 1 wff ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  kbfval  29151
  Copyright terms: Public domain W3C validator