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Mirrors > Home > HSE Home > Th. List > df-kb | Structured version Visualization version GIF version |
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 29626, 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.) |
Ref | Expression |
---|---|
df-kb | ⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ck 28733 | . 2 class ketbra | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | chba 28695 | . . 3 class ℋ | |
5 | vz | . . . 4 setvar 𝑧 | |
6 | 5 | cv 1532 | . . . . . 6 class 𝑧 |
7 | 3 | cv 1532 | . . . . . 6 class 𝑦 |
8 | csp 28698 | . . . . . 6 class ·ih | |
9 | 6, 7, 8 | co 7155 | . . . . 5 class (𝑧 ·ih 𝑦) |
10 | 2 | cv 1532 | . . . . 5 class 𝑥 |
11 | csm 28697 | . . . . 5 class ·ℎ | |
12 | 9, 10, 11 | co 7155 | . . . 4 class ((𝑧 ·ih 𝑦) ·ℎ 𝑥) |
13 | 5, 4, 12 | cmpt 5145 | . . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)) |
14 | 2, 3, 4, 4, 13 | cmpo 7157 | . 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥))) |
15 | 1, 14 | wceq 1533 | 1 wff ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥))) |
Colors of variables: wff setvar class |
This definition is referenced by: kbfval 29728 |
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