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Definition df-kb 22423
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 22422, 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  e.  ~H , 
y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y
)  .h  x ) ) )
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-kb
StepHypRef Expression
1 ck 21529 . 2  class  ketbra
2 vx . . 3  set  x
3 vy . . 3  set  y
4 chil 21491 . . 3  class  ~H
5 vz . . . 4  set  z
65cv 1627 . . . . . 6  class  z
73cv 1627 . . . . . 6  class  y
8 csp 21494 . . . . . 6  class  .ih
96, 7, 8co 5819 . . . . 5  class  ( z 
.ih  y )
102cv 1627 . . . . 5  class  x
11 csm 21493 . . . . 5  class  .h
129, 10, 11co 5819 . . . 4  class  ( ( z  .ih  y )  .h  x )
135, 4, 12cmpt 4078 . . 3  class  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) )
142, 3, 4, 4, 13cmpt2 5821 . 2  class  ( x  e.  ~H ,  y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) ) )
151, 14wceq 1628 1  wff  ketbra  =  ( x  e.  ~H , 
y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y
)  .h  x ) ) )
Colors of variables: wff set class
This definition is referenced by:  kbfval  22524
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