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

Definition df-kb 22431
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 22430, 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 21537 . 2  class  ketbra
2 vx . . 3  set  x
3 vy . . 3  set  y
4 chil 21499 . . 3  class  ~H
5 vz . . . 4  set  z
65cv 1622 . . . . . 6  class  z
73cv 1622 . . . . . 6  class  y
8 csp 21502 . . . . . 6  class  .ih
96, 7, 8co 5858 . . . . 5  class  ( z 
.ih  y )
102cv 1622 . . . . 5  class  x
11 csm 21501 . . . . 5  class  .h
129, 10, 11co 5858 . . . 4  class  ( ( z  .ih  y )  .h  x )
135, 4, 12cmpt 4077 . . 3  class  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) )
142, 3, 4, 4, 13cmpt2 5860 . 2  class  ( x  e.  ~H ,  y  e.  ~H  |->  ( z  e.  ~H  |->  ( ( z  .ih  y )  .h  x ) ) )
151, 14wceq 1623 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  22532
  Copyright terms: Public domain W3C validator