/sci/ - Science & Math

[Anonymous No.10049416]

Can someone please explain to a brainlet how do you go from the matrix representation of an operator, to the bra ket notation?

Is there something i'm missing?

[Anonymous No.10049424]

http://www.mathpages.com/home/kmath638/kmath638.htm

[Anonymous No.10049432]

>>10049424
I mean i know what they represent, what i'm asking is if there's a way to just analytically look at the matrix and say "Yeah, if i make outer product with these vectors i will surely get this matrix".

Basically what i'm asking is if writing P in its bra ket notation (starting from the matrix representation) should be trivial

[Anonymous No.10049435]

>>10049416
kets are just fancy ways of listing basis vectors
|0>= [1; 0; 0; 0; 0....]
|1>= [0; 1; 0; 0; 0....]
|3>= [0; 0; 0; 1; 0....]
...

bras are just the conjugate transpose (row vectors, dual space, etc)
<0| = [1, 0, 0, ....]
<1| = [0, 1, 0, ....]
....


|0><1| = e0 * e1' = [1; 0]* [0, 1] = [0, 0; 1, 0] = e0,1

[Anonymous No.10049445]

nvm now i understand it, thanks folks

[Anonymous No.10049449]

>>10049432
this guy has it
>>10049435
but bras and kets are much more general since they apply to differential operators and wavefunctions, which take the role of matrices and vectors when your basis of eigenstates becomes infinite-dimensional.

like e.g. the position basis. you could try to write down eigenvectors where you have a unit vector for each point along the x coordinate, but then your matrix becomes infinitely big. so you use dirac delta functions for eigenstates and make superpositions of those to form e.g. eigenstates of momentum, which conveniently becomes an operator with a derivative in it instead of a matrix thingy

[Anonymous No.10049451]

>>10049432
The numbers bras and kets are just the lines and colums of the matrix as you can see. That comes from the identity A_ij = <i|A|j>, it's just basic linear algebra