Dual hilbert space

1. Mar 1, 2014

wasi-uz-zaman

Hi, if ket is 2+3i , than its bra is 2-3i , my question is 2+3i is in Hilbert space than 2-3i can be represented in same hilbert space, but in books it is written we need dual Hilbert space for bra?

2. Mar 1, 2014

tiny-tim

hi wasi-uz-zaman!
it's not the same, it's back-to-front!

you can't add a ket to a bra

it's like ordinary 3D vectors and pseudovectors (a 3D pseudovector is a cross product of two 3D vectors)

you can't add a vector to a pseudovector …

they look as if they exist in the same space, but in fact the two spaces are back-to-front

3. Mar 1, 2014

Staff: Mentor

Kets are elements of a vector space, bras are linear functions defined on the kets - entirely different things.

They are, with a few caveats such as Rigged Hilbert Spaces, isomorphic via the Rietz-Fisher Theorem - but that doesn't mean they are the same.

Thanks
Bill

4. Mar 2, 2014

stevendaryl

Staff Emeritus
The easiest Hilbert space to deal with (or maybe the second easiest) is that of spin-1/2 states. Then the states are (or can be represented as) column matrices with 2 elements. The dual states are the row matrices with 2 elements. Obviously it doesn't make any sense to add a row and a column.