Newbie problem about <a|b> and |a> <b|

[member 141513]

[URL]http://upload.wikimedia.org/math/3/1/d/31dd2919c01a33cbe4e007cd3d027167.png[/URL]
my teacher said this means the integral of psi* rho dx

but how about the |psi> <rho|?
does it hv a integral form?

thx for any help!

 

[HallsofIvy]

\left&lt;\psi||\rho\right&gt;
is a number. It is, as you say, the integral
\int \psi(x)\rho(x)dx
integrated over the "universe".

|\psi\left&gt;\right&lt;\rho|
is an operator that changes one quantum state into another. Specifically, it changes the state \left|\phi\right&gt; into a\left|\psi\right&gt; where a is the number
\left&lt;\rho||\theta\right&gt;.

 

[dextercioby]

The advantage of using Dirac's bracket formalism is that by |\psi\rangle\langle\rho| you denote both an operator acting on the Hilbert space and on its dual (or both on kets and bras). It doesn't have an explicit integral form (the integrals actually appear when the abstract Hilbert space is chosen to be L^2(\Omega, dx)).

 

[The_Duck]

You might think of \langle \psi | \rho \rangle as representing something like

\psi(x) \int dy [\rho^\star(y) \bullet ]

where the dot gets filled in with whatever \langle \rho | acts on.

 

[A. Neumaier]

[URL]http://upload.wikimedia.org/math/3/1/d/31dd2919c01a33cbe4e007cd3d027167.png[/URL]
my teacher said this means the integral of psi* rho dx

but how about the |psi> <rho|?
does it hv a integral form?

thx for any help!

If you are familiar with matrix algebra, you can think of a bra as a row vector and a ket as a column vector. Then bra times ket gives a number, while ket times bra gives a matrix.
Dirac's notation is essentially an infinite-dimensional version of this, where sums are replaced by integrals.