Is Dirac Notation Appropriate for Proving Hermitian Conjugates of Operators?

[natski]

This didn't seem appropriate for College level so I thought I'd post it here. I'm struggling to find a way to prove that the product of two operators P and Q written PQ have the hermitian conjugate Q*P* where the star denotes hermitian conjugate. Really just can't get off the first line with it. Any help would be a appreciated.

Natski

 

[matt grime]

<PQx,y> = <Qx,P*y>

the rest is left as an exercise for the reader...

 

[Fredrik]

As you can see the problem is trivial if you're using the notation that Matt's using. Maybe the problem is that you're using Dirac notation. The Dirac notation isn't very good when you're trying to prove things like this. Actually, you have to prove a bunch of "things like this" to understand why the Dirac notation even makes sense.

Note that the equation that defines the Hermitean conjugate is just <P*x,y>=<x,Py> in the notation that Matt's using. In Dirac notation (with |x> instead of x, and |y> instead of y), this equation takes the form (<x|P) |y>=<x| (P|y>), which looks more like an associativity rule than anything else.

If you don't understand exactly why this is so (and probably even if you do), it's better to stay away from the Dirac notation when you're doing proofs.