# Projector in an arbitrary basis, howto compute?

1. Jul 3, 2009

### keen23

Hello!
once again I got trouble with Dirac-notation:

1. The problem statement, all variables and given/known data
Given an non-orthonormal basis. Measurement via the projection operator should not give an definit answer, in which state the system was due to the overlap. Geometrical thats clear, but I'm unable to compute that... :(

I tried it for the following states:
$$|\psi_0\rangle=|0\rangle$$
$$|\psi_1\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$$

2. Relevant equations

3. The attempt at a solution
My projector looks the following way (I am not sure about the coefficients):

$$P=|0\rangle\langle0|+\frac{1}{2} (|0\rangle+|1\rangle)(\langle 0|+\langle 1|)$$
And now I am confused. When I plug state psi1 on both sides I get a sum of several 0s and 1s I don't know how to interprete.

Any hints? Thank you!

2. Jul 3, 2009

### Avodyne

What exactly are you trying to do?

3. Jul 4, 2009

### George Jones

Staff Emeritus
Does $P^2 = P$? If it doesn't (I don't think that it does), then $P$ is not a projector.

I, too, would like to know the answer to this.

4. Jul 6, 2009