Projector in an arbitrary basis, howto compute?

[keen23]

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

Homework Statement

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 that's clear, but I'm unable to compute that... :(

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

Homework Equations

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!

 

[Avodyne]

What exactly are you trying to do?

 

[George Jones]

My projector looks the following way (I am not sure about the coefficients):

$$P=|0\rangle\langle0|+\frac{1}{2} (|0\rangle+|1\rangle)(\langle 0|+\langle 1|)$$

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

What exactly are you trying to do?

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

 

[keen23]

At the moment I am reading about general measurements and POVMs.
They say, that you can't definitely distinguish two non-orthonormal states by a "normal" measurement, meaning by a projection onto the states (since each has an overlap with the other).
(That's why you need the POVM to do that.)

I just would like to see the math.

Or is that the point, that I can't construct an projection-operator from my non-orthonormal states? But actually you can always project a vector onto its constructing basis vectors, no matter if they are orthogonal or not.

I hope this makes my confusion a bit clearer.