Ah.. got it, sorry, matrix notation getting me confused.
A measurement of an observable will be one of the eigenvalues:
(A-λI)\psi = 0
\left|\begin{array}{cc} 1-λ & 1 \\ 1 & -1-λ \end{array}\right| = 0
λ = ±√2
I'm now struggling to understand what this means.
If I operate C on u+, or rather matrix multiply C with the vector, I get the vector:
\widehat{C}u_{+} = \left(\begin{array}{c} 1 \\ 1 \end{array} \right)
Which clearly has a magnitude \pm\sqrt{2} (which I have in the solutions for this...
It would appear that B is:
\widehat{B} = \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)
and thus C:
\widehat{C} = \left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array} \right)
There is probably a much easier way to obtain those matrices than the way I just used (trial and...
The matrix method sounds useful, so I'm reading up on it.
As far as I can see so far I can represent each eigenfunction as a vector in terms of the 'u' basis:
u_{+} = \left(\begin{array}{c} 1 \\ 0 \end{array}\right)
u_{-} = \left(\begin{array}{c} 0 \\ 1 \end{array}\right)
v_{+} =...
Thank you for your response.
I'm a few weeks into this QM module, I've yet to cover the matrix representations and thus my understanding is lacking in this approach. Since you've suggested this I have realized that this method is outlined a few pages after what I had previously studied in my...
Homework Statement
Observable \widehat{A} has eigenvalues \pm1 with corresponding eigenfunctions u_{+} and u_{-}. Observable \widehat{B} has eigenvalues \pm1 with corresponding eigenfunctions v_{+} and v_{-}.
The eigenfunctions are related by:
v_{+} = (u_{+} + u_{-})/\sqrt{2}
v_{-} =...