- #1
sgsurrey
- 8
- 0
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_{-} = (u_{+} - u_{-})/\sqrt{2}
Show that \widehat{C} =\widehat{A} + \widehat{B} is an observable and find the possible results of a measurement of \widehat{C}.
Find the probability of obtaining each result when a measurement of \widehat{C} is performed on an atom in the state u_{+} and the corresponding state of the atom immediately after the measurement in terms of u_{+} and u_{-} .
Homework Equations
\widehat{A}u_{\pm}=\pm u_{\pm}
\widehat{B}v_{\pm}=\pm v_{\pm}
v_{+} = (u_{+} + u_{-})/\sqrt{2}
v_{-} = (u_{+} - u_{-})/\sqrt{2}
u_{+} = (v_{+} + v_{-})/\sqrt{2}
u_{-} = (v_{+} - v_{-})/\sqrt{2}
The Attempt at a Solution
Showing that C is an observable I have assumed is as simple as the fact that it is a linear combination of the A and B operators, which are observables matching the requirement to have real eigenvalues.
I'm slightly puzzled by the measurement part of the question; I have tried simply:
\widehat{C}v_{+} = \widehat{A}v_{+} + \widehat{B}v_{+}<br /> = \widehat{A}(u_{+} + u_{-})/\sqrt{2} + v_{+} = v_{-} + v_{+} = \sqrt{2}u_{+}
I'm not sure I fully understand this however; I expected a result of the format:
\widehat{C}v_{+} = c_{+}v_{+}
...where c is a 'result of the measurement', the eigenvalue of C. I'm not sure if it is correct to say that \sqrt{2} is the eigenvalue if the resulting eigenfunction is not the same as the starting eigenfunction.
The answer I have been given for this part of the question is \pm\sqrt{2}, but I feel I have now simply found the simplest way to get this in an answer, without understanding why (had I not been lost from the outset I would have avoided looking at the answer in the first place).
For the later part of the question my attempted answer is so far from the solution that I am clueless on how to approach it at this point. I am hoping that with some insight into this first part of the question I might understand how to approach the rest of the question. I would very much appreciate any guidance.