1. MATRIX REPRESENTATION OF OPERATORS Consider a spin 1/2

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1. MATRIX REPRESENTATION OF OPERATORS
Consider a spin 1/2 particle. Let me introduce a new operator A
represented (in the usual Sz-basis) by the matrix
1+1
A= (₁ ²₁ ¹+¹)
0
(1) (2) Is it possible that A is an observable operator? Prove it quickly
using the adjoint.
(2) (2) If you measure A, what values can you get and why? (Hint:
this requires you to diagonalize A and interpret.)
(3) (2) Write A in the basis of its own eigenstates. (You'll compute
one of these eigenstates in the next step explicitly, you don't need
it yet.)
(4) (3) Since you almost found it for the last step, take one step further
and show that one eigenstate of A in the Sz-basis is
(1-1))
(1/√(3)).
1
(5) (3) Are A and Sz (the z-component of spin) compatible observables?
How do you decide?
(a) (2) Explain in your own words what this means in terms of
simultaneous physical measurements of A and Sz.