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Advanced Physics Homework Help
Finding state vectors for pure states
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[QUOTE="Pouyan, post: 6053172, member: 549817"] [h2]Homework Statement [/h2] Is the following matrix a state operator ? and if it is a state operator is it a pure state ? and if it is so then find the state vectors for the pure state. [ATTACH=full]230379[/ATTACH] If you don't see image here is the matrix which is 2X2 in MATLAB code: [9/25 12/25; 12/25 16/25] [h2]Homework Equations[/h2] To be a state operator, if we have a operator ρ we know : Tr(ρ)=1 ρ=ρ[SUP]t[/SUP] (self-adjoint) <u|ρ|u> >= 0 for all vectors |u> and these means : the sum of eigenvalues must be 1 and eigenvalues must be greater or equal to zero For pure state what do I know are these: ρ=|ψ><ψ| where |ψ> is the unit-normed vector called state vector. The average value of an observable R in this pure state is: <R> = Tr(|ψ><ψ|R) = <ψ|R|ψ> The other condition is : ρ[SUP]2[/SUP]=ρ (which is possible for 1 or 0 but the sum of eigenvalues must be 1) The third condition is : Tr(ρ[SUP]2[/SUP])=1 [h2]The Attempt at a Solution[/h2] This matrix has eigenvalues 1 and 0. And this means it is a state operator. In my solution I do see that this matrix is a pure state and it has the vector state : (3/5 4/5). But I don't know how I can use conditions for pure state to see that if a matrix or an operator is a pure state and I can not either get the state vectors. What do I know is that : WWith eigenvalue 1 we get vector (-(4/3) 1). I do see that (3/5 4/5) is the norm of the diagonal of the matrix, [9/25 12/25; 12/25 16/25], that is in the first place in the matrix we have 9/25 and √(9/25)= 3/5. In the last place of this matrix we have 16/25 and √(16/25) = 4/5 ofcourse 16/25 + 9/25 =1 But is that correct to think so ? [/QUOTE]
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Finding state vectors for pure states
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