Problem%20Set%207%202008.1 - R =0. (f) Show that the...

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Problem set #7 (Due June 9 12:00 pm) 1. Consider a two-state system. Examples include a spin 1/2 particle, the polarization of a photon, two discrete energy levels of an atom, etc. (a) Show that any operator A on this space can be written as A = 2 1 ( B 0 I + ), where I is the identity operator, σ x , σ y , σ z are the Pauli spin operators and B 0 , B x , B y , B z are scalars. σ r r B (b) Show that B i = Tr( σ i A), where i=0,x,y,z, with σ 0 = I. (c) Given the density operator ρ describing the state of the system, show that ρ = 2 1 (1 + ), where r r R ) ( σρ r r Tr R = and show ) ( 2 1 0 R B B A r r + = The vector R is known as the Bloch vector, and completely characterizes the state for a two dimensional Hilbert space. (d) Show that | R | 2 1, with | R | 2 = 1 for pure state and | R | 2 < 1 for mixed state. Thus, the vector R lies somewhere within a sphere of unit radius, with those states on the surface of the sphere describing pure states. (e) For pure states with R e z , and R =a e x +b e y (with a 2 +b 2 =1) give the state vector | ψ >. Also, describe physically the state with
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Unformatted text preview: R =0. (f) Show that the projection operator onto the spin in the direction can be expressed as |n;+&gt;&lt;n;+| = n ˆ ) ˆ ( 2 1 r ⋅ + n I . Can an arbitrary pure state of the spin 1/2 particle be expressed this way? 2. Suppose we have an ensemble of spin 1/2 particles that consists of a statistical mixture with 1/3 of the ensemble in the state |z;+&gt; and 2/3 of the ensemble in the state |z;-&gt;. (a) Find the matrix of the density operator in the basis {|z;+&gt;, |z;-&gt;}, and in the basis of eigenstates of S x , {|x;+&gt;, |x;-&gt;} (b) Find, &lt;S x &gt;, &lt;S y &gt;, &lt;S z &gt; for this state, and compare it to the result for the mixed state |) ; ; | | ; ; (| 2 1 − &gt;&lt; − + + &gt;&lt; + = z z z z ρ . Please comment on your findings. (c) Find n S ˆ ⋅ r for an arbitrary axis , for the 1/3-2/3 mixture and for the completely incoherent state with a 1/2-1/2 mixture. Please comment n ˆ 3. Sakurai, Chap. 3. 25...
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