Extra HW 1. Prove the following properties of the density matrix. (a) ? is a Hermitian...
1. Prove that the matrices are hermitian or anti-hermitian. 2. Prove that in order for the Dirac equation to be covariant (i.e., form-invariant) under the Lorentz group of trans formations with the constraint that the gamma-matrices are unchanged, the following relation must hold: , where is the Lorentz-transformation matrix for the 4- vectors (x'=Lx) and S is a unitary matrix that transforms the spinor as
27. Some More Facts about Density Operators: Let ? be a density operator acting in an N-dimensional complex vector space. (a) Show that ?-?2 if and only if ? is a pure state. (b) show that Trlo l 1, with equality if and only if p is a pure state. Thus, Tr ?2 is one convenient m easure of the purity or impurity of the state. (Borrowing from the interpretation of mixed states for spin-systems, this quantity is sometimes referred...
1. Prove that the time evolution of the density operator p (in the Schrödinger picture) is given by: (1) p(t) = U(t, top(tout(t, to) 2. Suppose we have a pure ensemble at t = 0. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrödinger equation.
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3.1 Rotations and Angular-Momentum Commutation Relations 159 We are particularly interested in an infinitesimal form of Ry: (3.1.4) where terms of order & and higher are ignored. Likewise, we have R0= ° :- R(E) = 1 (3.1.5) and (3.1.5b) - E01 which may be read from (3.1.4) by cyclic permutations of x, y, zthat is, x y , y → 2,2 → x....