Derive the density matrix \(ho\) for (i) a free particle and (ii) a linear harmonic oscillator in the momentum representation and study its main properties along the lines of Section 5.3.

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5.3.A An electron in a magnetic field We consider, for illustration, the case of a single electron that possesses an intrinsic spin ho and a magnetic moment B, where is the Pauli spin operator and B = eh/2m is the Bohr magneton. The spin of the electron can have two possible orientations, or , with respect to an applied magnetic field H. If the applied field is taken to be in the direction of the z-axis, the configurational Hamiltonian of the spin takes the form =-o(H)=-MHz, (1) where o is the permeability of the vacuum; see Appendix H. In the representation that makes & diagonal, namely, = by= -()-(-)-(6-9). = = the density matrix in the canonical ensemble would be = (2) p = (e-B) Tr(e-B) 1 (3) 0 B 0 - We thus obtain for the expectation value z - e-BH (oz) = Tr(po)=BHH in perfect agreement with the findings of Sections 3.9 and 3.10. = = tanh(Pu, RoH), (4)