852_2010hw1

# 852_2010hw1 - PHYS852 Quantum Mechanics II Fall 2008...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PHYS852 Quantum Mechanics II, Fall 2008 HOMEWORK ASSIGNMENT 1: Density Operator 1. [10pts] The trace of an operator is deﬁned as T r {A} = m m|A|m , where {|m } is an arbitrary basis set. Introduce a second arbitrary basis set, and use it to prove that the trace is independent of the choice of basis. 2. [10pts] Prove the linearity of the trace operation by proving T r {aA + bB } = aT r {A} + bT r {B }. 3. [10pts] Prove the cyclic property of the trace by proving T r {ABC } = T r {BCA} = T r {CAB }. 4. [10pts] Which of the following density matrices correspond to a pure state? ρ1 = 2 7 0 0 ρ2 = 5 7 ρ4 = 1 5 √ 2 5 √ 2 5 4 5 1 4√ −i 3 4 i √ 3 4 3 4 ρ5 = 00 01 ρ3 = 1 9 2 9 2 9 2 9 4 9 4 9 2 9 4 9 4 9 d 5. [10 pts] Derive the equation of motion, dt ρ(t) = − i [H, ρ(t)], using Schr¨dinger’s equation and o the most general form of the density operator, ρ = j Pj |ψj ψj |. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online