Density Matrix for Harmonic Oscillator

More accurately this is the density matrix in position-basis 1 .With this density matrix we can evaluate the expectation value of the momentum 2p = −iħ ∂∂x : ∫〈p〉 =(dx −iħ ∂ ) ∫∂x ψ(x) ψ ∗ (x) =(dx −iħ ∂ )ρ(x, x ′ ) ∣ ∂xx ′ =x(6)Note that ρ(x) = ρ(x, x).From the time-dependant Schrödinger eq. and its complex conjugate:i∂ t ψ(x) = Ĥψ(x), −i∂ t ψ ∗ (x) = ψ ∗ (x)Ĥ (7)we find the equation of motion for the density matrixi∂ t ρ(x, x ′ ) = ( i∂ t ψ(x) ) ψ ∗ (x ′ ) + ψ(x) ( i∂ t ψ ∗ (x ′ ) )= Ĥρ(x, x ′ ) − ρ(x, x ′ )Ĥ= [Ĥ,ρ](8)which is called the quantum Liouville equation.1.2 Quantum StatisticsSo far we have only discussed the intrinsic randomness due to the probabilistic natureof QM. In general we also have randomness due to initial conditions etc. We canimagine that our system is in one of many states |ψ n 〉 (or in postion basis ψ n (x)).The statistical average is then to be taken wrt an ensemble of states |ψ n 〉 weighted bysome probability w n :〈x〉 = ∑ w n 〈x〉 ψn = ∑ ∫w n dxψ n (x)xψ ∗ n (x) (9)nnIf we then define a new statistical density matrixρ(x, x ′ ) = ∑ nw n ψ n (x)ψ ∗ n (x′ ) (10)1 The more general form of the density matrix of a pure state is ˆρ = |ψ〉〈ψ| and we can defineaverages as 〈A〉 = Tr( ˆρ) = ∫ dx ∫ dx ′ 〈x ′ |Â|x〉〈x|ψ〉〈ψ|x ′ 〉 = ∫ dx ∫ dx ′ A(x ′ , x)ψ(x)ψ ∗ (x ′ ) =∫dx∫dx ′ A(x, x ′ )ρ(x, x ′ ) which for  = ˆx with 〈x ′ | ˆx|x〉 = xδ(x − x ′ ) gives us 〈x〉 = ∫ dxxρ(x, x)2 In the Dirac Bra-Ket notation we have 〈x ′ | ˆp|x〉 = δ(x − x ′ )(−iħ ∂∂x) so that∫〈p〉 = Tr( ˆp ˆρ) =∫dx∫dx ′ 〈x ′ | ˆp|x〉〈x| ˆρ|x ′ 〉 =∫dxdx ′ δ(x−x ′ )(−iħ ∂∂x ρ(x, x′ )) =∫(dx −iħ ∂ )∂x ρ(x, x′ )x ′ =x2