Density Matrix for Harmonic Oscillator

The average value of some operator  can in this basis be written〈A〉 = 1 Z Tr[Âρ] = 1 ∑〈ψ n |Âψ n 〉e −βE n(33)Zwhere |ψ n 〉 is an energy eigenstate: H|ψ n 〉 = E n |ψ n 〉. For example, we haven〈x〉 = 1 Z Tr[xρ] = 1 ∑〈ψ n |xψ n 〉e −βE nZ n〈p〉 = 1 Z Tr[pρ] = 1 ∑〈ψ n |pψ n 〉e −βE nZn(34)In order to evaluate the averages we need to know the matrix elements 〈ψ n |xψ n 〉 and〈ψ n |pψ n 〉. This is where the ladder operators come in.Starting from the Hamiltonian:ħω(− ∂2)2 ∂X 2 + X 2 = ħω 2(K 2 + X 2) (35)where we define K = −i ∂∂X. From the commutation relations [x, p] = iħ we find[X ,K ] = i (36)The ladder operators are defined asa = 1 2(X + iK )a † = 1 2(X − iK )⇐⇒X = 1 ) (a † + a2K = i ) (37)(a † − a2Inserting this into the Hamiltonian:H = ħω [− 1 ()a † a † − a † a − aa † + aa + 1 () ]a † a + a † a + aa † + aa = ħω [a † a + aa †]2 222(38)Here we need to keep track of the ordering because a † and a do not commute sinceX ,K do not commute, in fact:[X ,K ] = i =⇒ [a, a † ] = aa † − a † a = 1 (39)Using this in the Hamiltonian we get(H = ħω a † a + 1 )2(40)6