QM: Purity, two-outcome measurements

alc95 · Nov 13, 2017

Homework Statement

Typeset exam questions: Question 1 defines the thermal state gamma_T = (1/Z) exp(-H/(k_B T)) and partition function Z = Tr(exp(-H/(k_B T))), with parts (a) to (c) on purity; Question 2 defines projectors P0 and P1 and states psi and rho for parts (a) and (b).

Homework Equations

in addition to those provided in the questions, I used the following:

Tr(B) = sigma<x_j|B|x_j>
purity = Tr(rho^2)

The Attempt at a Solution

Handwritten derivation of the spectral decomposition of the Hamiltonian H as a sum over E_j |E_j><E_j|, and the density operator exp(-H/k_B T) and partition function Z = Tr exp(-H/k_B T) written out in bra-ket notation.

Handwritten working for Q2(a) computing projection probabilities: P(P0) = Tr(rho |psi><psi|) = <psi|rho|psi> = |<psi|phi>|^2 = 1/2, and P(P1) = |<psi|phi>|^2 = 1/2.

[/B]
I find calculating trace and purity very confusing. Am I on the right track with question 1? With question 2, I am really unsure about how to proceed further. Any help would be greatly appreciated.

Thanks in advance.

DrClaude · Nov 14, 2017

In Q1, you didn't specify what ##|x_k \rangle## is.

For Q2, just go step by step, either by replacing the density operator for its bracket representation then taking the trace, or the other way around.

QM: Purity, two-outcome measurements

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

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