Calculate density of probability.

[Helpless guy]

Homework Statement

Given a quantum density ρ=w1|φ1><φ1|+w2|φ2><φ2|
where φ1=c^(1/2)*exp[-(x-b)²/2c²]
φ2=c^(1/2)*exp[-(x+b)²/2c²]
find the density of probability of finding x₀ if X(operator) is measured

Homework Equations

P_A=Σw_i(Σ|<a;α|φ_i>|²)

The Attempt at a Solution

I know that the probability of measuring a x₀ is Tr(ρΣ|x₀><x₀|), but I don't know how to calculate the projector.

All help is welcome.

 

[mark726]

Thank you for your question. The projection operator for a given state |x0> is simply the outer product |x0><x0|, which can be written as a matrix with a 1 in the (x0,x0) position and 0s everywhere else. Therefore, the probability of measuring x0 is given by Tr(ρ|x0><x0|), which is equivalent to Σwi|<x0|φi>|^2. I hope this helps.
Scientist