# Expectation Values in the Quantum Trajectory Formalism

If I want to know the expectation value of an operator O in the quantum trajectory formalism, I average over $$N$$ trajectories, where I call one such trajectory $$\Psi_n$$:

$$$$\langle O \rangle = \frac{1}{N}\sum_{n=1}^N \langle \Psi_n | O |\Psi_n \rangle,$$$$ correct?

If so, my question is: Is that still correct if the different $$\Psi_n$$ are not orthogonal/parallel to each other?

For example: If I am interested in the average probabiltiy density of the position distribution $$\rho$$, do I get it by calculating:

$$$$\rho(x) = \frac{1}{N}\sum_{n=1}^N \Psi^*_n(x) \Psi_n(x),$$$$

even if $$\langle \Psi_n(x)|\Psi_m(x) \rangle \neq \delta_{n,m}$$?

If not, what would be the correct formula?

• yes, i believe what i wrote should be correct, but i want to be sure – Luke Oct 11 '20 at 18:55