Quantum HW: Find Operator P_2 Given psi and c1c1*+c2c2*=1

[quantum99]

Homework Statement

I can't figure this one out: A two-state quantum system is psi = c1|1> + c2|2> and the states are orthonormal and c1 and c2 are complex and normalized condition is c1c1* + c2c2* = 1

where * denotes complex conjugate

The probability of measuring the quantum system to be in state |2> is given by the expectation value of a certain operator P_2

What is the operator P_2

Homework Equations

standard expectation value and operator equations

The Attempt at a Solution

I think i figured that the probability to measure the system in state |2> is c2c2*

then c2c2* = <P_2> = <psi|P_2|psi> but going through that I can't figure out how to determine P_2 ... i think i am missing something simple and any help would be very much appreciated! :)

 

[OlderDan]

Have you done anything with projection operators?

 

[StatMechGuy]

Let's try thinking about this just a bit.

What's the expectation value of an operator for a state $$| a\rangle$$? It's $$\langle a | \hat{\mathcal{O}} | a \rangle$$. Well, we can write this as $$\sum_i \langle a | o_i \rangle \langle o_i \rangle \hat{\mathcal{O}} | a \rangle$$ where $$\mathcal{O}$$ is the observable in question, and the $$|o_i \rangle$$ are the eigenstates of the observable. The the above sum becomes

$$\langle \hat{\mathcal{O}} \rangle = \sum_i | \langle o_i | a \rangle |^2 o_i$$

If we interpret this probabilistically, this is the sum over the probability of measuring the value $$o_i$$ times this value, which is pretty much the definition of an average.

So what's the $$\hat{\mathcal{O}}$$ that you want?