How Do You Calculate the Expectation Value of an Observable for a Quantum State?

[sunrah]

Homework Statement

given
$\mid \psi \rangle = \frac{1}{\sqrt{2}} (\mid1\rangle + \mid2\rangle )$
where $\mid1\rangle, \mid2\rangle$ are orthonormal
calculate
i)density operator
ii) $\langle A \rangle$ where A is an observable

Homework Equations

The Attempt at a Solution

i) $\rho = \frac{1}{2} (\mid1\rangle\langle1\mid + \mid2\rangle\langle2\mid)$

ii) $\langle A \rangle = \frac{1}{2}\langle1\mid A\mid1\rangle + \frac{1}{2}\langle2\mid A\mid2\rangle + \frac{1}{2}\langle2\mid A\mid1\rangle + \frac{1}{2}\langle1\mid A\mid2\rangle$
i guess $\frac{1}{2}\langle2\mid A\mid1\rangle + \frac{1}{2}\langle1\mid A\mid2\rangle$ = 0 (because first 2 terms are each half the expected value but surely depends on operator therefore not necessarily zero??)

also I would like to use the trace to solve this:
$\langle A \rangle = tr[\rho A] = tr[\frac{1}{2}\mid1\rangle \langle1\mid A + \frac{1}{2}\mid2\rangle \langle2\mid A]$ but what does this mean?

 

[vela]

Homework Statement

given
$\mid \psi \rangle = \frac{1}{\sqrt{2}} (\mid1\rangle + \mid2\rangle )$
where $\mid1\rangle, \mid2\rangle$ are orthonormal
calculate
i)density operator
ii) $\langle A \rangle$ where A is an observable

Homework Equations

The Attempt at a Solution

i) $\rho = \frac{1}{2} (\mid1\rangle\langle1\mid + \mid2\rangle\langle2\mid)$

This is the density matrix for a mixed state, where half the particles are in state $\vert 1 \rangle$ and half are in state $\vert 2 \rangle$. The state you were given is a pure state.

ii) $\langle A \rangle = \frac{1}{2}\langle1\mid A\mid1\rangle + \frac{1}{2}\langle2\mid A\mid2\rangle + \frac{1}{2}\langle2\mid A\mid1\rangle + \frac{1}{2}\langle1\mid A\mid2\rangle$
i guess $\frac{1}{2}\langle2\mid A\mid1\rangle + \frac{1}{2}\langle1\mid A\mid2\rangle$ = 0 (because first 2 terms are each half the expected value but surely depends on operator therefore not necessarily zero??)

You can't really say anything about those matrix elements since you don't know anything about $\hat{A}$ except that it's an observable.

also I would like to use the trace to solve this:
$\langle A \rangle = tr[\rho A] = tr[\frac{1}{2}\mid1\rangle \langle1\mid A + \frac{1}{2}\mid2\rangle \langle2\mid A]$ but what does this mean?

Say you have some basis and you find the matrix representing $\rho\hat{A}$. The trace is simply the sum of the diagonal elements. How would you write that in Dirac notation?

 

[sunrah]

thanks, I have since worked out that the density matrix I gave is not complete (I assumed that the outer product of two orthogonal vectors would be zero!)