The interference term. Confused.

[LostConjugate]

Can someone explain what the difference is between the expectation value of an observable in a pure vs a mixed state. The equations are identical. For example with spin up and down $$|A|^2<S_u|O|S_u> + |B|^2<S_d|O|S_d>$$

 

[Bill_K]

What you've quoted is <O> for the case where the system is in an incoherent mixture of |S_u> and |S_d>. This would be a classical mixture, like heads and tails on a coin, and is represented by a density matrix rather than a wavefunction.

More typically what we mean by a mixed state is a coherent mixture, |S> = A |S_u> + B |S_d>, in which case the expectation value <S|O|S> will need to include interference terms A*B <S_u|O|S_d> and B*A <S_d|O|S_u>.

 

[LostConjugate]

What you've quoted is <O> for the case where the system is in an incoherent mixture of |S_u> and |S_d>. This would be a classical mixture, like heads and tails on a coin, and is represented by a density matrix rather than a wavefunction.

More typically what we mean by a mixed state is a coherent mixture, |S> = A |S_u> + B |S_d>, in which case the expectation value <S|O|S> will need to include interference terms A*B <S_u|O|S_d> and B*A <S_d|O|S_u>.

If you could quickly reference page 358 of this pdf http://www.physics.sfsu.edu/~greensit/book.pdf

It shows that your equation is equivalent to mine...

 

[kith]

Can someone explain what the difference is between the expectation value of an observable in a pure vs a mixed state. The equations are identical. For example with spin up and down $$|A|^2<S_u|O|S_u> + |B|^2<S_d|O|S_d>$$

That's true only if your states are eigenstates of O (so the interference terms vanish).

This means, you can't distinguish between a coherent superposition and an incoherent mixture by measuring in one basis alone. But you don't get identical expectation values in all bases!

Consider a superposition of 50:50 spin up and spin down along a given axis. If you change your measurement axis (corresponding to another basis) you get other ratios. In the case of incoherent mixing, the ratio is 50:50 regardless of basis. You can verify this by an easy calculation of the corresponding density matrices.

 

[LostConjugate]

That's true only if your states are eigenstates of O (so the interference terms vanish).

What are the terms that vanish. If the system is mixed or pure it does not have any extra terms.

Edit: I see the term now, I looked right over it, thanks!