Quick Braket notation question

[Crosshash]

I'm a complete noob with Braket and I've only just started getting to grips with it.

For completeness' sake though (from the book I'm currently reading), I can't seem to find a definition for:

\langle J_z \rangle

Would this just be the "magnitude" of J_z?

Thanks

 

[Fredrik]

It's an expectation value, i.e. the average result in a (long) series of measurements of J_z on identically prepared systems.

...the average result in a series of measurements of A on identically prepared systems is

\langle A\rangle=\sum_a P(a)a=\sum_a a|\langle a|\psi\rangle|^2=\sum_a\langle a|\psi\rangle\langle\psi|A|a\rangle=\mbox{Tr}(\rho A) ...and also =\langle\psi|\Big(\sum_a|a\rangle\langle a|\Big)A|\psi\rangle=\langle\psi|A|\psi\rangle

Note that it depends on what state the system is in.

This post should be useful if you're learning bra-ket notation.

 

[diazona]

Actually it's the expectation value of J_z - that is,
\langle J_z \rangle = \langle \psi \vert J_z \vert \psi \rangle
In order to actually evaluate that expression, you would have to have some quantum state \vert\psi\rangle, since the expectation value of any operator depends on the quantum state.

 

[Crosshash]

Ah crud, I completely forgot that's how you write Expectation value.

so, just to confirm I have a grip on this,

\langle x \rangle = \langle \psi \vert {x} \vert \psi \rangle = \int_{-\infty}^{\infty} \psi^* x \psi dx

Is that right? Assuming the limits are from infinity to minus infinity.

Thanks