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