The Uncertainty Principle - question within Griffiths' Text

[Sparky_]

Hello,

In Griffiths (2nd edition) pgs 110-111 - deriving the uncertainty principle

I have 2 questions

1)
I am stuck on a point ...

(h = ^ hat )

<f | g > = < ( Ah - <A>) ψ | ( Bh - <B>) ψ >

= <Ψ | ( Ah - <A>) ( Bh - <B>) Ψ>

FOIL


= <ψ | AhBh ψ> - <B><ψ | Ah ψ> - <A>< ψ | Bh ψ> + <A><B>< ψ | ψ>

I do see where < ψ | Ahψ > = <A> so <B>< ψ | Bh ψ> = <A><B>

I don't understand how / why : < ψ | AhBh ψ> = <AhBh> the expectation of A hat times B hat = <AhBh>

why would it not be <AB> instead <AhBh>
like expectation of A hat = <A> not <Ah>

Meaning with the single operator A-hat or B-hat the result is <A> and <B> respectively

the double < ψ | AhBh ψ>, Griffiths has = <AhBh>2) I want to confirm I am correct with this ...

the book shows <B><A> - <A><B> + <A><B>
the result is <A><B>
(I want to say <B><A> instead)

Am I correct that <B><A> = <A><B>

(thinking of the expectations as a resulting number or average)

Thanks
-Sparky

 

[DrClaude]

I do see where < ψ | Ahψ > = <A> so <B>< ψ | Bh ψ> = <A><B>

I don't understand how / why : < ψ | AhBh ψ> = <AhBh> the expectation of A hat times B hat = <AhBh>

why would it not be <AB> instead <AhBh>
like expectation of A hat = <A> not <Ah>

Meaning with the single operator A-hat or B-hat the result is <A> and <B> respectively

the double < ψ | AhBh ψ>, Griffiths has = <AhBh>

Different authors use different conventions. Personally, I prefer to always indicate operators with hats and to write it as the expectation value of the operator,
$$
\langle \hat{A} \rangle \equiv \langle \psi | \hat{A} | \psi \rangle
$$
but many authors will use the symbol for the eigenvalue when expressing the expectation values. You will often see it for energy as $\langle E \rangle$, meaning
$$
\langle E \rangle = \langle \psi | \hat{H} | \psi \rangle
$$
Since there is no symbol for the eigenvalue of the operation of two operators, that particular notation breaks down and one has to revert to using operators in the expectation value. Using $\langle A B \rangle$ instead $\langle \hat{A} \hat{B} \rangle$ is misleading, since it looks like
$$
\langle A B \rangle = \langle A \rangle \langle B \rangle
$$
which is only true if $\hat{A}$ and $\hat{B}$ commute.

2) I want to confirm I am correct with this ...

the book shows <B><A> - <A><B> + <A><B>
the result is <A><B>
(I want to say <B><A> instead)

Am I correct that <B><A> = <A><B>

(thinking of the expectations as a resulting number or average)

$\langle A \rangle$ and $\langle B \rangle$ are real numbers, so commutativity applies.