Few questions on Dirac's Principles of QM.

[DiracRules]

Hi there,

I'm reading Dirac's Principles of QM, but I think I miss something...

1)When he derives the Poisson quantic bracket, he states that u_1v_1-v_1u_1=i\hbar[u_1,v_1] and says that hbar must be real since we introduced the imaginary unit. The thing is, since uv-vu is real, because we are talking about dynamics variable that are real linear operator, this means that [u,v] must be pure imaginary. But that doesn't fit well with what Dirac says a few lines below, where he hypothesizes that [u,v] in QM has the same value than in classical mechanic (and I derive from here that it is real).
What am I missing?

2)Why is so important a representation where the observables are diagonal? And why it is so important to build Schroedinger's representation?

3)I can't follow his derivation of the conjugate transpose of d/dq :
giving that \frac{d}{dq}\left.\psi\right>=\left.\frac{d\psi}{dq}\right>
<\psi\frac{d}{dq}=-<\frac{d\psi}{dq}
he makes these consideration: the conjugate transpose of \frac{d}{dt}\psi>\,\,\mathrm{is}\,\,<\frac{d\bar{\psi}}{dq} and, from the previous equation, <\frac{d\bar{\psi}}{dq}=-<\bar{\psi}\frac{d}{dq} hence the conjugate transpose of d/dq -d/dq.
My question is:why \bar{\frac{d\psi}{dq}>}=<\frac{d\bar{\psi}}{dq}? I thought that \forall |P>, \bar{|P>}=<P|and this gives \bar{\frac{d\psi}{dq}>}=<\frac{d\psi}{dq}
Again, what am I missing?

4) What does it mean that \mathbf{p}=i\hbar\mathbf{d}, where d is the translation operator? Better, what is the physical importance and meaning?

Thank you in advance, more questions coming :D

 

[Bill_K]

I think much of your question is a result of terminology. Where you say a physical quantity needs to be represented by a "real linear operator", what you mean is Hermitian or self-adjoint. The Hermitian conjugate involves a complex conjugate and an order reversal, probably what you're calling conjugate transpose. Note that if u and v are Hermitian, then uv - vu will be anti-Hermitian. This explains the appearance of i in question 1.

It also explains question 3, since ∂/∂q is anti-Hermitian also. When you take the adjoint of ∂/∂q |ψ> you'll pick up a minus sign.

 

[DiracRules]

Ok, I get (probably) the answer to question (1), but not to the others.

Just to clarify: is the equation \overline{\alpha|P>}=<P|\overline{\alpha}(*) correct, the bar meaning taking the complex conjugate and alpha being an hermitian operator?
If so, then I can't understand why the following passage is right
\overline{\frac{d}{dq}\psi>}=\overline{\frac{d\psi}{dq}>}=<\frac{d\overline{\psi}}{dq}

If we consider in (*) |P>=dP/dq>, then it should be \overline{\frac{d\psi}{dq}>}=<\frac{d\psi}{dq} w/o the complex conjugate on psi.

Can it be this way:
\overline{\frac{d}{dq}\psi>}=<\psi\overline{\frac{d}{dq}}
\overline{\frac{d}{dq}\psi>}=\overline{\frac{d\psi}{dq}>}=<\frac{d\psi}{dq}=-<\psi\frac{d}{dq} hence \overline{\frac{d}{dq}}=-\frac{d}{dq}? (This demonstration is different from Dirac's but arrives to the same conclusion).