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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 [itex]u_1v_1-v_1u_1=i\hbar[u_1,v_1][/itex] 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 [itex]\frac{d}{dq}\left.\psi\right>=\left.\frac{d\psi}{dq}\right>[/itex]

[itex]<\psi\frac{d}{dq}=-<\frac{d\psi}{dq}[/itex]

he makes these consideration: the conjugate transpose of [tex]\frac{d}{dt}\psi>\,\,\mathrm{is}\,\,<\frac{d\bar{\psi}}{dq}[/tex] and, from the previous equation, [itex]<\frac{d\bar{\psi}}{dq}=-<\bar{\psi}\frac{d}{dq}[/itex] hence the conjugate transpose of d/dq -d/dq.

My question is:why [itex]\bar{\frac{d\psi}{dq}>}=<\frac{d\bar{\psi}}{dq}[/itex]? I thought that [itex]\forall |P>, \bar{|P>}=<P|[/itex]and this gives [itex]\bar{\frac{d\psi}{dq}>}=<\frac{d\psi}{dq}[/itex]

Again, what am I missing?

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

Thank you in advance, more questions coming :D

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# Few questions on Dirac's Principles of QM.

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