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## Main Question or Discussion Point

I'm following a derivation (p85 of

"...we have

[tex]\left[\mathcal{G}_p,\mathbf{r}_i\right] &=& \mathbf{v}_0t\mathcal{G}_p, [/tex]

[tex]\left[\mathcal{G}_r,\mathbf{p}_i\right] &=& \mathbf{v}_0m_i\mathcal{G}_r,[/tex]

that is,

[tex]-i\hbar\frac{\partial\mathcal{G}_p}{\partial p_{ik}} = v_{0k}t\mathcal{G}_p,[/tex]

[tex]i\hbar\frac{\partial\mathcal{G}_r}{\partial r_{ik}} = v_{0k}m_i\mathcal{G}_r.[/tex]"

In the book's notation, [tex]\mathcal{G}_p[/tex] and [tex]\mathcal{G}_r[/tex] are unknown unitary operators which depend only on the momentum and position respectively; [tex]\mathbf{r}_i[/tex], [tex]\mathbf{p}_i[/tex] and [tex]m_i[/tex] are the position, momentum and mass of the [tex]i[/tex]th particle in the system; [tex]\mathbf{v}_0[/tex] is a constant velocity; and [tex]t[/tex] is the time.

I follow the derivation up to this point, and continue to follow it afterwards, but do not understand how the second pair of equations follows from the first. It clearly seems to assert that

[tex]-i\hbar\frac{\partial\mathcal{G}_p}{\partial \mathbf{p}_{i}} = \left[\mathcal{G}_p,\mathbf{r}_i\right], [/tex]

which reminds me of the Ehrenfest relation, and I also note that the derivatives are with respect to the canonical conjugates of the quantities in the commutators, but I'm not sure if that's just a coincidence or how to show that last equality. I have a feeling that it may be something rather trivial that I've just overlooked, but I'm out of ideas right now... Can anybody explain this step, or point me to a theorem that may help?

Thanks,

Mike

*Symmetry Principles in Quantum Physics*by Fonda & Ghirardi, for anyone who has it) in which the following assertion is made:"...we have

[tex]\left[\mathcal{G}_p,\mathbf{r}_i\right] &=& \mathbf{v}_0t\mathcal{G}_p, [/tex]

[tex]\left[\mathcal{G}_r,\mathbf{p}_i\right] &=& \mathbf{v}_0m_i\mathcal{G}_r,[/tex]

that is,

[tex]-i\hbar\frac{\partial\mathcal{G}_p}{\partial p_{ik}} = v_{0k}t\mathcal{G}_p,[/tex]

[tex]i\hbar\frac{\partial\mathcal{G}_r}{\partial r_{ik}} = v_{0k}m_i\mathcal{G}_r.[/tex]"

I follow the derivation up to this point, and continue to follow it afterwards, but do not understand how the second pair of equations follows from the first. It clearly seems to assert that

[tex]-i\hbar\frac{\partial\mathcal{G}_p}{\partial \mathbf{p}_{i}} = \left[\mathcal{G}_p,\mathbf{r}_i\right], [/tex]

which reminds me of the Ehrenfest relation, and I also note that the derivatives are with respect to the canonical conjugates of the quantities in the commutators, but I'm not sure if that's just a coincidence or how to show that last equality. I have a feeling that it may be something rather trivial that I've just overlooked, but I'm out of ideas right now... Can anybody explain this step, or point me to a theorem that may help?

Thanks,

Mike