- #1

jeebs

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I have this density operator [tex] \rho(t) = \sum_a |\psi_a(t)\rangle P_a \langle\psi_a(t)|[/tex] and I am supposed to be showing that "the equation of motion satisfies [tex]i\hbar\frac{\partial\rho(t)}{\partial t} = [H,\rho(t)][/tex].

I'm not making much progress though, this is all the info I'm given.

I'm thinking I have to use the product rule here, ie. [tex]\frac{\partial\rho(t)}{\partial t} = \sum_a (\frac{\partial}{\partial t}|\psi_a(t)\rangle) P_a \langle\psi_a(t)| + \sum_a |\psi_a(t)\rangle P_a \frac{\partial}{\partial t}\langle\psi_a(t)|[/tex]

also if [tex]H = i\hbar\frac{\partial}{\partial t}[/tex] then [tex] H\rho = i\hbar(\sum_a (\frac{\partial}{\partial t}|\psi_a(t)\rangle) P_a \langle\psi_a(t)| + \sum_a |\psi_a(t)\rangle P_a \frac{\partial}{\partial t}\langle\psi_a(t)| [/tex]

and also I know the commutator is just [tex] [H,\rho] = H\rho - \rho H [/tex]

so that gives me [tex] [H,\rho] = i\hbar(\sum_a (\frac{\partial}{\partial t}|\psi_a(t)\rangle) P_a \langle\psi_a(t)| + \sum_a |\psi_a(t)\rangle P_a \frac{\partial}{\partial t}\langle\psi_a(t)| - \sum_a |\psi_a(t)\rangle P_a \langle\psi_a(t)|i\hbar\frac{\partial}{\partial t} [/tex]

but I can't see how I'm supposed to get any further. I mean, I don't see what's wrong with saying [tex] i\hbar\frac{\partial\rho}{\partial t} = H\rho [/tex]. I don't see where the commutator comes from at all, unless for some reason we can say that [tex]\rho H = 0[/tex]

I'm not making much progress though, this is all the info I'm given.

I'm thinking I have to use the product rule here, ie. [tex]\frac{\partial\rho(t)}{\partial t} = \sum_a (\frac{\partial}{\partial t}|\psi_a(t)\rangle) P_a \langle\psi_a(t)| + \sum_a |\psi_a(t)\rangle P_a \frac{\partial}{\partial t}\langle\psi_a(t)|[/tex]

also if [tex]H = i\hbar\frac{\partial}{\partial t}[/tex] then [tex] H\rho = i\hbar(\sum_a (\frac{\partial}{\partial t}|\psi_a(t)\rangle) P_a \langle\psi_a(t)| + \sum_a |\psi_a(t)\rangle P_a \frac{\partial}{\partial t}\langle\psi_a(t)| [/tex]

and also I know the commutator is just [tex] [H,\rho] = H\rho - \rho H [/tex]

so that gives me [tex] [H,\rho] = i\hbar(\sum_a (\frac{\partial}{\partial t}|\psi_a(t)\rangle) P_a \langle\psi_a(t)| + \sum_a |\psi_a(t)\rangle P_a \frac{\partial}{\partial t}\langle\psi_a(t)| - \sum_a |\psi_a(t)\rangle P_a \langle\psi_a(t)|i\hbar\frac{\partial}{\partial t} [/tex]

but I can't see how I'm supposed to get any further. I mean, I don't see what's wrong with saying [tex] i\hbar\frac{\partial\rho}{\partial t} = H\rho [/tex]. I don't see where the commutator comes from at all, unless for some reason we can say that [tex]\rho H = 0[/tex]

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