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barcodeIIIII
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Homework Statement
The unitary time evolution of the density operator is given by
$$\rho(t)=\textrm{exp}(-\frac{i}{\hbar}Ht)\,\rho_0 \,\textrm{exp}(\frac{i}{\hbar}Ht)$$
General definition of entropy is
$$S=-k_B\,Tr\,\{\rho(t) ln \rho(t)\}$$
Proof: $$\frac{dS}{dt}=0$$
Homework Equations
I am not very clear how does the chain rule works in matrix derivative, also how do I differentiate 'ln' of a matrix?
$$\frac{d}{dt} ln\,\rho(t)=?$$
Any insights would be great, Thank you
The Attempt at a Solution
$$ \frac{dS}{dt}=-k_B \,Tr \, \{\frac{d}{dt}[\rho(t)\,ln\,\rho(t)]\} $$
$$\frac{dS}{dt}=-k_B\,Tr\, \{ \frac{ d}{dt} \rho(t) ln\,\rho(t)+ \frac{d}{dt} [ln\, \rho(t) ] \rho(t) \}$$
where $$\frac{d}{dt} \rho (t)=-\frac{i}{\hbar} [H, \rho(t)]$$