In[22]:= \[Rho][t_] := {{a[t], b[t]}, {c[t], d[t]}}
 H[t_] := PauliMatrix[1]
 DSolve[{I \[HBar] \[Rho]'[t] ==
    H[t].\[Rho] - \[Rho].H[t], \[Rho][0] == PauliMatrix[3]}, {a, b, c,
   d}, t]
 
 During evaluation of In[22]:= DSolve::nolist: List encountered within {{{I \[HBar] (a^\[Prime])[t],I \[HBar] (b^\[Prime])[t]},{I \[HBar] (c^\[Prime])[t],I \[HBar] (d^\[Prime])[t]}}==-\[Rho].{{0,1},{1,0}}+{{0,1},{1,0}}.\[Rho]}. There should be no lists on either side of the equations. >>
 
 Out[24]= DSolve[{{{I \[HBar] Derivative[1][a],
     I \[HBar] Derivative[1][b]}, {I \[HBar] Derivative[1][c],
     I \[HBar] Derivative[1][d][
       t]}} == -\[Rho].{{0, 1}, {1, 0}} + {{0, 1}, {1,
       0}}.\[Rho], {{a[0], b[0]}, {c[0], d[0]}} == {{1,
     0}, {0, -1}}}, {a, b, c, d}, t][/b]

In[59]:= \[Rho][t_] := {{a[t], b[t]}, {c[t], d[t]}}

In[60]:= H[t_] := PauliMatrix[1]

In[61]:= DSolve[{I \[HBar] \[Rho]'[t] ==
H[t].\[Rho][t] - \[Rho][t].H[t], \[Rho][0] == PauliMatrix[3]}, {a, b, c,
d}, t]

Out[61]= {{a -> Function[{t}, Cos[(2 t)/\[HBar]]],
b -> Function[{t}, I Sin[(2 t)/\[HBar]]],
c -> Function[{t}, -I Sin[(2 t)/\[HBar]]],
d -> Function[{t}, -Cos[(2 t)/\[HBar]]]}}

In[62]:= \[Rho][t_] = \[Rho][t] /. First[%]

Out[62]= {{Cos[(2 t)/\[HBar]],
I Sin[(2 t)/\[HBar]]}, {-I Sin[(2 t)/\[HBar]], -Cos[(2 t)/\[HBar]]}}