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Von Neumann equation

Posted 11 years ago
I'm trying to symbolically integrate the Von Neumann equation for a two-level system with some basic Hamiltonians. Thus, I want to solve
I \[HBar] \[Rho]'[t] = H[t].\[Rho] - \[Rho].H[t]
Where
\[Rho]
is a 2x2 matrix.

In the simplest case,
H[t]
will be the Pauli sigma X matrix.

What's the proper way of setting this problem up using DSolve?
POSTED BY: Thomas Harty
5 Replies
Posted 11 years ago
Perfect, thank you.
POSTED BY: Thomas Harty
You can also try this:
\[Rho]M[t_] = Table[\[Rho][i, j][t], {i, 2}, {j, 2}]

H = PauliMatrix[1]

sol1 = DSolve[{I \[HBar] \[Rho]M'[t] ==
    H.\[Rho]M[t] - \[Rho]M[t].H, \[Rho]M[0] ==
    Table[\[Rho]0[i, j], {i, 2}, {j, 2}]}, Flatten[\[Rho]M[t]], t]
or more elegantly
sol2 = MatrixExp[-I t H/\[HBar]].Table[\[Rho]0[i, j], {i, 2}, {j,
    2}].Simplify[ConjugateTranspose[MatrixExp[-I t H/\[HBar]]],
   Element[{t, \[HBar]}, Reals]]
POSTED BY: Michael Trott
Posted 11 years ago
They key point is that you can't just define H and ? to be interpreted as matrix values functions, since by default symbols are interpreted as complex numbers. So you have to define ? and H to be matrices, like this
In[59]:= \[Rho][t_] := {{a[t], b[t]}, {c[t], d[t]}}

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

You can then type in commutation relation pretty much as expected, except you must remember that your unknowns are a,b,c, and d.
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]]]}}
You can now substitute this into to get your solution. Note the use if Set (=) rather than SetDelayed (:=) to make sure the substitution is done immediately, rather than each time you evaluate ?.
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]]}}
POSTED BY: Itai Seggev
Posted 11 years ago
Thanks for the help.
Itai, that looks like exactly what I wanted. However, I still seem to get an error when I run the code you suggested
 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]

Do you know what this means?
POSTED BY: Thomas Harty
Posted 11 years ago
There was a copy/paste problem.  On the RHS of the equation, it should have been
H[t].\[Rho][t] - \[Rho][t] . H[t]
(Notice that the ? is now ?[ t ]).  I have fixed my original post, so copy and pasting it out should work now.
POSTED BY: Itai Seggev
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