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So I am trying to use DSolve to solve the following system of equations:

eqns = {c1'[
 t] == -I/
  h (H12 c2[t] E^(-I W12 t ) + 
   H13 c3[t] E^(-I W13 t ) + 
   H14 c4[t] E^(-I W14 t )), c2'[t] == -I/
  h (H12 c1[t] E^(I W12 t ) + 
   H23 c3[t] E^(-I W23 t ) + 
   H24 c4[t] E^(-I W24 t )), c3'[t] == -I/
  h (H23 c2[t] E^(I W23 t ) + 
   H13 c1[t] E^(I W13 t ) + 
   H34 c4[t] E^(-I W34 t )), c4'[t] == -I/
  h (H24 c2[t] E^(I W24 t ) + 
   H34 c3[t] E^(I W34 t ) + 
   H14 c1[t] E^(I W14 t )), c1[0] == a,  c2[0] == b, c3[0] == c, c4[0] == d}; DSolve[eqns, {c1, c2, c3, c4}, t]

But Mathematica returns the input and doesn't do anything. Can someone help me out? Thanks! (Sorry for the bad layout, I don't know how to seperate the equations here without getting them out of the code section)

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    $\begingroup$ It usually means that it can't find a solution. Do you know if a symbolic solution is known? $\endgroup$
    – Michael E2
    Feb 4 at 19:37
  • $\begingroup$ @MichaelE2 no, unfortunately I have no idea what's the solution or even if there exists one. That's what I was hoping to find out... :) $\endgroup$ Feb 4 at 20:01
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    $\begingroup$ Well, it solves the system given some numeric values for the parameters, but not this one: Block[{H12, H13, H14, H23, H24, H34, h, W12, W13, W14, W23, W24, W34}, {H12, H13, H14, H23, H24, H34, h, W12, W13, W14, W23, W24, W34} = {1, 1, 1, 1, 1, 1, 1(*h*), 2, 0, 0, 0, 0, 0}; DSolve[eqns, {c1, c2, c3, c4}, t] ] -- one's hopes are diminished. Sorry. $\endgroup$
    – Michael E2
    Feb 4 at 20:45
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It looks like you are trying to propagate the time-dependent Schrödinger equation in the interaction picture. If this is indeed the case, could it be that your W parameters can be written as $W_{ij}=w_i-w_j$? If yes, then the solution consists of going back to the Schrödinger picture, where the Hamiltonian becomes time-independent:

W12 = w1 - w2;
W13 = w1 - w3;
W14 = w1 - w4;
W23 = w2 - w3;
W24 = w2 - w4;
W34 = w3 - w4;
c1[t_] = a1[t] E^(-I w1 t);
c2[t_] = a2[t] E^(-I w2 t);
c3[t_] = a3[t] E^(-I w3 t);
c4[t_] = a4[t] E^(-I w4 t);

Your equations can now be written as a time-dependent Schrödinger equation with a time-independent Hamiltonian $H$ for the $a_i(t)$,

y[t_] = {a1[t], a2[t], a3[t], a4[t]};
H = {{-h w1, H12,   H13,   H14  },
     {H12,   -h w2, H23,   H24  },
     {H13,   H23,   -h w3, H34  },
     {H14,   H24,   H34,   -h w4}};
y'[t] == -(I/h) H . y[t] // Thread
(*    four differential equations equivalent to yours    *)

The solution is now found by matrix exponentiation,

Y[t_] = MatrixExp[-(I/h) H t, {a, b, c, d}]

which contains RootSum terms that you can convert to radicals with ToRadicals:

Y[t] // ToRadicals
(*    huge result    *)

From this solution for the $a_i(t)$ you can compute the $c_i(t)$ with the aforementioned formulas.

Alternatively, with the time-independent Hamiltonian $H$ you can go the more traditional route of using the time-independent Schrödinger equation and simply look for the (energy) eigenvalues of $H$,

Eigenvalues[H]

or

Eigenvalues[H, Quartics -> True]
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