system of ordinary differential equation in Matlab

Question

Suppose we have this Hamiltonian:

n = 10;
H = ones(n,n);

The density matrix is:

Ro = sym('r',[n,n]);%Density matrix

The equation of motion is:

 H*Ro-Ro*H

The above equation of motion is the right hand side of the equation, the left hand side is the time derivative of density matrix.

How can I solve the equation of motion in Matlab without the symbolic math toolbox? I need to change the value of n. It can be up to 100.

Answer 1

In your dynamics function, reshape between vectors and matrices in order to use MATLAB's standard ode functions , which (to my knowledge) require vector inputs. Note, the symbolic toolbox isn't used anywhere in this solution. R can be any size n-by-n , within the constraints of your machine's memory.

function dR = dynfun(R,H)
                                %// R    = n^2-by-1 vector
                                %// H    = n-by-n matrix
    n  = sqrt(length(R));       
    R  = reshape(R,[n,n]);      %// reshape R to n-by-n matrix
    dR = H*R-R*H;              
    dR = dR(:);                 %// reshape dR to n^2-by-1 vector


end

Call the ODE solver:

[tout,Rout] = ode45(@(t,R) dynfun(R,H), [0,T], R0(:));

where T is final time, R0 is n-by-n initial condition, tout are the output time steps, and Rout is the solution trajectory. Note due to the reshaping, Rout will be k-by-n^2 , where k is the number of time steps. You'll need to reshape each row of Rout if you want to have the actual matrices over time.