CODEWITHSUNDEEP
matlabode
MOON
MOON

Reputation: 2801

system of ordinary differential equation in Matlab

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.

Upvotes: 1

Views: 185

Answers (1)

Geoff
Geoff

Reputation: 1222

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.

Upvotes: 0

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