Need help solving first order DiffyQ analytically with matrix

I'm given the equation . This can also be written as . Where where and is a 2x1 matrix with initial value of . I need to solve this problem analytically with Euler method, I've already solved it numerically. I need to plot time vs. intensity.

my code attempt is as follows:

gamma=0.5;
H=[(gamma*i) -1;-1 -i*gamma];
G=(2:2);
psi=(2:1);
psi(1,1)=1;
psi(2,1)=0;
T=10;
N=1000;
deltat=T/N;
x=[];
y=[];
t=[];
x(1)=0.5;
y(1)=0;
t(1)=0;
t1=[];

for t=1:N
    t1(end+1)=t;
    x(t+1)=x(t)+y(t)*deltat;
    y(t+1)=y(t)+deltat*(-i*H*psi);
    
    intensity=abs(y).^2;
    int1(end+1)=intensity(1,1);
    int2(end+1)=intensity(2,1);
end

plot (t1,int1,'-*');
hold on;
plot (t1,int2,'-*');
hold off;

I keep getting the error 'Unable to perform assignment because the left and right sides

have a different number of elements.' for line 36: y(n+1)=n*y(n)+deltat*(-i*H*psi);

I am very confused by this and could use the help!

[Chuguang Pan]

The right n*y(n)+delta*(-i*H*psi) is 2x1 matrix, but left y(n+1) is scalar!

[Eva Carrillo]

How would I change y to be a matrix?

Answer 1

[Ridwan Alam]

If I understood it right, your x and y are supposed to be 2x1 vectors.

gamma=0.5;
H=[(gamma*i) -1;-1 -i*gamma];
% G=(2:2); % where is it used?
psi=zeros(2,1);
psi(1)=1;
psi(2)=0;
T=10;
N=1000;
deltat=T/N;
x1=0.5;
y1=0;
t=0;
x=[x1; x1];
y=[y1; y1];

t1=[]; int1 = []; int2 = [];
for t=1:N
    t1(end+1)=t;
    x(:,t+1)=x(:,t)+y(:,t)*deltat;
    y(:,t+1)=y(:,t)+deltat*(-i*H*psi);
    
    intensity=abs(y(:,t+1)).^2;
    int1(end+1)=intensity(1);
    int2(end+1)=intensity(2);
end
plot (t1,int1,'-*');
hold on;
plot (t1,int2,'-*');
hold off;

Hope this helps!