# QM: system with two lin. independent states

1. Dec 28, 2007

### jacobrhcp

[SOLVED] QM: system with two lin. independent states

1. The problem statement, all variables and given/known data

Imagine a system with just two linear independent states:

$$|1>=(1,0)$$ and $$|2>=(0,1)$$ (these are actually column matrices, but I don't know how to type those in tex)

$$|\Psi>=a|1>+b|2>=(a,b)$$, also $$|a|^2+|b|^2=1$$

suppose the hamiltonian is a 2x2 matrix with entries j,g above and g,j below, $$(g,j \in R>0)$$.

$$H |\Psi>=i h/(2 pi) d/dt(|\Psi>)$$

a) find the eigenvalues and eigenvectors of this hamiltonian
b) suppose the system starts out at t=0 in |1>, what is the state at time t?

3. The attempt at a solution

I thought of solving the time-independent SE;

$$ja+gb= i h/(2 pi) d/dt(a)$$
$$ga+jb= i h/(2 pi) d/dt(b)$$

but does this mean g=0? And if so, how do I get any further. I'm stuck.

Last edited: Dec 28, 2007
2. Dec 28, 2007

### Dick

Change to the basis where H is diagonal.

3. Jan 1, 2008

### jacobrhcp

solved it with a friend today =)