jacobrhcp
Dec28-07, 08:47 AM
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).
The time-independent schroding equation reads
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.
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).
The time-independent schroding equation reads
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.