QM: system with two lin. independent states

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

Homework Statement



Imagine a system with just two linear independent states:

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

[tex]|\Psi>=a|1>+b|2>=(a,b)[/tex], also [tex]|a|^2+|b|^2=1[/tex]

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

The time-independent schroding equation reads

[tex]H |\Psi>=i h/(2 pi) d/dt(|\Psi>)[/tex]

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?

The Attempt at a Solution



I thought of solving the time-independent SE;

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

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

  • #2
Dick
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Change to the basis where H is diagonal.
 
  • #3
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solved it with a friend today =)
 

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