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Homework Statement
let [1> and [2> mutually orthogonal states (eigenstates of some Hermitian operator).
the Hamiltonian operator is given by H=c[1><2]+c[2><1], where c is a real number.
(a) calculate the eigenstates and corresponding eigenvalues of H
(b) if the initial state of the system is [1>, how does the state evolve with time?
Homework Equations
H[ψ>=ih[itex]\frac{∂[ψ>}{∂t}[/itex]
The Attempt at a Solution
for (a), I first earned hamiltonian matrix as below
0 c
c 0
from this, I earned two eigenvalues +c and -c.
and earned [itex]\sqrt{1/2}[/itex]( 1 1 ) and [itex]\sqrt{1/2}[/itex]( 1 -1 ) as two eigenstates of hamiltonian matrix.
however what I don't know is the answer of (b)
the problem I was given does not have any information of the state [1>. Is there anyway to solve this problem? if I assume [1> as (1 0) (and [2> as (0 1)) this problem can be solved. but I don't know more... can anyone help me?