- #1
B3NR4Y
Gold Member
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Homework Statement
[itex]|1>[/itex] and [itex]|2>[/itex] form an orthonormal basis for a two-level system. The Hamiltonian of this system is given by:
[tex]
\hat{H} = \epsilon
\begin{pmatrix}
1 & i \\
-i & 1
\end{pmatrix}
[/tex]
a.) Is this Hamiltonian hermitian? What is the significance of a hermitian operator?
b.) Find the eigenvalues and eigenvectors of this hamiltonian.
c.) Suppose a particle is in the state [itex]|1>[/itex]. An energy measurement is performed on the particle. What are the possible outcomes of such a measurement, and what are the probabilities?
Homework Equations
All that should be necessary is the Schrodinger equation.
[tex]\hat{H} |\psi> = E|\psi>[/tex]
The Attempt at a Solution
For part a I checked that the eigenvalues of the hamiltonian are real. This kills two birds with one stone because part b asks for the eigenvectors. They were real, and given by 2ε and 0. The significance of an operator being hermitian is that hermitian operators correspond to observables.
For part b I just went through the rigor of finding the eigenvectors.
They were
[itex] |1> =
\begin{pmatrix}
i \\
1
\end{pmatrix}
[/itex] and
[itex]
|2> =
\begin{pmatrix}
-i \\
1
\end{pmatrix}
[/itex] The first corresponds to the eigenvalue 2ε and the second corresponds to the eigenvalue 0.
For part C I used Schrodinger's equation and said that since |1> is an eigenvector of the hamiltonian it has a constant definite energy given by 2ε with 100% probability. This is where I feel like I am wrong, because I think this violates an uncertainty principle.
For part D I am not sure what to do. I know I should multiple by some exponent with time, but I'm not sure how to find this.