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I am currently taking a course in Quantum Mechanics and I am getting confused by the start of this question, I'm sure it's fairly simple but I'm not getting it.

The question is:

A quantum mechanical system is described by a two dimensional Hilbert space of states, spanned by an orthonormal basis $ |\alpha\rangle,|\beta\rangle $, with the following Hamiltonian:

$$ H|\alpha\rangle = 4|\alpha\rangle + |\beta\rangle , H|\beta\rangle = |\alpha\rangle+4|\beta\rangle $$

I need to first find the eigenvalues and eigenstates of the hamiltonian. My intuition tells me this is possible without directly working out the matrix, but I would also like to know the matrix. I also know that $|\alpha\rangle$ and $|\beta\rangle$ are orthogonal, since a quantum hamiltonian is hermitian and admits and orthogonal basis.

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I'm a little rusty at physics but this this looks like a simple enough linear algebra problem. Your intuition is correct in that it's relatively easy to guess an eigenstate.

\begin{align*} H(|\alpha\rangle + |\beta\rangle)&= H|\alpha\rangle+H|\beta\rangle= 4|\alpha\rangle+|\beta\rangle+|\alpha\rangle+4|\beta\rangle= 5(|\alpha\rangle + |\beta\rangle)\\ H(|\alpha\rangle - |\beta\rangle)&= H|\alpha\rangle-H|\beta\rangle= 4|\alpha\rangle+|\beta\rangle-(|\alpha\rangle+4|\beta\rangle)= 3(|\alpha\rangle - |\beta\rangle)\\ \end{align*}

The matrix is $\left[\begin{matrix}4 & 1\\1&4\end{matrix}\right]$

It is constructed by stacking column vectors made from the coefficients in the equations of the Hamiltonian you wrote.

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    $\begingroup$ My initial gut feeling was to contract the matrix in the way you mentioned at the bottom, I actually did this in order to complete the rest of the question. It didn't really sit right with my however, so thought I'd clarify. Thanks for the response! $\endgroup$ – user2662468 Nov 17 '16 at 17:45

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