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.
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.