I'm trying to understand if there's a more systematic approach to build the matrix associated with the Hamiltonian in a quantum system of finite dimension. For example, I know that for the ammonia molecule (which has two states based on the position of the nitrogen), it should be:
$$
\begin{pmatrix}
E & -A \\
-A & E
\end{pmatrix}
$$
but the explanation tends to be just "based on symmetry". Or another example that we did with a cyclobutadiene molecule (the four states correspond to a $\pi$ electron on each of the four carbons), the Hamiltonian matrix was given as:
$$
\begin{pmatrix}
0 & b & 0 & b \\
b & 0 & b & 0 \\
0 & b & 0 & b \\
b & 0 & b & 0 \\
\end{pmatrix}
$$
Again, it makes sense based on symmetry, but I was wondering if there's an actual systematic approach to this, or if it's just "feeling".
Best Answer
In general this is a hard problem and physical hamiltonians are usually far more complicated, infinite-dimensional objects. For the cases you mention and other simple ones one can often assume the system to be in (superpositions of) a finite number of states. This brings the dimension of the hamiltonian matrix down to the finite size of your basis, but it still could be anything, provided it's hermitian.
In general, the procedure one should (in principle) do is to list the relevant physical interactions, formulate the corresponding operators, and calculate the matrix elements as the relevant inner products.
In the examples you mention, the symmetry of the problem provides additional constraints on what the hamiltonian's matrix can look like, reducing the number of parameters involved. Thus one can equate certain matrix elements to each other or to zero, but these considerations can never provide the actual numbers, so at least one number - such as the $b$ in your example - remains indeterminate and must be either calculated or measured through experiment. This is reasonable enough as one can often say a lot about the structure of the resulting eigenstates without actually having the parameters (and if there's only one then all it does is provide an energy scale without affecting the structure).