In any introduction to quantum mechanics, an explanation for why the Hamiltonian $H$ is a Hermitian operator is given. This explanation usually relies on the spectral theorem, which guarantees that $H$ has a complete orthonormal basis of eigenvectors, and real eigenvalues. Then, the eigenvectors of $H$ specify the states of well-defined energy, and the eigenvalues are the corresponding energies.
However, I am wondering if there is a simple physical explanation for why the Hamiltonian matrix elements must be symmetric (up to complex conjugation). To clarify what I mean, suppose I fix a basis $B = \{|1\rangle,\ldots, |n\rangle\}$ of my Hilbert space, and I have a Hamiltonian $H$. Suppose moreover that, when I express my Hamiltonian in the basis $B$, there is a term of the form
$$c_{ij}|i\rangle \langle j|.$$
Then, by Hermiticity, there must also be a term of the form
$$c_{ij}^*|j\rangle\langle i|.$$
My intuitive understanding of terms of the form $|i\rangle\langle j|$ is that it is "like a force" in the Hamiltonian which sends $| j \rangle$ to $|i \rangle$, since, when $H$ acts on $|j\rangle$, it will replace $|j\rangle$ with $|i\rangle$. Then, insisting that both $|i\rangle\langle j|$ and $|j\rangle \langle i|$ are present in the Hamiltonian is like saying that the forces must always go in both directions. Is this a correct interpretation, and if so, why must this be true?
Best Answer
I set $\hbar = 1$
Another way to come at this is to consider the Hamiltonian to be the generator of unitary time evolution. That is, say we have an initial state $|\psi(0)\rangle$. We can take it as a postulate of quantum mechanics that the state at a later time is given by
$$ |\psi(t)\rangle = U(t) |\psi(0)\rangle $$
Where $U$ is a unitary operator. By taking the time derivative of this expression we arrive at the Schrodinger equation
$$ \frac{d}{dt}|\psi(t)\rangle = -i H(t) |\psi(t)\rangle $$
Where I've defined the Hamiltonian
$$ H(t) = i \frac{dU}{dt} U^{\dagger} $$
From this definition and unitary of $U(t)$ it can be proven that
$$ H(t) = H^{\dagger}(t) $$
and, if $H(t)$ happens to be time-independent:
$$ U(t) = e^{-i H t} $$
What you seem to be gesturing at in your question is that the Hermiticity of $H$ seems to intuitively follow from the reversibility of dynamics/interactions. This reversibility is exactly encoded in the unitarity of the time evolution operator, so deriving Hermiticity of $H$ from this unitarity should then, maybe, be satisfying to you.