Energy eigenvalue of Hamiltonian operator Hamiltonian is the total energy of the system. Then, is its eigenvalue $E$ also total energy of the system? What is the difference between them? Both of them are energy.
 A: For bound states:
$$\mathcal{H}\psi_n=E_n\psi_n$$
Where $\mathcal{H}$ is the Hamiltonion operator, $\psi_n$ a set of wave (eigen)functions and $E_n$ the corresponding eigenvalues.
If you're not familiar with eigenvalues, consider the simple quantum system of a single particle in a 1D box with infinite potential walls and zero potential inside the box.
The Hamiltonian operator of the system is:
$$\mathcal{H}=-\frac{\hbar^2}{2m}\frac{\operatorname{d^2} }{\operatorname{d}x^2}$$
The Schrödinger equation (SE) becomes:
$$-\frac{\hbar^2}{2m}\frac{\operatorname{d^2} }{\operatorname{d}x^2}\psi_n(x)=E_n\psi_n(x)$$
Solving, as per the link above, we get:
$$\psi_n(x)=C\sin\Big(\frac{n\pi x}{a}\Big)$$
Where $n=1,2,3,...$ and $a$ is the length of the box.
The eigenvalues (allowed energy levels) compute (as per the same link) to:
$$E_n=\frac{n^2\pi^2\hbar^2}{2ma^2}$$
Where $n=1,2,3,...$
$\psi_n(x)$ are the eigenfunctions of the SE and $E_n$ are its eigenvalues.
A: You need to review operators. if $\mathcal{H}$ is an Hamiltonian, and $\phi(t,x)$ is some wave vector, then
$\mathcal{H}\phi=\sum a_i\phi_i$
So, the operator is what you act with (operate) on a vector to change it to another vector, often represented as a sum of base vecotrs as I have written.
In the special case:
$\mathcal{H}\phi=E_i\phi_i$
$\phi$ is such a base vector, and $E$ the eigenvalue. Now we need to consider on the physical interpretation of the operator (this is more complicated to define), and in this case we say E is the energy - a scalar.
