Wave function energy calculation

The Energy of a Wave function is defined to be $E\in \mathbb{R}$ such that $H|\psi ⟩=E|\psi ⟩$. We can use this to calculate the Energy of a generic Wave function as the Expectation ${⟨H⟩}_{\psi }$, with $\psi$ having Norm 1. This holds for an Eigenstate $\psi$ of the Hamiltonian $H$ we have:

$$\begin{array}{rll}{⟨H⟩}_{\psi }& =⟨\psi |H|\psi ⟩& \text{This holds for an eigenstate of the Hamiltonian}\\ & =⟨\psi |E\psi ⟩& \text{Since E∈R is a scalar}\\ & =E⟨\psi |\psi ⟩& \text{By definition of the norm}\\ & =E\Vert \psi \Vert & \text{Since ψ is normalized}\\ & =E& \end{array}$$

If $\psi$ is not an Eigenstate of $H$, we then use ${⟨H⟩}_{\psi }$ not as the Energy, but as the expected Energy of the system upon Measurement.