The correlation amplitude is defined in [*Modern Quantum Mechanics* by JJ Sakurai][1], chapter 2 Eq. (2.1.65) as

$$
C(t)=\langle\alpha|\alpha,t_0=0;t\rangle=\langle\alpha|\mathcal{U}(t,0)|\alpha\rangle \, .
$$

In Eq. (2.1.66), for an initial state $|\alpha\rangle$ represented by a superposition of $|a'\rangle$, i.e, $\sum_{a'} c_{a'}|a'\rangle$ the correlation amplitude is
$$
C(t)=\sum_{a'}|c_{a'}|^2 \exp\left(\frac{-iE_{a'}t}{\hbar}\right) \, .
$$

Consider a state ket which is a superposition of many energy eigenkets with similar energies, i.e, a quasi-continuous spectrum. Here we replace
$$
\sum_{a'}\rightarrow\int dE\rho(E)
\\ c_{a'} \rightarrow g(E) \Bigg|_{E\approx E_0}
$$
where $\rho(E)$ characterizes the density of energy eigenstates. The correlation amplitude becomes
$$
C(t)=\int dE|g(E)|^2\rho(E) \exp\left(\frac{-iEt}{\hbar}\right)
$$
and subjected to the normalization condition
$$
\int dE|g(E)|^{2}\rho(E)=1 \, .
$$

**My understanding:**

For a continuous spectrum
$$
\sum_{a'}|a'\rangle\langle{a'}|\alpha\rangle=\sum_{a'}\mathcal{C}_{a'}|a'\rangle\rightarrow \int d\mathcal{E'}|\mathcal{E'}\rangle\langle\mathcal{E'}|\alpha\rangle=\int d\mathcal{E'}\mathcal{C}_{\mathcal{E'}}|\mathcal{E'}\rangle
$$

$$
\sum_{a'}|\langle{a'}|\alpha\rangle|^{2}=1 \rightarrow \int d\mathcal{E'}|\langle{\mathcal{E'}}|\alpha\rangle|^2=1 \, .
$$

**Doubt:**

How do we define the correlation amplitude $C(t)$ using $\rho(E)$ and $g(E)$ instead of $\mathcal{C}(E)$?

What is the physical meaning of this expression and of $\rho(E)$ and $g(E)$?


  [1]: http://www.fisica.net/quantica/Sakurai%20-%20Modern%20Quantum%20Mechanics.pdf