Correlation amplitude and continuous spectra

Question

The correlation amplitude is defined in Modern Quantum Mechanics by JJ Sakurai, chapter 2 as

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

In Eq. (2.1.65), 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:

Why 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 $g(E)$?

Answer 1

Correlation amplitude determines to what degree the two state vectors representing the same physical system are similar. By same physical system I mean, in general, two different states of the same system. For example if you consider the time evolution operator. This operator tells what will be the state of the system in your hand after a certain time, provided you know it's state at a any initial time. Once you know the state ket of a system at any instant then using the time evolution operator you could find the state ket at any instant.

In general, since the state ket is a function of both position and time (i.e, allowed to vary with both space and time co-ordinates), the state ket changes after a certain time. So while operating with the time evolution operator you just changed the state of the system (note that it may remain unchanged also).

The correlation amplitude tells you how much this latter system resembles the former. If it's value is one it means that the system has no change. You will get the exact state ket from which you started. In general since we expect some change (i.e, an increase in the degree of mismatching between the two states of a physical system), it's value will be less than one. I.e, it's value changes from 1 (corresponding to the same initial state) to some value less than 1. If the amplitude value is closer to one, then the degree of resemblence between the two states is too good. Now let's see what this $g(E)$ means? If you refer to the text book (Modern Quantum Mechanics by J.J.Sakurai) there the author states that the state ket is assumed to be a linear combination of so many eigen kets with "similar" (which means they are close but not equal) energy eigen values. So it could represent a quasi-continuos spectrum. Hence we replace the summation by integral (changing from discrete to continuous). Next, since we are dealing with the summation of linear combination of eigen vectors with similar energy values we place inside the integral the energy density $ρ(E)$ at different points represented by the eigen vectors with the variable $dE$. Now the complex number Ca′ should be replaced by $g(E)$. $g(E)$ represents the degeneracy (or degenerative multiplicity). Degeneracy of an eigenvalue correspond to how many times the eigenvalues repeated. For example when I solve the equation $X|a\rangle=a|a\rangle$, if I get the eigen values as $a=-1,2,-1$, it means the value $a=-1$ has a degeneracy of 2. The eigen vectors corresponding to them will be identical. The reason for using degeneracy here is because we postulated that the energy eigenvalues are similar. I hope you understand.