How Does One Derive Equation (8.27) from (8.26) in Gerry's Quantum Optics Text?

[Tspirit]

Homework Statement

In Gerry's book "Introductory quantum optics", Chapter 8.5 Decoherence, Page 207, there is a calculation which doesn't give the detail derivation, who can give me the calculation from (8.26) to (8.27)?

Homework Equations

The coherent state is defined by $$\hat{a}|\alpha>=\alpha|\alpha>$$.

The Attempt at a Solution

Traditionally, we can calculate the differential equation of a complex function. However, it is about an equation of operator and the order of the operators is very important. I have no idea about it.

 

[Gene Naden]

They are saying that if you substitute their formula for $\rho$ into the left side of their equation equation for $\frac{d\rho}{dt}$ then it equals the right-hand side. Of course $\frac{d\rho}{dt}=\frac{d|\alpha e^{-\frac{\gamma t}{2}}>}{dt}<\alpha e^{-\frac{\gamma t}{2}}|+ |\alpha e^{\frac{-\gamma t}{2}}>\frac {d <\alpha e^ {-\frac{\gamma t}{2}}| } {dt}$
Can you provide a bit of background? What is the time dependence of $|\alpha>$?