A Time evolution of coherent state with vacuum

[deepalakshmi]

TL;DR Summary

Here I am having a Hamiltonian as a†b + b†a and I want to evolve |0,α⟩ with e^(−iHt)

I have attached my work here. I don't know how to proceed further?

 

[PeterDonis]

Summary:: Here I am having a Hamiltonian as a†b + b†a and I want to evolve |0,α⟩ with e^(−iHt)

I have attached my work here.

Attachments are not allowed. Please post your equations directly in the thread using the PF LaTeX feature. You will see a LaTeX Guide link at the bottom left of the post window.

 

[deepalakshmi]

The initial state is
$|ψ(0)⟩ = |0⟩ |α⟩$
The time evolved state is
$|ψ(t)⟩ = e^(−iHt) |0⟩|α⟩$
$|ψ(t)⟩ = e^(−iHt) e^(αa†−α∗a) |0⟩|0⟩$
$|ψ(t)⟩ = e^(−iHt) e^(αa†−α∗a)e^(−iHt) e^(iHt) |0⟩|0⟩$
$|ψ(t)⟩ = e^(−iHt) e^(αa†−α∗a)e^(iHt) |0⟩|0⟩$
Using BCH formula
$|ψ(t)⟩ = {e^(αa†−α∗a)+(-it)[ b†a+a†b, e^(αa†−α∗a)]+(it)^2{\2}[b†a+a†b,[b†a+a†b, e^(αa†−α∗a)]]+...}|0⟩|0⟩$
$|ψ(t)⟩={e^(αa†−α∗a)+(-it)((e^(αa†−α∗a) αb†+α∗b)+(it)^2{\2}((αa†−α∗a)+(αb†+α∗b)^2)e^(αa†−α∗a)+...}|0⟩|0⟩$
How to simplify further?