A Cat state acting on given Hamiltonian

[deepalakshmi]

TL;DR Summary

Can a Hamiltonian which contain superposition of (a^†)b act on even coherent state

For example if I consider H = (a^†)b+a(b^†), how will it act on even coherent state i.e. |α⟩+|-α⟩?. I know that |α⟩ don't act on (a^†) because |α⟩ is a eigenstate of lowering operator.

 

[PeterDonis]

I know that |α⟩ don't act on (a^†) because |α⟩ is a eigenstate of lowering operator.

First, you are saying this backwards: operators act on states, not states on operators.

Second, the fact that a state is an eigenstate of one operator certainly does not mean that no other operators can act on it. All it means is that other operators will change the state.