# Definition of state of a quantum system

In QM, we solve for the eigen kets of the Hamiltonian operator $\hat{H}$ and say that the state of my system lies in a linear superposition of these eigenstates $\{|n\rangle\}$ as the relation implies $|\psi\rangle=\sum_{n=0}^{\infty}\left |n\rangle\langle n|\psi \right \rangle$ and then everything about the system, its momentum, position etc. can be inferred by the action of corresponding operators on $|\psi\rangle$. Similar is done for the case of a simple harmonic oscillator.

1. But, what then is the meaning of coherent states in this context and what relation do coherent states hold with the state of the system $|\psi\rangle$? Do coherent states just provide us with basis to express out state in?

2. Why is the Hamiltonian operator so special in the case of generating basis or solving for the state of a quantum system? why can't we write the state of the system in the momentum basis for example without solving for the eigenbasis of $\hat{H}$? Because if we can do so, for any system, $\hat{p}|\phi\rangle=\phi|\phi\rangle$ would give us the state of the system as $|\psi\rangle=\int |\phi\rangle\left \langle \phi|\psi \right \rangle d\phi$ would give us the same state for every system.

• Thanks for the answer @ggcg. What about this "why can't we write the state of the system in the momentum basis for example without solving for the eigenbasis of $\hat{H}$? Because if we can do so, for any system, $\hat{p}|ϕ⟩=ϕ|ϕ⟩$would give us the state of the system as |ψ⟩=∫|ϕ⟩⟨ϕ|ψ⟩dϕ would give us the same state for every system. " – Naman Agarwal Jun 5 '18 at 12:14