I would like to question this statement. How do you know such modification doesn't change the size of the Hilbert's space?DrClaude said:At first glance, I don't think that perturbation theory is necessary. The ##\ket{n}## form a complete basis, even for the modified Hamiltonian.
A simple harmonic oscillator is a system that exhibits periodic motion, where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction of the displacement. Examples of simple harmonic oscillators include a mass on a spring and a pendulum.
A modification to the simple harmonic oscillator is any change made to the system that affects its behavior and characteristics. This can include changes to the mass, spring constant, or damping factor.
Changing the mass of a simple harmonic oscillator affects the period of the oscillation. A larger mass will result in a longer period, while a smaller mass will result in a shorter period.
The spring constant affects the stiffness of the spring and therefore the force applied to the mass. A higher spring constant will result in a stronger restoring force and a shorter period, while a lower spring constant will result in a weaker restoring force and a longer period.
Damping is the process of reducing the amplitude of the oscillation over time. This can be caused by external forces such as friction or air resistance, or by internal forces such as energy dissipation within the system. Damping can be either underdamped, critically damped, or overdamped, depending on the amount of damping present.