Sudden perturbation approximation for oscillator

[jsc314159]

Homework Statement

An oscillator is in the ground state of $$H = H^0 + H^1$$, where the time-independent perturbation $$H^1$$ is the linear potential (-fx). It at t = 0, $$H^1$$ is abruptly turned off, determine the probability that the system is in the nth state of $$H^0$$ .

Homework Equations

First, since the perturbation is turned off abruptly, the sudden perturbation approximation may be used.

$$P(n) = |<n|n^0>|^2$$.

The Attempt at a Solution

I am not sure where to start. Is this best done in the coordinate basis or the energy basis?

In either basis, I need some type of representation of |n> but I am not sure where to start with that.

$$n^0$$ can be represented as |0> in the energy basis or $$\psi(t) = A_0 * e^{y{2}}$$ in the coordinate basis.

jsc

 

[Jhero]

ientist: Hi there! Let's start by defining our terms. The ground state of a system is the lowest energy state that the system can be in. In this case, the system is described by the Hamiltonian H, which is made up of two parts: H^0 and H^1. H^0 represents the energy of the system in the absence of any perturbations, while H^1 represents the linear potential (-fx) that is acting on the system.

Now, since we are dealing with a sudden perturbation, we can use the sudden perturbation approximation. This means that the system's wavefunction will evolve according to the unperturbed Hamiltonian (H^0) until the perturbation (H^1) is turned off at t=0. After that, the system's wavefunction will continue to evolve according to H^0.

To determine the probability that the system is in the nth state of H^0, we can use the formula P(n) = |<n|n^0>|^2, where |n> represents the nth state of H^0 and |n^0> represents the ground state of the system described by H^0.

In order to calculate P(n), we need to know the representation of |n> in either the coordinate basis or the energy basis. This will depend on the specific system and its potential. Once we have the representation of |n>, we can plug it into the formula and solve for P(n).

I hope this helps get you started! Let me know if you have any further questions.