# Time Dependent Perturbation of Harmonic Oscillator

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• Diracobama2181
In summary, the conversation discusses the application of first-order time dependent perturbation theory to a one-dimensional system of a charged harmonic oscillator under the influence of an incident electric field. The resulting amplitude for finding the system in an excited state at time t under the condition that it starts in the ground state at t = -∞ is discussed. In part b, the expectation value of the momentum of the oscillator is considered, using the position operator and a general particle state.
Diracobama2181
TL;DR Summary
I am struggling a bit with the following problem, primarily with the momentum expectation value.
An electric field E(t) (such that E(t) → 0 fast enough as t → −∞)
is incident on a charged (q) harmonic oscillator (ω) in the x direction,
which gives rise to an added ”potential energy” V (x, t) = −qxE(t).
This whole problem is one-dimensional.
(a) Using first-order time dependent perturbation theory, write down
the amplitude cn(t) for finding the system in excited state n at time t
if the system starts in n = 0 at t → −∞.
(b) Find the expectation of the momentum of the oscillator as a function of time.Now part a) seemed straightforward enough.
For that, I got $$C_n=(i/\hbar)\int_{−∞}^{T}(<n|(-qxE(t))|0>)e^{\frac{(-in)(\omega)}{\hbar}}dt$$
Using the position operator $$x=\frac{\hbar}{2m\omega}^{1/2}(a+a^{'})$$,
I obtain $$C_n=(i/\hbar)\int_{−∞}^{T}(-qE(t))(<n|1>)\frac{\hbar}{2m\omega}^{1/2}e^{\frac{(-in)(\omega)}{\hbar}}dt$$
which leaves only $$C_1=(i/\hbar)\int_{−∞}^{T}(-qE(t))\frac{\hbar}{2m\omega}^{1/2}e^{\frac{(-i)(\omega)}{\hbar}}dt$$

Part b is where my issue lies.

I know I can represent the particle state as $$|\psi>=\sum_{n=0}^{\infty}c_n(t)e^{\frac{-iE_nt}{\hbar}}|n>$$
I also know $$p=-(i)\frac{m\hbar\omega}{2}^{1/2}(a-a^{'})$$

When I apply the operator on psi, I get
$$p|\psi>=-(i)\frac{m\hbar\omega}{2}^{1/2}(c_0(t)0-|1>)e^{\frac{-iE_0t}{\hbar}}+...+c_n(t)(\sqrt(n)|n-1>-\sqrt(n+1)|n+1>)e^{\frac{-iE_nt}{\hbar}}$$
When I attempt the expectation value, I find

$$<\psi|P|\psi>=-(i)\frac{m\hbar\omega}{2}^{1/2}(C_0(t)^{*}C_1(t)e^{\frac{i(E_0-E_1)t}{\hbar}}-C_1(t)^{*}C_10t)e^{\frac{i(E_1-E_0)t}{\hbar}}+...C_{n-1}(t)^{*}C_n(t)e^{\frac{i(E_{n-1}-E_n)t}{\hbar}}-C_{n}(t)^{*}C_{n-1}(t)e^{\frac{i(E_{n}-E_{n-1})t}{\hbar}}$$

I suppose I just want to know whether my reasoning has been fine thus far, and if so, how could I go about reducing this equation?

Why are you considering a general state in part b?

## 1. What is time dependent perturbation?

Time dependent perturbation is a method used in quantum mechanics to study the behavior of a system that is subject to an external force that varies with time. It allows us to analyze the effects of this force on the system's energy levels and wave functions.

## 2. How is the harmonic oscillator affected by time dependent perturbation?

The harmonic oscillator is a system that exhibits simple harmonic motion, and is commonly used as a model system in quantum mechanics. When subjected to time dependent perturbation, the energy levels of the harmonic oscillator can shift and the wave function can become distorted.

## 3. What is the perturbation theory used in time dependent perturbation?

The perturbation theory used in time dependent perturbation is a mathematical method that allows us to approximate the behavior of a system that is subject to a small perturbation. It involves expanding the system's Hamiltonian in a series and solving for the perturbed energy levels and wave functions.

## 4. How is the perturbation parameter chosen in time dependent perturbation?

The perturbation parameter in time dependent perturbation is typically chosen to be small, so that the perturbation theory is valid. It can be a physical quantity, such as the strength of an external force, or a mathematical parameter, such as the coefficient in the perturbation series expansion.

## 5. What are some applications of time dependent perturbation in physics?

Time dependent perturbation is used in various fields of physics, such as quantum mechanics, solid state physics, and atomic and molecular physics. It is commonly used to study the effects of electromagnetic fields on atoms and molecules, and to analyze the behavior of quantum systems under time-varying potentials.

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