# Time Evolution operator in Interaction Picture (Harmonic Oscillator)

• Xyius
In summary, the conversation discusses a time-dependent harmonic oscillator with Hamiltonian and perturbation theory. The conversation includes equations for the interaction picture and solving for the ladder operators. The goal is to transform the expression for V into the interaction picture and integrate it using Equation 1 to obtain an expression for ##\hat{U}_S(t,0)## to second order perturbation theory.
Xyius

## Homework Statement

Consider a time-dependent harmonic oscillator with Hamiltonian

$$\hat{H}(t)=\hat{H}_0+\hat{V}(t)$$
$$\hat{H}_0=\hbar \omega \left( \hat{a}^{\dagger}\hat{a}+\frac{1}{2} \right)$$
$$\hat{V}(t)=\lambda \left( e^{i\Omega t}\hat{a}^{\dagger}+e^{-i\Omega t}\hat{a} \right)$$

(i) Compute ##\hat{U}_S(t,0)## using the interaction representation formula (Equation 1 in next section) to second order perturbation theory.
(ii) Compute ##\hat{U}_S(t,0)## using (Equation 2 in next section) to second order perturbation theory

## Homework Equations

EQUATION 1:
$$U_I(t,0)=1-\frac{i}{\hbar}\int_0^t dt' V_I(t')+\left( \frac{-i}{\hbar} \right)^2 \int_0^t dt' \int_0^{t'} V_I(t')V_I(t'') + \dots$$

EQUATION 2:

$$U(t,0)=1+\sum_{n=1}^{∞}\left( \frac{-i}{\hbar} \right)^n\int_0^t dt_1 \int_0^{t_1} dt_2 \dots \int_0^{t_{n-1}}dt_n H(t_1)H(t_2)\dots H(t_n)$$

## The Attempt at a Solution

So I know that for the interaction picture the transformation of the operator ##\hat{V}_I## is..

$$\hat{V}_I=e^{\frac{i}{\hbar}\hat{H}_0 t} \hat{V} e^{\frac{-i}{\hbar}\hat{H}_0 t}$$

I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time.

$$\frac{d\hat{a}}{dt}=\frac{1}{i\hbar}\left[ \hat{a},\hbar \omega \left( \hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$

$$\frac{d\hat{a}^{\dagger}}{dt}=\frac{1}{i\hbar}\left[ \hat{a}^{\dagger},\hbar \omega \left( \hat{a}^{\dagger}\hat{a} + \frac{1}{2} \right) \right]$$

I then got..

$$\hat{a}(t)=\hat{a}(0)e^{-i\omega t}$$
$$\hat{a}^{\dagger}(t)=\hat{a}^{\dagger}(0)e^{i\omega t}$$

I plugged these into the expression for V to get..

$$\hat{V}=\lambda \left[ \hat{a}^{\dagger}(0)e^{i(\Omega + \omega)t} + \hat{a}(0)e^{-i(\Omega + \omega)t} \right]$$

So now what needs to be done, is to transform this into the interaction picture and then plug it into Equation 1 from above and integrate. But this seems very messy and I am having doubts if this is the correct way to go about this problem.

If anyone can shed some light onto this I would really appreciate it!

I'm pretty sure your very last line is V_I; you don't need to transform it any more. Just plug it into Equation 1.

## 1. What is the Time Evolution operator in Interaction Picture?

The Time Evolution operator in Interaction Picture is a mathematical tool used in quantum mechanics to describe the evolution of a system over time. It takes into account both the time-dependence of the system and the interactions between different parts of the system.

## 2. How does the Time Evolution operator differ from the Schrödinger Picture and the Heisenberg Picture?

In the Schrödinger Picture, the time evolution of a system is described by a time-dependent wavefunction. In the Heisenberg Picture, the time-dependence is transferred to the operators instead of the wavefunction. The Interaction Picture is a combination of these two, where the time-dependence is split between the operators and the wavefunction.

## 3. What is a Harmonic Oscillator and why is it important in the study of Time Evolution?

A Harmonic Oscillator is a system that exhibits periodic motion around an equilibrium point. It is important in the study of Time Evolution because it is a simple and well-understood system that can be used as a model for more complex systems. It also allows for the application of mathematical techniques and concepts, such as the Time Evolution operator, to be easily applied and understood.

## 4. How is the Time Evolution operator used to calculate the evolution of a Harmonic Oscillator in the Interaction Picture?

The Time Evolution operator in the Interaction Picture is used to calculate the evolution of a Harmonic Oscillator by first breaking down the system into its constituent parts, such as the position and momentum operators. Then, using the time-dependent part of the wavefunction and the time-dependent operators, the Time Evolution operator is applied to calculate the evolution of the system at different points in time.

## 5. Can the Time Evolution operator in the Interaction Picture be applied to other systems besides the Harmonic Oscillator?

Yes, the Time Evolution operator in the Interaction Picture can be applied to any quantum mechanical system. However, it is most commonly used for systems that can be modeled as the sum of simple harmonic oscillators, such as molecular vibrations or the quantized electromagnetic field.

Replies
1
Views
781
Replies
0
Views
740
Replies
10
Views
1K
Replies
24
Views
1K
Replies
2
Views
1K
Replies
1
Views
2K
Replies
2
Views
3K
Replies
1
Views
815