Equation of motion and operators in the interaction picture.

[Denver Dang]

Homework Statement

I have a question that says:
What is the equation of motion for a general operator in the interaction picture. I.e. how does the time derivative of the operators behaves ? Show this.

And then I have to find the time development for the annihilation and creation operator ($\hat{a}$ and $\[{{\hat{a}}^{\dagger }}\]$) in the interaction picture.

Homework Equations

The Attempt at a Solution

The first question I THINK this is how it is supposed to be done, but I'm not sure.
I have that:

$$\[\frac{d{{A}_{I}}}{dt}=\frac{1}{i\hbar }\left[ {{A}_{I}},{{H}_{0}} \right]\]$$

where:

$${{A}_{I}}={{e}^{i{{H}_{0}}t/\hbar }}{{A}_{s}}{{e}^{-i{{H}_{0}}t/\hbar }}$$

So my thought is, that I just take the derivative of ${{A}_{I}}$, and then I think, if my math is correct, I end up with something where I'm able to write the commutator as in the first equation. And then I get what is says. And if I'm now mistaken that is the equation of motion I need to find ?

The second question I'm not entirely sure about how to do.
For the Harmonic Oscillator, which is what I'm working with here, the commutator of the two operators is $\left[ {{a}_{-}},{{a}_{+}} \right]=1$.
But then I don't know what my next step is.

So if someone could help I would be grateful :)Regards

 

[mark726]

,

Hello,

Thank you for your question. The equation of motion for a general operator in the interaction picture is indeed given by:

\[\frac{d{{A}_{I}}}{dt}=\frac{1}{i\hbar }\left[ {{A}_{I}},{{H}_{0}} \right]\]

where {{A}_{I}} is the operator in the interaction picture, {{H}_{0}} is the Hamiltonian of the system, and \hbar is the reduced Planck constant.

To find the time development for the annihilation and creation operators in the interaction picture, we can start with the definition of these operators:

{{a}_{-}}={{e}^{-i{{H}_{0}}t/\hbar }}{{a}_{s}}{{e}^{i{{H}_{0}}t/\hbar }}
{{a}_{+}}={{e}^{i{{H}_{0}}t/\hbar }}{{a}_{s}}{{e}^{-i{{H}_{0}}t/\hbar }}

where {{a}_{s}} is the annihilation operator in the Schrödinger picture. Now, we can take the time derivative of these equations:

\[\frac{d{{a}_{-}}}{dt}=\frac{1}{i\hbar }{{e}^{-i{{H}_{0}}t/\hbar }}\left[ {{a}_{s}},{{H}_{0}} \right]{{e}^{i{{H}_{0}}t/\hbar }}=-\frac{1}{i\hbar }{{e}^{-i{{H}_{0}}t/\hbar }}{{H}_{0}}{{a}_{s}}{{e}^{i{{H}_{0}}t/\hbar }}=-\frac{1}{i\hbar }{{e}^{-i{{H}_{0}}t/\hbar }}{{H}_{0}}{{a}_{-}}\]
\[\frac{d{{a}_{+}}}{dt}=\frac{1}{i\hbar }{{e}^{i{{H}_{0}}t/\hbar }}\left[ {{a}_{s}},{{H}_{0}} \right]{{e}^{-i{{H}_{0}}t/\hbar }}=\frac{1}{i\hbar }{{e}^{i{{H}_{0}}t/\hbar }}{{H}_{0}}{{a}_{s}}{{e}