Equation of motion and operators in the interaction picture.

H}_{0}}t/\hbar }}=\frac{1}{i\hbar }{{e}^{i{{H}_{0}}t/\hbar }}{{H}_{0}}{{a}_{+}}\]Therefore, the time development of the annihilation and creation operators in the interaction picture is given by:\[\frac{d{{a}_{-}}}{dt}=-\frac{1}{i\hbar }{{H}_{0}}{{a}_{-}}\]\[\frac{d{{a}_{+}}}{dt}=\frac{1}{i\hbar }{{H}_{0}}{{a}_{+}}\]In summary, to find the equation of motion for a general operator
  • #1
Denver Dang
148
1

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 ([itex]\hat{a}[/itex] and [itex]\[{{\hat{a}}^{\dagger }}\][/itex]) 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:

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

where:

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

So my thought is, that I just take the derivative of [itex]{{A}_{I}}[/itex], 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 [itex]\left[ {{a}_{-}},{{a}_{+}} \right]=1[/itex].
But then I don't know what my next step is.

So if someone could help I would be grateful :)Regards
 
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  • #2
,

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}
 

FAQ: Equation of motion and operators in the interaction picture.

What is the equation of motion in the interaction picture?

The equation of motion in the interaction picture is a mathematical tool used to describe the time-evolution of a quantum system, specifically in the presence of an external potential or force. It takes into account the effects of the external forces on the system and allows for the calculation of the system's state at any given time.

What are operators in the interaction picture?

Operators in the interaction picture are mathematical entities used to describe the properties and behavior of a quantum system. They represent physical observables such as position, momentum, and energy, and allow for the calculation of their values at different points in time.

How is the interaction picture related to the Schrödinger picture?

The Schrödinger picture and the interaction picture are two different mathematical formulations of quantum mechanics. The Schrödinger picture uses the time-dependent Schrödinger equation to describe the time-evolution of a quantum system, while the interaction picture uses the equation of motion to take into account the effects of external forces on the system. The two pictures are related through a transformation known as the unitary transformation.

What is the role of the Hamiltonian in the interaction picture?

The Hamiltonian is a fundamental concept in quantum mechanics and plays a crucial role in the interaction picture. It represents the total energy of a quantum system and is used to calculate the time-evolution of the system in the presence of external forces. In the interaction picture, the Hamiltonian is split into two parts: the free Hamiltonian, which describes the system without external forces, and the interaction Hamiltonian, which accounts for the external forces.

How is the interaction picture used in practical applications?

The interaction picture is a powerful tool that is used in many practical applications in quantum mechanics. It is commonly used in calculations related to quantum field theory, quantum optics, and particle physics, among others. It allows for the analysis of complex systems and the calculation of physical observables in the presence of external forces, making it an essential concept in many areas of quantum physics.

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