Interaction picture equation from Heisenberg equation

In summary: It seems like this approach may not work then. Have you tried a different method or have any other ideas? In summary, the conversation is about the derivation of the interaction-picture equation of motion from the Heisenberg-picture equation. The speaker has been attempting to go directly from the Heisenberg equation to the interaction equation but has encountered difficulties due to the presence of the interaction term in the original equation. They have also considered using the Schrodinger picture to derive the equation, but are unsure if it should work the other way as well. The conversation ends with a discussion about the limitations of this approach and potential alternatives.
  • #1
copernicus1
99
0
The standard Heisenberg picture equation of motion is $$i\hbar\frac d{dt}A_H=[A_H,H],$$ assuming no explicit ##t##-dependence on the Heisenberg-picture operator ##A_H##. I've been trying to go directly from this equation to the corresponding interaction-picture equation, $$i\hbar\frac d{dt}A_I=[A_I,H_0],$$ (see Sakurai 5.5.12) which I thought at first would be simple, but I keep coming up with $$i\hbar\frac d{dt}A_I=[A_I,H_0]+[A_I,V_I],$$ where ##V_I## is the interaction part of the hamiltonian in the interaction picture. The basic problem is that in the original equation ##H## contains both ##H_0## and ##V## and I don't know how to get rid of the ##V## part. Has anyone been through this calculation?

Thanks!

P.S. I know I could just start with ##A_I(t)=e^{iH_0t/\hbar}A_Se^{-iH_0t/\hbar}##, where ##A_S## is in the Schrodinger picture, and I can derive the equation this way, but I feel like it should work the other way too.
 
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  • #2
Could you please show your derivation?
 
  • #3
I start with $$i\hbar\frac{}d{dt}A_H(t)=[A_H(t),H_H(t)]$$ (subscript means Heisenberg picture) and plug in ##A_H(t)=e^{iH_St/\hbar}A_Se^{-iH_St/\hbar}## and ##H_H(t)=e^{iH_St/\hbar}H_Se^{-iH_St/\hbar}##. (I then replace ##H_S=H_{0,S}+V_S## everywhere and transform both sides of the original Heisenberg equation using $$i\hbar\frac{}d{dt}e^{-iV_St/\hbar}A_H(t)e^{iV_St/\hbar}=e^{-iV_St/\hbar}[A_H(t),H_H(t)]e^{iV_St/\hbar}.$$ Simplify and I'm left with $$i\hbar\frac{d}{dt}A_I(t)=A_I(t)e^{iH_{0,S}t/\hbar}H_Se^{-iH_{0,S}t/\hbar}-e^{iH_{0,S}t/\hbar}H_Se^{-iH_{0,S}t/\hbar}A_I(t),$$ but ##H_S\neq H_{0,S}##. If it did I would be done. Instead it's ##H_S=H_{0,S}+V_S##. This is where I get stuck.
 
  • #4
copernicus1 said:
I then replace ##H_S=H_{0,S}+V_S## everywhere
I guess this is where it doesn't work. Since ##[H_0, V] \neq 0##, ##e^{i (H_0 + V) t/ \hbar} \neq e^{i H_0 t/ \hbar} e^{i V t/ \hbar}##.
 
  • #5
Ah, interesting! Thanks for pointing that out.
 

1. What is the interaction picture equation and how is it related to the Heisenberg equation?

The interaction picture equation is a mathematical tool used in quantum mechanics to simplify the analysis of time-dependent systems. It is related to the Heisenberg equation because it is derived from it and allows for a separation of the time-dependent and time-independent parts of a system's evolution.

2. How is the interaction picture equation different from the Schrödinger and Heisenberg equations?

The interaction picture equation is different from the Schrödinger and Heisenberg equations in that it introduces an additional term known as the interaction Hamiltonian, which accounts for the time-dependence of a system. The Schrödinger equation describes the time evolution of a quantum system in terms of its state vector, while the Heisenberg equation describes the time evolution of observables.

3. What types of systems can be analyzed using the interaction picture equation?

The interaction picture equation can be used to analyze any time-dependent quantum system, including systems with external forces or interactions between particles. It is particularly useful for studying systems undergoing perturbations, such as an electron in an electromagnetic field or a particle in a potential well.

4. How is the interaction picture equation applied in practice?

In practice, the interaction picture equation is applied by using it to transform the state vector and operators from the Schrödinger picture to the interaction picture. This allows for a separation of the time-dependent and time-independent parts of the system, making it easier to analyze and solve for the time evolution of the system.

5. What are the benefits of using the interaction picture equation in quantum mechanics?

The interaction picture equation offers several benefits in quantum mechanics, including simplifying the analysis of time-dependent systems, allowing for a separation of time scales, and providing a more intuitive understanding of the time evolution of a system. It also makes it easier to incorporate perturbations into the analysis of a system, making it a valuable tool for studying a wide range of physical phenomena.

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