# Heisenberg's equation of motion

• noospace
In summary, the conversation discusses the equation of motion for an observable A and how it changes when the representation is changed via a unitary transformation. It is mentioned that the transformed operator may satisfy the first equation of motion but not the second, depending on the generator of the unitary transform. The question of deriving the Heisenberg equation of motion is also brought up, and the source of information recommended is the Messiah QM vol 2.
noospace
The equation of motion for an observeable A is given by $\dot{A} = \frac{1}{i \hbar} [A,H]$.

If we change representation, via some unitary transformation $\widetilde{A} \mapsto U^\dag A U$ is the corresponding equation of motion now

$\dot{\widetilde{A}} = \frac{1}{i \hbar} [\widetilde{A},U^\dag H U]$
or
$\dot{\widetilde{A}} = \frac{1}{i \hbar} [\widetilde{A},H]$?

I'm asking because I want to write the time derivative of the Dirac representation of the position operator in the Foldy-Wouthusyen representation.

Last edited:
If you know how to derive Heisenberg eq of Motion, then you should have no problem to find the answer.

They're the same, the first equation of motion for the operator UAUt gives the second EOM for A.

Are you saying that the transformed operator satisfies the first equation but not the second?

If the generator of the unitary transform U depends on t -- like going from Schrodinger picture to the Interaction Picture -- then noospace, you have left out a term. Standard stuff, can be found in most QM or QFT texts.
Regards,
Reilly Atkinson

noospace said:
I'm asking because I want to write the time derivative of the Dirac representation of the position operator in the Foldy-Wouthusyen representation.

see Messiah QM vol 2.

## What is Heisenberg's equation of motion?

Heisenberg's equation of motion is a fundamental equation in quantum mechanics that describes how the state of a quantum system changes over time.

## What does the equation describe?

The equation describes the time evolution of the operator that represents the state of the system, known as the Heisenberg operator. It relates the change in the operator to the Hamiltonian, which represents the total energy of the system.

## How is Heisenberg's equation of motion different from Schrödinger's equation?

While Schrödinger's equation describes the time evolution of the state of a quantum system, Heisenberg's equation describes the time evolution of the operator that represents the state. This means that in Heisenberg's equation, the operators are time-dependent while in Schrödinger's equation, the operators are time-independent.

## What is the significance of Heisenberg's equation of motion?

Heisenberg's equation of motion is significant because it allows us to calculate the time evolution of quantum systems and make predictions about their behavior. It also helps us understand the relationship between observables and their corresponding operators in quantum mechanics.

## What are some applications of Heisenberg's equation of motion?

Heisenberg's equation of motion is used in many areas of physics, including quantum field theory, solid-state physics, and particle physics. It is also essential in the development of quantum computing and other quantum technologies.

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