# Ehrenfest theorem and Heisenberg quantum mechanics

• luxxio
In summary, the Ehrenfest theorem and the Heisenberg picture of quantum mechanics differ in their approach to expressing the equation of motion for observables. While the Ehrenfest theorem focuses on the mean value of observables, the Heisenberg picture uses operators and eliminates the need for a Schroedinger equation. The Ehrenfest theorem can be derived without using the Heisenberg picture, but it becomes easier with its use.
luxxio
Wich are the differences between the theorem of ehrenfest and the Heisenberg's rappresentation of quantum mechanics?

Last edited:
The Ehrenfest theorem states what are the equation of motion for the mean value of observables like Position and Momentum

The Heisenberg picture moves the time dependence of the system to operators instead of state vector. So it provides the equation of motions for the operators not for their mean value over a state. In Heisenberg picture there is no more need of a Schroedinger equations (for the state vector) because it is now substituted by the Heisenberg equation of motions (for the operators).

ruleva1983 said:
The Ehrenfest theorem states what are the equation of motion for the mean value of observables like Position and Momentum

The Heisenberg picture moves the time dependence of the system to operators instead of state vector. So it provides the equation of motions for the operators not for their mean value over a state. In Heisenberg picture there is no more need of a Schroedinger equations (for the state vector) because it is now substituted by the Heisenberg equation of motions (for the operators).

yes i know, but my question is to be intended in differrent manner. is the theorem of ehrenfest a consequence of the equation of motion of Heisenberg or they can be differents?

Ehrenfest theorem can be derived even without using the Heisenberg picture, but if that picture is used then the derivation is trivial. Is it an answer to your question?

Demystifier said:
Ehrenfest theorem can be derived even without using the Heisenberg picture, but if that picture is used then the derivation is trivial. Is it an answer to your question?
yes it is

## 1. What is the Ehrenfest theorem?

The Ehrenfest theorem is a mathematical concept in quantum mechanics that relates the time evolution of the expectation values of physical observables to the corresponding equations of motion. It states that the rate of change of an observable's expectation value is equal to the expectation value of the observable's corresponding commutator with the Hamiltonian operator.

## 2. Who was Paul Ehrenfest and why is the theorem named after him?

Paul Ehrenfest was a Dutch physicist who made significant contributions to the development of quantum mechanics in the early 20th century. He worked closely with Niels Bohr and Albert Einstein and is best known for his work on the Ehrenfest theorem, which he published in 1927. The theorem is named after him in recognition of his pioneering work in the field.

## 3. What is the difference between Heisenberg quantum mechanics and Schrödinger quantum mechanics?

Heisenberg quantum mechanics and Schrödinger quantum mechanics are two different formulations of quantum mechanics. In Heisenberg's formulation, the fundamental quantities are operators that represent physical observables, while in Schrödinger's formulation, the fundamental quantities are wave functions that describe the state of a system. Additionally, Heisenberg's formulation is based on matrix mechanics, while Schrödinger's formulation is based on wave mechanics.

## 4. How does the Heisenberg uncertainty principle relate to the Ehrenfest theorem?

The Heisenberg uncertainty principle is a fundamental principle in quantum mechanics that states that it is impossible to know both the position and momentum of a particle with absolute certainty. This principle is related to the Ehrenfest theorem because the time evolution of the expectation values of observables, as described by the theorem, is subject to the limitations of the uncertainty principle. In other words, the uncertainty principle places constraints on the precision with which we can measure the expectation values of observables.

## 5. What are some practical applications of the Ehrenfest theorem in quantum mechanics?

The Ehrenfest theorem is a useful tool for calculating the time evolution of expectation values of physical observables in quantum mechanical systems. It has applications in various areas of physics, including atomic and molecular physics, solid state physics, and nuclear physics. It is also used in theoretical studies of quantum systems, such as quantum computing and quantum information theory.

Replies
13
Views
1K
Replies
18
Views
2K
Replies
2
Views
1K
Replies
10
Views
1K
Replies
4
Views
898
Replies
7
Views
2K
Replies
8
Views
2K
Replies
2
Views
403
Replies
8
Views
1K
Replies
3
Views
986