Ehrenfest theorem

The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, helps us understand how tiny particles, like atoms, move and change over time. It connects two important ideas in quantum mechanics—the position and momentum of a particle—to the force acting on that particle. This theorem shows that even though quantum particles behave in strange ways, their average behavior can still follow rules that look familiar from classical physics.

This theorem is a special case of a broader rule that links quantum operators—mathematical objects that represent physical quantities—to their "commutators," which measure how they interact with each other. In the Heisenberg picture of quantum mechanics, the Ehrenfest theorem appears as the expectation value of the Heisenberg equation of motion, giving it a clear and direct meaning.

Ehrenfest's theorem is closely related to Liouville's theorem from classical Hamiltonian mechanics. It suggests that quantum mechanics, when written with commutators, often mirrors classical mechanics, where similar ideas use Poisson brackets instead. This connection helps explain how quantum systems can sometimes behave like their classical counterparts, especially when the energy of the system depends in a simple way on position and momentum. Even when this isn’t perfectly true, the theorem can still give good approximations if changes are small.

Relation to classical physics

The Ehrenfest theorem shows how quantum physics connects to classical physics. It describes how the average position and momentum of particles change over time. However, these averages do not always follow Newton’s classical laws exactly.

In simple cases, like a quantum harmonic oscillator, the averages do match classical motion perfectly. But for more complex systems, there are small differences due to quantum uncertainty. When a particle’s position is very certain, these averages come close to classical predictions.

Derivation in the Schrödinger picture

Imagine you have a tiny particle described by quantum physics. The Ehrenfest theorem helps us understand how certain properties of this particle, like its position and speed, change over time.

To study this, scientists look at something called a "quantum state," which tells us everything about the particle. By using special math and the Schrödinger equation — which is like a rule book for how quantum particles behave — they show that the change in the particle’s properties connects to another important part of quantum physics called the Hamiltonian. This helps explain how quantum ideas match up with the physics we see in everyday life.

Derivation in the Heisenberg picture

In the Heisenberg picture, the Ehrenfest theorem is easy to understand. This picture shifts the time changes of a system to operators rather than the states themselves. By using the Heisenberg equation of motion, we can see how these operators change over time.

The theorem shows that the change in an operator's expectation value over time depends on two parts: how the operator itself changes and how it interacts with the system's energy. This helps connect quantum mechanics with classical physics, showing how quantum behaviors can look like normal physics under certain conditions.

General example

For a tiny particle moving in a special kind of field, scientists use something called the Ehrenfest theorem to understand how its position and speed change over time. This theorem helps connect ideas from quantum physics — the study of very small particles — to the physics we see every day.

The theorem shows that while tiny particles don’t always follow the exact same rules as bigger objects, their average behavior can still look a lot like classical physics when they are very close to one place. This helps explain why the rules of the very small world sometimes seem to match the rules of the big world we live in.

Derivation of the Schrödinger equation from the Ehrenfest theorems

The Ehrenfest theorems can help us understand how the Schrödinger equation works. These theorems show links between tiny particle movements and forces. By using math rules, we can move from these links back to the Schrödinger equation, which describes how particles behave in quantum physics.

This process shows a key idea: quantum physics and classic physics differ mainly in how we describe tiny particles' positions and motions. The Schrödinger equation comes from assuming these properties do not commute — meaning changing position then momentum is different from changing momentum then position. If we assume they do commute, we get classical physics instead. This highlights a core difference between quantum and classical views of the world.

Main article: Schrödinger equation

Main articles: Koopman–von Neumann classical mechanics, Hilbert space

Further information: derivation of the Koopman–von Neumann mechanics