Dirac correspondence principle

The Dirac correspondence principle, also known as Dirac's theorem, is a fundamental concept in the bridge between quantum mechanics and classical mechanics. It describes how the quantum mechanical description of a physical system transforms into the classical description as the scale of the system becomes large.

Paul Dirac proposed this principle as a way to connect the two theories and show that quantum mechanics naturally gives rise to classical mechanics under certain conditions. The correspondence principle helps us understand how classical physics emerges from the underlying quantum mechanical behaviour of particles and systems.

In simple terms, the Dirac correspondence principle states that the predictions of quantum mechanics should closely match the predictions of classical mechanics when the quantum numbers involved (like energy levels, angular momentum, etc.) become large. This means that for macroscopic systems or systems with high energies, quantum behaviour tends to resemble classical behaviour.

Here are the key points of the Dirac correspondence principle:

1. Large Quantum Numbers: As quantum numbers (like energy or angular momentum) become large, the quantum mechanical equations start to resemble the classical equations. This is why we see classical behaviour in our everyday world—classical mechanics is an approximation of quantum mechanics for large systems.

2. Wavefunctions to Classical Trajectories: In the classical limit, the wavefunction of a quantum system "collapses" to a classical trajectory. This trajectory follows the path of least action, similar to the principle of least action in classical mechanics.

3. Quantum Averages to Classical Values: The expectation values of quantum observables (like position and momentum) approach the classical values that we're familiar with.

4. Consistency: Quantum mechanics does not contradict classical mechanics in the classical limit; instead, it becomes consistent with it.

In essence, the Dirac correspondence principle reassures us that the seemingly strange behaviour of quantum mechanics at small scales gives rise to the well-understood behaviour of classical mechanics when we consider larger systems or systems with higher energies. This principle provides an important connection between these two branches of physics, allowing us to better understand the transition from the quantum to the classical world.

The operator equations of Quantum mechanics reduce to the laws of classical mechanics if for any pair of observables A and B, we make the replacement

In Quantum mechanics

d< X >/ dt = 1/ iℏ [ X, H]

In classical mechanics

d< X >/ dt = { X, H}

In Quantum mechanics

In classical mechanics

dPx/dt" represents the rate of change of the expectation value of momentum "Px" with respect to time "t" for a quantum system. The momentum operator Px corresponds to the momentum in the x-direction.

In quantum mechanics, the expectation value of an observable (like momentum) gives you the average value you would expect to measure if you perform many measurements on identical systems in the same state.

Significance

The expression "dPx/dt" is significant in the context of the Dirac correspondence principle because it shows how the change in momentum expectation value over time in a quantum system approaches the classical concept of force.

In classical mechanics, "dPx/dt" corresponds to the force acting on an object, according to Newton's second law (F = dPx/dt). This connection between momentum change and force is a fundamental aspect of classical physics.

This note is a part of the Physics Repository.