The classical limit in quantum mechanics refers to the behaviour of quantum systems becoming more similar to the predictions of classical physics when certain conditions are met, such as when the system's mass is large or when quantum effects become negligible.

In the realm of everyday experiences and larger objects, classical physics provides accurate predictions for the behaviour of physical systems. Classical physics describes the motion of objects, the interactions of forces, and various other phenomena we encounter in our macroscopic world.

Quantum mechanics, on the other hand, deals with the behaviour of particles on extremely small scales, like atoms and subatomic particles. It introduces wave-like properties and uncertainty, which can lead to phenomena like particle-wave duality, tunneling, and entanglement.

The classical limit arises when the quantum effects become negligible and the behaviour of a quantum system becomes more and more like that described by classical physics. This usually happens under the following conditions:

Large Mass and Size: For macroscopic objects with large mass and size, the wavelength associated with their motion becomes extremely small, and quantum effects become insignificant. These systems behave classically.
High Temperatures: At high temperatures, the thermal energy can dominate over the quantum effects, leading to classical behaviour.
Large Quantum Numbers: When quantum states have large quantum numbers (like energy levels in atoms), the spacing between energy levels becomes very small, making the quantum behaviour less pronounced.
Decoherence: When a quantum system interacts with its environment, the quantum coherence can be lost, leading to classical-like behaviour.

In the classical limit, the predictions of quantum mechanics converge toward the predictions of classical physics. This doesn't mean that quantum mechanics becomes invalid; rather, it reflects that the quantum effects become less noticeable and less relevant as the system's properties change.

In classical limit

Note that < -∇V(R) > is not the force acting at the centre of the wave packet (or the force acting at the particle). So < -∇V(R) > is not equal to classical force acting at the particle. In one dimension,

<X> = centre of the wave packet

If the wave packet is highly localised such that (x - <x>) is extremely small then

Force acting at the centre of the wave packet which is force acting at the classical particle.

In such case

d < x > / dt = <Px> / m

d < Px > / dt = - ( ∂V/ ∂x) = classical force = centre of centre of classical particle

If the wave packet is highly localised such that potential energy function varies slowly in the region of wave packet then the motion of centre of the wave packet obeys the classical laws.

When Δx is extremely small ( wave packet is highly localized)

ΔpΔx ∼ ℏ. Δp is high. So de- Broglie wavelength is small, in comparison to the characteristic size of the system the quantum result reduces to classical result. Quantum mechanics is relevant when de Broglie wavelength 𝜆 is large in comparison to the characteristic size of the system. This is correspondence theorem.

The principle of first quantization and the correspondence between quantum mechanics and the classical mechanics.

For any two function M and N of coordinated qi and momentum

pi ; i = 1,2,3.....n

The Poisson Bracket

Hamiltonian equation

dqi/ dt = ∂H/ ∂pi and dpi/ dt =- ∂H/ ∂qi

For a physical quantity

A(qi, pi, i = 1,2,3......n)

In quantum mechanics, physical quantity is replaced by corresponding operator A. We have, the equation of motion

d <A> / dt = 1/ iℏ < [ A, H]>

= < 1/ iℏ [ A, H]>

If we denote the operator whose expectation value is equal to d<A>/ dt by dA/ dt

Then we can write

dA/ dt = 1/ iℏ [ A, H]

We say that the equation dA/ dt = 1/ iℏ [ A, H] describes the quantization of classical equation dA/ dt = { A, H}. This is the first quantization.

The classical limit is an important concept because it helps us understand how the quantum world transitions to the classical world that we are more familiar with in our everyday experiences. It also helps to bridge the gap between these two fundamentally different yet interconnected realms of physics.

This note is a part of the Physics Repository.

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