The Border Between Classical and Quantum Worlds: When Does Physics Go Quantum?

Quantum confinement effect: controlled dance of electrons at microscopic level

The Quantum Evolution: Quantum Mechanics Developed Just Like Quantum Mechanics Works

Have you heard of Max Planck? He was the scientist who laid the foundations of quantum mechanics. While studying blackbody radiation, he made a revolutionary assumption [1]: energy is not emitted or absorbed continuously, but in discrete packets. He called these packets quanta.

To express the relationship between the energy of these quanta and the frequency of electromagnetic radiation, he introduced a proportionality constant, now known as Planck’s constant:

E = h f

Where E is energy, h is Planck’s constant and f is frequency of the wave.

The value of this constant is very small = 6.626 x 10⁻³⁴ J⋅s. At the time, no one fully understood the significance of this extremely small constant, but later we learned that it is this constant that sets the boundary between classical and quantum mechanics. In other words, this seemingly insignificant constant, with the dimension of energy x time, determines whether we need classical physics or quantum mechanics to describe a system.

To understand this better, let me introduce you a concept called Action, designated by S, which also has the dimension of energy × time, defined as

Action (S) = ∫(T−V) dt

where T is kinetic energy (mv²/2) and V is potential energy.

So, action (S) is integral of the difference between kinetic energy (T) and potential energy (V) over time.

In case of a  moving particle, potential energy V= 0, and then we get

S = ∫ T dt

This can be further rewritten as-

S ≈ ∫ p dx,

where p is momentum of the particle.

Thus, action is essentially the integral of momentum over distance.

Now the basic thing is this: The relation between the action S with the Planck constant (h) decides whether we are in the quantum regime or the classical regime.

→When 𝑆 ≫ ℎ

the system follows classical mechanics, where momentum and position are well defined and can be measured simultaneously. In this case, the objects follow a deterministic path and Newton’s equation of motion governs it.

→When S≈h

the system enters the quantum realm and in that case we must consider the wave-like behavior of the particle, where momentum and position cannot be measured simultaneously with precision, as stated by Heisenberg’s uncertainty principle. In this case, there is no deterministic path but rather probabilistic path, which is governed by Schrödinger’s equation of motion.

So, remembering that action is momentum over distance, we say-

For low momentum, short distance, short time – quantum effects dominate

For large momentum, longer distance, longer time – classical mechanics applies

Example to understand the boundary

To understand it better, let me give you an example. Assume a particle is placed in a large box, where it is free to move in all possible directions. In this case, due to high momentum and freedom of movement, the action would be large, so classical physics applies.

Now shrink the box and restrict the motion of the particle in all the direction so that it becomes trapped in a position. In this scenario, momentum of the particle decreases, action becomes small and the particle is now confined. This particle has to be considered as a quantum particle, which would show wave-like properties. This is precisely what happens in quantum dots, where confinement of electron’s momentum leads to a small action, and intresting quantum properties emerge.

What about real Life?

In our everyday experience, the value of the action is much greater than the Planck constant, so everything around us behaves classically.

Classical to quantum transition

We can also change some systems to move from classical to quantum regime by lowering the temperature. Lowering the temperature can reduce the action in some systems to Planck’s constant level, which pushes the system into quantum regime. This is how we get unique phenomena like superconductivity, quantized conductance and Bose-Einstein condensation.

How to visualize it?

Have you zoomed into an image to the level where you start to notice its smallest element called a pixel. Think of Planck’s constant as the smallest pixel in the universe. Quantum effects are only visible when the “motion” (action) is at or near this pixel size, i.e. at low speeds, short distances, low temperatures and short times – precise ‘pixel level’ motion. On the other hand, high momentum, large distances lead to blurry, noisy motion in the classical regime.

So, Where Is the Boundary?

The boundary between classical and quantum physics is not a fixed line but depends on the scale of action. The Planck constant, although small, plays a key role in defining this boundary. As we move into the quantum era, it is important to appreciate the fundamental principles that separate the classical world from the quantum realm.

Stay curious and keep exploring the quantum world!