Quantum

Quantum: Definition and Mathematical Representation

Introduction

In physics, the term quantum refers to the smallest possible discrete unit of any physical property. The concept originates from quantum mechanics, a fundamental theory that describes the behavior of matter and energy on atomic and subatomic scales.

The term "quantum" (plural: "quanta") was first introduced in the early 20th century to explain phenomena that classical physics could not, such as blackbody radiation and the photoelectric effect.

Key Concepts

1. Quantization

Many physical quantities, such as energy or angular momentum, are not continuous but occur in discrete levels. For example, the energy of an electron in a hydrogen atom is quantized.

$E_n=-\frac{13.6\text{eV}}{n^2},n=1,2,3,\dots$

Where:

• $E_n$ is the energy of the $n$-th level,
• $n$ is the principal quantum number.

2. Planck’s Quantum Hypothesis

Energy is emitted or absorbed in discrete packets (quanta), given by:

$E=hf$

Where:

• $E$ is the energy of a quantum,
• $h$ is Planck’s constant $(6.626\times 10^{-34}\text{Js})$,
• $f$ is the frequency of the radiation.

3. Wave-Particle Duality

Particles such as electrons exhibit both wave-like and particle-like properties.

• de Broglie wavelength:

$\lambda =\frac{h}{p}$

Where:

• $\lambda$ is the wavelength,
• $p$ is momentum,
• $h$ is Planck’s constant.

4. Heisenberg Uncertainty Principle

It is impossible to simultaneously know the exact position and momentum of a particle:

$\mathrm{\Delta }x\cdot \mathrm{\Delta }p\ge \frac{\hslash }{2}$

Where:

• $\mathrm{\Delta }x$ is the uncertainty in position,
• $\mathrm{\Delta }p$ is the uncertainty in momentum,
• $\hslash =\frac{h}{2\pi }$ is the reduced Planck’s constant.

Schrödinger Equation

The central equation of non-relativistic quantum mechanics describes how the quantum state evolves over time:

$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(x,t)=\hat{H}\mathrm{\Psi }(x,t)$

Where:

• $\mathrm{\Psi }(x,t)$ is the wavefunction,
• $\hat{H}$ is the Hamiltonian operator,
• $i$ is the imaginary unit.

The solution $\mathrm{\Psi }$ gives the probability amplitude. The probability density is:

$P(x,t)=|\mathrm{\Psi }(x,t){|}^2$

Quantum Numbers

Each quantum system is described using a set of quantum numbers:

• Principal quantum number $n$
• Angular momentum quantum number $l$
• Magnetic quantum number $m_l$
• Spin quantum number $m_s$

These define the allowed states of electrons in atoms.

Applications of Quantum Theory

• Atomic structure and spectra
• Semiconductors and transistors
• Quantum computing
• Superconductivity
• Lasers
• Nuclear and particle physics

See Also

• Quantum Mechanics
• Wave-Particle Duality
• Planck's Constant
• Heisenberg Uncertainty Principle
• Schrödinger Equation
• Quantum Numbers
• Photon
• Electron Configuration