Editing Quantum (section)

= 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.

<math>
E_n = -\frac{13.6\, \text{eV}}{n^2}, \quad n = 1, 2, 3, \ldots
</math>

Where:
* <math>E_n</math> is the energy of the <math>n</math>-th level,
* <math>n</math> is the principal quantum number.

=== 2. Planck’s Quantum Hypothesis ===
Energy is emitted or absorbed in discrete packets (quanta), given by:

<math>
E = h f
</math>

Where:
* <math>E</math> is the energy of a quantum,
* <math>h</math> is Planck’s constant <math>(6.626 \times 10^{-34}\, \text{Js})</math>,
* <math>f</math> 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:

<math>
\lambda = \frac{h}{p}
</math>

Where:
* <math>\lambda</math> is the wavelength,
* <math>p</math> is momentum,
* <math>h</math> is Planck’s constant.

=== 4. Heisenberg Uncertainty Principle ===
It is impossible to simultaneously know the exact position and momentum of a particle:

<math>
\Delta x \cdot \Delta p \geq \frac{\hbar}{2}
</math>

Where:
* <math>\Delta x</math> is the uncertainty in position,
* <math>\Delta p</math> is the uncertainty in momentum,
* <math>\hbar = \frac{h}{2\pi}</math> is the reduced Planck’s constant.

== Schrödinger Equation ==

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

<math>
i\hbar \frac{\partial}{\partial t} \Psi(x, t) = \hat{H} \Psi(x, t)
</math>

Where:
* <math>\Psi(x, t)</math> is the wavefunction,
* <math>\hat{H}</math> is the Hamiltonian operator,
* <math>i</math> is the imaginary unit.

The solution <math>\Psi</math> gives the probability amplitude. The probability density is:

<math>
P(x, t) = |\Psi(x, t)|^2
</math>

== Quantum Numbers ==

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

* Principal quantum number <math>n</math>
* Angular momentum quantum number <math>l</math>
* Magnetic quantum number <math>m_l</math>
* Spin quantum number <math>m_s</math>

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]]