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Quantum Mechanics

Quantum Mechanics Equations

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Time-independent Schrödinger Equation

The Time-independent Schrödinger Equation:

\hat{H}\Psi = E\Psi

, where \( \hat{H} \) is the Hamiltonian operator, \( \Psi \) is the wave function, and E is the energy of the system, describes the stationary states of quantum systems.

Born Rule

The Born Rule:

P(x) = |\Psi(x)|^2

, where P is the probability density function, and \( \Psi(x) \) is the wave function of the particle, gives the probability of finding a particle at a particular position.

Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers.

Schrödinger Equation

The Schrödinger equation:

i\hbar\frac{\partial}{\partial t}\Psi (\mathbf{r},t) = \hat{H}\Psi (\mathbf{r}, t)

, describes how the quantum state of a physical system changes over time.

Dirac Equation

The Dirac Equation:

\left(\beta mc^2 + c\sum_{n=1}^{3}\alpha_n p_n \right)\Psi = i\hbar\frac{\partial \Psi}{\partial t}

, combines quantum mechanics and special relativity to describe particles like electrons.

De Broglie Wavelength

The De Broglie Wavelength formula:

\lambda = \frac{h}{p}

, where \( \lambda \) is the wavelength, h is Planck's constant, and p is momentum, describes the wave nature of moving particles.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle:

\Delta x \Delta p \geq \frac{\hbar}{2}

, represents the limit to the precision with which pairs of physical properties such as position (x) and momentum (p) can be known simultaneously.

Quantum Field Theory Propagator

The Quantum Field Theory (QFT) Propagator represents the probability amplitude for a particle to travel from one point to another and is a fundamental concept rather than a single equation.

Planck-Einstein Relation

The Planck-Einstein Relation:

E = h\nu

, relates the energy (E) of a photon to its frequency (\( \nu \)) using Planck's constant (h).

Einstein's Energy-Mass Equivalence

Einstein's Energy-Mass Equivalence:

E = mc^2

, states that the energy (E) of a system is equal to its mass (m) times the speed of light squared (c^2).

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