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Quantum Mechanics

Quantum Mechanics Equations

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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).

Pauli Exclusion Principle

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

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.

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.

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.

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.

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).

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.

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.

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