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Quantum Mechanics

Quantum Mechanics Theorems

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Heisenberg Uncertainty Principle

States that the position and momentum of a particle cannot be simultaneously known with arbitrarily high precision.

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

This sets a fundamental limit to the precision with which certain pairs of physical properties can be known.

Energy-Time Uncertainty Principle

A variant of the Heisenberg Uncertainty Principle, it states that the uncertainty in energy and the uncertainty in time are inversely related.

\Delta E \Delta t \geq \frac{\hbar}{2}

Schrodinger's Equation

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

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

This equation is the basis of quantum mechanics and determines how wavefunctions evolve.

Born Rule

Gives the probability that a measurement on a quantum system will yield a given result. For a quantum state $\Psi$ , the probability density function is $\left|\Psi\right|^2$ .

Bell's Theorem

Demonstrates that no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics. It has profound implications for the nature of reality, including the issues of quantum entanglement and locality.

De Broglie Hypothesis

Suggests that particles of matter can exhibit wave-like behavior, and relates the observed wavelength of matter to its momentum.

\lambda = \frac{h}{p}

, where

\lambda

is wavelength,

h

is Planck's constant, and

p

is momentum.

No-Cloning Theorem

States that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state. This theorem is fundamental for quantum cryptography and quantum information theory.

Ehrenfest Theorem

Shows that the expectation value of the quantum observables follow the classical equations of motion. This creates a bridge between classical mechanics and quantum mechanics, revealing the correspondence principle.

Fermi's Golden Rule

Provides the transition rate from one energy eigenstate of a quantum system to another when perturbed, commonly used to calculate the probability of transitions due to interaction with a perturbative Hamiltonian. The formula is

W = \frac{2\pi}{\hbar} \left| \langle f| \hat{H}' |i\rangle \right|^2 \rho(E_f)

, where

W

is the transition rate,

|i\rangle

and

|f\rangle

are the initial and final states,

\hat{H}'

is the perturbation Hamiltonian, and

\rho(E_f)

is the density of states.

Pauli Exclusion Principle

Two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously. This principle explains a wide variety of physical phenomena including the structure of the Periodic Table.

Wigner's Theorem

States that quantum symmetries are represented by either unitary or antiunitary operators on a Hilbert space, providing a mathematical foundation to the notion of symmetry in quantum mechanics.

Superposition Principle

A physical system exists partly in all its particular, theoretically possible states simultaneously; but when measured or observed, it gives a result corresponding to only one of the possible configurations. This is commonly known as the quantum superposition of states.

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