Quantum mechanics

Quantum mechanics consists of the breakthrough in physics in the 1920s in understanding how particles behave inside atoms. Quantum mechanics posits that an electron (or any other sub-atomic particle) behaves as both a wave and a particle.

the essence of quantum theory is energy quantization

Classical mechanics, as initially discovered by Isaac Newton, cannot explain atomic behavior. Erwin Schrodinger is generally credited with the formulation of the Schrodinger equation, around 1926. Other contributions were from Werner Heisenberg, Niels Bohr, John von Neumann, and Hermann Weyl.

Classical mechanics would predict that an electron orbits a proton just as planets orbit the sun. Classical electromagnetism would predict that the orbiting electron would emit a time-varying electrical field just as a radio station does. But the electron would lose energy as it emits this radiation, and would orbit closer and closer to the proton, until it collapses into the proton! Such a model cannot be correct.

As a result of the wave nature of the electron, the position of the electron can never be precisely known. Whenever it is attempted to be measured, knowledge of the electron's velocity is lost. Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the Heisenberg Uncertainty Principle.

Quantum mechanics forms the basis for our understanding of chemical reactions, as well as all computers and electronic devices today.

An important aspect of Quantum Mechanics is the predictions it makes about the radioactive decay of isotopes. Radioactive decay processes, controlled by the wave equations, are random events. A radioactive atom has a certain probability of decaying per unit time. As a result, the decay results in an exponential decrease in the amount of isotope remaining in a given sample as a function of time. The characteristic time required for 1/2 of the original amount of isotope to decay is known as the "half-life" and can vary from quadrillionths of a second (⁹B) to quintillions of years (¹⁸⁶W).

Mathematics

The mathematics of Quantum mechanics can be formulated in a number of ways: the "matrix mechanics" of Werner Heisenberg, the "path integrals" of Richard Feynman, or the "wave mechanics" of Erwin Schrodinger. Wave mechanics is the most common formulation. It uses the language of infinite dimensional Hilbert Spaces; observables such as position and momentum are operators on such Hilbert Spaces.

Postulates of Quantum Mechanics

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• A physical state corresponds to a unit vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} in a Hilbert space.
• Observable physical properties such as position, energy, and momentum are represented by self-adjoint operators on the Hilbert space.
• If we measure an observable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} of a physical state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} , the result of our measurement will be an eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} for the operator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} . The probability that the measurement will yield the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is given by the norm-squared of the projection of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi} onto the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} -eigenspace of $Q$.
• If we measure the observable $Q$ and produce the value $\lambda$, the state $\Psi$ collapses to its projection onto the $\lambda$-eigenspace of $Q$.
• A state $\Psi$ evolves in time according the equation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(t)=e^{iHt/\hbar}\Psi(0)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is the energy operator.

External Links

For an excellent discussion of quantum mechanics, see: http://www.chemistry.ohio-state.edu/betha/qm/

See also: Momentum (operator)