# Quantum mechanics

### Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, or quantum theory) is a fundamental branch of physics which deals with physical phenomena at nanoscopic scales where the action is on the order of the Planck constant. It departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. Quantum mechanics provides a substantially useful framework for many features of the modern periodic table of elements including the behavior of atoms during chemical bonding and has played a significant role in the development of many modern technologies.

In advanced topics of quantum mechanics, some of these behaviors are macroscopic (see macroscopic quantum phenomena) and emerge at only extreme (i.e., very low or very high) energies or temperatures (such as in the use of superconducting magnets). In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons, and other atomic-scale objects.

The mathematical formulations of quantum mechanics are abstract. A mathematical function, the wavefunction, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Mathematical manipulations of the wavefunction usually involve bra–ket notation which requires an understanding of complex numbers and linear functionals. The wavefunction formulation treats the particle as a quantum harmonic oscillator, and the mathematics is akin to that describing acoustic resonance. Many of the results of quantum mechanics are not easily visualized in terms of classical mechanics. For instance, in a quantum mechanical model the lowest energy state of a system, the ground state, is non-zero as opposed to a more "traditional" ground state with zero kinetic energy (all particles at rest). Instead of a traditional static, unchanging zero energy state, quantum mechanics allows for far more dynamic, chaotic possibilities, according to John Wheeler.

The earliest versions of quantum mechanics were formulated in the first decade of the 20th century. About this time, the atomic theory and the corpuscular theory of light (as updated by Einstein)[1] first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of matter and electromagnetic radiation, respectively. Early quantum theory was significantly reformulated in the mid-1920s by Werner Heisenberg, Max Born and Pascual Jordan, (matrix mechanics); Louis de Broglie and Erwin Schrödinger (wave mechanics); and Wolfgang Pauli and Satyendra Nath Bose (statistics of subatomic particles). Moreover, the Copenhagen interpretation of Niels Bohr became widely accepted. By 1930, quantum mechanics had been further unified and formalized by the work of David Hilbert, Paul Dirac and John von Neumann[2] with a greater emphasis placed on measurement in quantum mechanics, the statistical nature of our knowledge of reality, and philosophical speculation about the role of the observer. Quantum mechanics has since permeated throughout many aspects of 20th-century physics and other disciplines including quantum chemistry, quantum electronics, quantum optics, and quantum information science. Much 19th-century physics has been re-evaluated as the "classical limit" of quantum mechanics and its more advanced developments in terms of quantum field theory, string theory, and speculative quantum gravity theories.

The name quantum mechanics derives from the observation that some physical quantities can change only in discrete amounts (Latin quanta), and not in a continuous (cf. analog) way.

## Contents

• History 1
• Mathematical formulations 2
• Mathematically equivalent formulations of quantum mechanics 3
• Interactions with other scientific theories 4
• Quantum mechanics and classical physics 4.1
• Relativity and quantum mechanics 4.2
• Attempts at a unified field theory 4.3
• Philosophical implications 5
• Applications 6
• Examples 7
• Notes 9
• References 10

## History

Scientific inquiry into the wave nature of light began in the 17th and 18th centuries when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations.[3] In 1803, Thomas Young, an English polymath, performed the famous double-slit experiment that he later described in a paper entitled "On the nature of light and colours". This experiment played a major role in the general acceptance of the wave theory of light.

In 1838, with the discovery of cathode rays by Michael Faraday, these studies were followed by the 1859 statement of the black-body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.[4] Planck's hypothesis that energy is radiated and absorbed in discrete "quanta" (or "energy elements") precisely matched the observed patterns of black-body radiation.

In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation,[5] known as Wien's law in his honor. Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations. However, it was valid only at high frequencies, and underestimated the radiance at low frequencies. Later, Max Planck corrected this model using Boltzmann statistical interpretation of thermodynamics and proposed what is now called Planck's law, which led to the development of quantum mechanics.

Among the first to study quantum phenomena in nature were Arthur Compton, C.V. Raman, and Pieter Zeeman, each of whom has a quantum effect named after him. Robert A. Millikan studied the Photoelectric effect experimentally and Albert Einstein developed a theory for it. At the same time Niels Bohr developed his theory of the atomic structure which was later confirmed by the experiments of Henry Moseley. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld.[6] This phase is known as Old quantum theory.

According to Planck, each energy element, E, is proportional to its frequency, ν:

E = h \nu\
Max Planck is considered the father of the Quantum Theory

where h is Planck's constant. Planck (cautiously) insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself.[7] In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[8] However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect in which shining light on certain materials can eject electrons from the material.

The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Niels Bohr, Werner Heisenberg, Louis de Broglie, Arthur Compton, Albert Einstein, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Enrico Fermi, Wolfgang Pauli, Max von Laue, Freeman Dyson, David Hilbert, Wilhelm Wien, Satyendra Nath Bose, Arnold Sommerfeld and others. In the mid-1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the "Old Quantum Theory". Out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons (1926). From Einstein's simple postulation was born a flurry of debating, theorizing, and testing. Thus the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927.

The other exemplar that led to quantum mechanics was the study of electromagnetic waves, such as visible and ultraviolet light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or "quanta", Albert Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the photon) with a discrete quantum of energy that was dependent on its frequency.[9] Einstein was able to use the photon theory of light to explain the photoelectric effect for which he won the 1921 Nobel Prize in Physics. This led to a theory of unity between subatomic particles and electromagnetic waves in which particles and waves are neither simply particle nor wave but have certain properties of each. This originated the concept of wave–particle duality.

While quantum mechanics traditionally described the world of the very small, it is also needed to explain certain recently investigated

Philosophy
• PHYS 201: Fundamentals of Physics II by Ramamurti Shankar, Open Yale Course
• Lectures on Quantum Mechanics by Leonard Susskind
• Everything you wanted to know about the quantum world — archive of articles from New Scientist.
• Quantum Physics Research from Science Daily
• Overbye, Dennis (December 27, 2005). "Quantum Trickery: Testing Einstein's Strangest Theory". The New York Times. Retrieved April 12, 2010.
• Audio: Astronomy Cast Quantum Mechanics — June 2009. Fraser Cain interviews Pamela L. Gay.
Media
• Many-worlds or relative-state interpretation.
• Measurement in Quantum mechanics.
FAQs
• Quantum Physics Database - Fundamentals and Historical Background of Quantum Theory.
• Doron Cohen: Lecture notes in Quantum Mechanics (comprehensive, with advanced topics).
• MIT OpenCourseWare: Chemistry.
• MIT OpenCourseWare: Physics. See 8.04
• Stanford Continuing Education PHY 25: Quantum Mechanics by Leonard Susskind, see course description Fall 2007
• 5½ Examples in Quantum Mechanics
• Imperial College Quantum Mechanics Course.
• Spark Notes - Quantum Physics.
• Quantum Physics Online : interactive introduction to quantum mechanics (RS applets).
• Experiments to the foundations of quantum physics with single photons.
• AQME : Advancing Quantum Mechanics for Engineers — by T.Barzso, D.Vasileska and G.Klimeck online learning resource with simulation tools on nanohub
• Quantum Mechanics by Martin Plenio
• Quantum Mechanics by Richard Fitzpatrick
• Quantum TransportOnline course on
Course material
• 3D animations, applications and research for basic quantum effects (animations also available in (Université paris Sud))
• Quantum Cook Book by R. Shankar, Open Yale PHYS 201 material (4pp)
• The Modern Revolution in Physics - an online textbook.
• J. O'Connor and E. F. Robertson: A history of quantum mechanics.
• Introduction to Quantum Theory at Quantiki.
• Quantum Physics Made Relatively Simple: three video lectures by Hans Bethe
• H is for h-bar.
• Quantum Mechanics Books Collection: Collection of free books