Mathematical formulation of quantum mechanics
In the 1890s, Planck was able to derive the blackbody spectrum, which was later used to avoid the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of electromagnetic radiation with matter, energy could only be exchanged in discrete units which he called quanta.In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were actual particles, which were later dubbed photons.They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of the Planck constant were actually allowed.The physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity.Werner Heisenberg's matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra.Schrödinger's formalism was considered easier to understand, visualize and calculate as it led to differential equations, which physicists were already familiar with solving.Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space.It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object.In his PhD thesis project, Paul Dirac[2] discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics.In his above-mentioned account, he introduced the bra–ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system.The first complete mathematical formulation of this approach, known as the Dirac–von Neumann axioms, is generally credited to John von Neumann's 1932 book Mathematical Foundations of Quantum Mechanics, although Hermann Weyl had already referred to Hilbert spaces (which he called unitary spaces) in his 1927 classic paper and book.In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.Separability is a mathematically convenient hypothesis, with the physical interpretation that the state is uniquely determined by countably many observations.The expectation value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector ψ ∈ H isIn the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first.A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see Naimark's dilation theorem).Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.In addition to their other properties, all particles possess a quantity called spin, an intrinsic angular momentum.For the best representation of the physical system, we expect this to be an eigenvector of P since exchange operator is not excepted to give completely different vectors in projective Hilbert space.Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics, the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit.where H is a densely defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and ħ is the reduced Planck constant.The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics.Summary: The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations.The Stone–von Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent.At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system.[13] The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment; it is not applicable to most present-day measurements within the quantum domain): For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues λi, with corresponding eigenvectors ψi.The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation.
Quantum mechanicsSchrödinger equationIntroductionGlossaryHistoryClassical mechanicsOld quantum theoryBra–ket notationHamiltonianInterferenceComplementarityDecoherenceEntanglementEnergy levelMeasurementNonlocalityQuantum numberSuperpositionSymmetryTunnellingUncertaintyWave functionCollapseBell's inequalityCHSH inequalityDavisson–GermerDouble-slitElitzur–VaidmanFranck–HertzLeggett inequalityLeggett–Garg inequalityMach–ZehnderPopperQuantum eraserDelayed-choiceSchrödinger's catStern–GerlachWheeler's delayed-choiceHeisenbergInteractionMatrixPhase-spaceSchrödingerSum-over-histories (path integral)Klein–GordonRydbergInterpretationsBayesianConsistent historiesCopenhagende Broglie–BohmEnsembleHidden-variableSuperdeterminismMany-worldsObjective-collapseQuantum logicRelationalTransactionalVon Neumann–WignerRelativistic quantum mechanicsQuantum field theoryQuantum information scienceQuantum computingQuantum chaosEPR paradoxDensity matrixScattering theoryQuantum statistical mechanicsQuantum machine learningAharonovBlackettde BroglieComptonDavissonEhrenfestEinsteinEverettFeynmanGlauberGutzwillerHilbertJordanKramersLandauMoseleyMillikanPlanckSimmonsSommerfeldvon NeumannWignerZeemanZeilingermathematical formalismsfunctional analysisHilbert spaceslinear spaceL2 spaceoperatorsobservablesenergymomentumfunctionsphase spaceeigenvaluesspectral valuesquantum statemodelsthought experimentnon-commutativitytheorymathematical analysiscalculusdifferential geometrypartial differential equationsProbability theorystatistical mechanicstheories of relativityclassical physicsSommerfeld–Wilson–Ishiwara quantizationblackbody spectrumultraviolet catastropheelectromagnetic radiationmatterquanta