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This formulation of quantum mechanics, called Canonical Quantization , continues to be used today, and still forms the basis of '' Ab-initio '' calculations in atomic, molecular and solid-state physics. At the heart of the description is an idea of ''quantum state'' which, for systems of atomic scale, is radically different from the previous models of physical reality. While the mathematics is a complete description and permits calculation of many quantities that can be measured experimentally, there is a definite limit to access for an observer with macroscopic instruments. This limitation was first elucidated by Heisenberg through a Thought Experiment , and is represented mathematically by the Non-commutativity of quantum observables.

Prior to the emergence of quantum mechanics as a separate Theory , the mathematics used in physics consisted mainly of Differential Geometry and Partial Differential Equation s; Probability Theory was used in Statistical Mechanics . Geometric intuition clearly played a strong role in the first two and, accordingly, Theories Of Relativity were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called Classical Physics , and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld-Wilson-Ishiwara Quantization rule, which was formulated entirely on the classical phase space.


HISTORY OF THE FORMALISM



The "old quantum theory" and the need for new mathematics

Main article: Old Quantum Theory

In the decade of 1890, Planck was able to derive the Blackbody Spectrum and solve the classical Ultraviolet Catastrophe by making the unorthodox assumption that, in the interaction of Radiation with Matter , energy could only be exchanged in discrete units which he called Quanta . Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, ''h'' is now called Planck's Constant in his honour.

In 1905, Einstein explained certain features of the Photoelectric Effect by assuming that Planck's light quanta were actual particles, which he called Photons .
observations for hydrogen atoms ]]
In 1913, Bohr calculated the spectrum of the Hydrogen Atom with the help of A New Model Of The Atom in which the Electron could orbit the Proton only on a discrete set of classical orbits, determined by the condition that angular momentum was an integer multiple of Planck's constant. Electrons could make Quantum Leap s from one orbit to another, emitting or absorbing single quanta of light at the right frequency.

All of these developments were Phenomenological and flew in the face of the theoretical physics of the time. Bohr And Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. 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 Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld-Wilson-Ishiwara Quantization . Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body Problem ) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.

In 1923 De Broglie proposed that Wave-particle Duality applied not only to photons but to electrons and every other physical system.

The situation changed rapidly in the years 1925-1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger and Werner Heisenberg and the foundational work of John Von Neumann , Hermann Weyl and Paul Dirac , and it became possible to unify several different approaches in terms of a fresh set of ideas.


The "new quantum theory"


Erwin Schrödinger 's Wave Mechanics originally was the first successful attempt at replicating the observed quantization of atomic spectra with the help of a precise mathematical realization of de Broglie's wave-particle duality. Schrödinger proposed an Equation (now bearing his name) for the wave associated to an electron in an atom according to de Broglie, and explained energy quantization by the well-known fact that differential operators of the kind appearing in his equation had a discrete spectrum. However, Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the (squared amplitude of the) wavefunction of an Electron must 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 probabilistic interpretation of the (squared amplitude of the) wave function as the probability distribution of the position of a ''pointlike'' object. With hindsight, Schrödinger's Wave Function can be seen to be closely related to the classical Hamilton-Jacobi Equation .

Werner Heisenberg 's Matrix Mechanics formulation, introduced contemporaneously to Schrödinger's wave mechanics and based on algebras of infinite matrices, was certainly very radical in light of the mathematics of classical physics. In fact, at the time Linear Algebra was not generally known to physicists in its present form.

The reconciliation of the two approaches is generally associated to Paul Dirac , who wrote a lucid account in his 1930 classic ''Principles of Quantum mechanics''. In it, he introduced the Bra-ket Notation , together with an abstract formulation in terms of the Hilbert Space used in Functional Analysis , and showed that Schödinger's and Heisenberg's approaches were two different representations of the same theory. Dirac 's method is now called Canonical Quantization . The first complete mathematical formulation of this approach 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 book. It was developed in parallel with a new approach to the mathematical Spectral Theory based on Linear Operator s rather than the Quadratic Form s that were David Hilbert 's approach a generation earlier.

Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John Von Neumann . 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.


Later developments


The application of the new quantum theory to electromagnetism resulted in Quantum Field Theory , which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the one presented here is a simple special case. In fact, the difficulties involved in implementing any of the following formulations cannot be said yet to have been solved in a satisfactory fashion except for ordinary quantum mechanics.

On a different front, von Neumann originally dispatched Quantum Measurement with his infamous postulate on the Collapse Of The Wavefunction , raising a host of philosophical problems. Over the intervening 70 years, the ''problem of measurement'' became an active research area and itself spawned some new formulations of quantum mechanics.

A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called Classical Limit Of Quantum Mechanics . Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, Quantization , namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.

Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called Hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of Quantum Optics .


MATHEMATICAL STRUCTURE OF QUANTUM MECHANICS


A physical system is generally described by three basic ingredients: phase space, observables are real-valued functions on it, time evolution is given by a one-parameter Group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of a Hilbert Space of states, observables are Self Adjoint Operator s on the space of states, time evolution is given by a One-parameter Group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.


Postulates of quantum mechanics


The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates.
  • Each physical system is associated with a Separable Complex Hilbert Space ''H'' with inner product \langle\phi\mid\psi angle. Rays (one-dimensional subspaces) in ''H'' are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in ''H'', where two vectors represent the same state if they differ only by a phase factor. ''Separability'' is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.

  • The Hilbert space of a composite system is the Hilbert space Tensor Product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

  • Physical symmetries act on the Hilbert space of quantum states Unitarily or Antiunitarily ( Supersymmetry is another matter entirely).

  • Physical Observable s are represented by densely-defined Self-adjoint Operator s on ''H''.

  If <math>\left\psi\left(t Ight) Ight Angle</math> Denotes The State Of The System At Any One Time ''t'', The Following "http://wwwinformationdelightinfo/encyclopedia/entry/Schrödinger_equation" class="copylinks">Schrödinger Equation holds:
  :<math> I\hbar Rac{d}{d T}\left\psi(t) Ight Angle H\left\psi(t) ight angle</math>
  :<math>\left\psi(t+s) Ight Angle U(t)\left\psi(s) ight angle</math>
  :<math>\left\psi Ight Angle \left\psi(0) ight angle</math>
  :<math> I\hbar Rac{d }{dt}\left\psi(t) Ight Angle \operatorname{H_0}\left\psi(t) ight angle</math>
  For Example, Suppose The State Space Is <math>\mathbb{C}^n</math>, And ''A'' Is A Hermitian Matrix With Eigenvalues <math> \lambda I </math>, With Corresponding Eigenvectors <math> \psi I Angle </math> The Projection Valued Measure Associated With A, Call It <math>P A</math> Is Now Then <math> P A (B I) \psi_i angle \langle \psi_i </math>, for any Borel set <math> B_i </math> containing only <math> \lambda_i </math> and no other <math> \lambda_j </math>'s Assume the system is prepared in state <math> \psi angle </math> The probability of a measurement returning the value <math>\lambda _i</math> can be calculated by integrating the spectral measure <math> \langle \psi, P_A \psi angle </math> over <math>B_i</math>, which gives trivially
  <math> \langle \psi \psi I Angle \langle \psi I \psi Angle \langle \psi \psi_i angle ^2 </math>
  A More General Formulation Replaces The Projection Valued Measure With A "http://wwwinformationdelightinfo/encyclopedia/entry/POVM" class="copylinks">Positive-operator Valued Measure(POVM) To illustrate, take again the finite dimensional case Here we would replace the rank-1 projections <math> \psi_i angle \langle \psi_i </math> by a finite set of positive operators <math> F_i F_i^ </math>, whose sum is still the identity operator as before Just like a set of possible outcomes <math>