Mathematical Formulation Of Quantum Mechanics History Of The

This formulation of quantum mechanics 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 mathematical 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 equationIn mathematics, and in particular calculus, a partial differential equation PDE is an equation involving partial derivatives of an unknown function. The idea is to describe a function indirectly by a relation between itself and its partial derivatives, ras; probability theoryProbability theory Discrete mathematics Mathematical analysis Probability theory is the mathematical study of probability. Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failure to occ was used in statistical mechanicsStatistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. It provides a fr. 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 physicsClassical physics is physics excluding quantum theory and occasionally excluding relativity as well. Roughly taken, the scale of classical physics is the level of isolated atoms and molecules on upwards, including the macroscopic and astronomical realm., 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.

1 History of the formalism

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

Main article: Old quantum theory

In the decade of 1890, PlanckMax Karl Ernst Ludwig Planck ( April 23, 1858 October 4, 1947) was a German physicist who is considered to be the inventor of quantum theory. Born in Kiel, Planck started his physics studies at Munich University in 1874, graduating in 1879 in Berlin. 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.

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 leaps 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 Schrödinger and Heisenberg and the foundational work of von Neumann, Weyl and Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas.

1.2 The "new quantum theory"

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 (amplitude squared) 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 (amplitude squared) 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. 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 physicist in its present form.

The reconciliation of the two approaches is generally associated to 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. The first complete mathematical formulation of this approach is generally credited to von Neumann's 1932 book Mathematical Foundations of Quantum Mechanics, although 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 operators rather than the quadratic forms 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 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.

1.3 Later developments

The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. The difficulties involved cannot be said yet to have been solved in as satisfactory a fashion. See:

quantum field theory
Feynman path integrals
axiomatic, algebraic and constructive quantum field theory
quantum field theory on curved spacetime
C* algebra formalism

2 Mathematical structure of quantum mechanics

A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries . A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic 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 operators 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.

The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's axioms.

Each system is associated with a separable complex Hilbert space H with inner product . Rays (one-dimensional subspaces) in H are associated with states of the physical system. These states can be identified to 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 detemine 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. If this action is not irreducible, H decomposes as a direct sum of irreducible representations of the physical symmetries. Each direct summand will be a so-called superselection sector. In brief, states in different superselection sectors cannot influence each other, and the relative phases between them are unobservable.
Physical observables are represented by densely-defined self-adjoint operators on H.

The expected value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector is

By spectral theory, we can associate a probability distribution to the values of A in any state ψ. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. In the special case A has discrete spectrum, the possible values of A in any state are its eigenvalues.

More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator normalized to be of trace 1. The expected value of A in the state is

If is the orthogonal projector onto the one-dimensional subspace of H spanned by , then

Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and density operators mixed states.

One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.

2.1 Pictures of dynamics

In the so-called Schrödinger picture of quantum mechanics, the dynamics is given as follows: if denotes the state of the system at any one time t, the following Schrödinger equation holds:

where H is a densely-defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.

Alternatively, one can state that there is a continuous one-parameter unitary group U(t): H → H such that

for all times s, t. The existence of a self-adjoint Hamiltonian H such that

is a consequence of Stone's theorem on one-parameter unitary groups.

The Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus:

It is then easily checked that the expected values of all observables are the same in both pictures

and that the time-dependent Heisenberg operators satisfy

The so-called Dirac picture or interaction picture has time-dependent states and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the states can be solved exactly, confining any complications to the evolution of the operators. For this reason, the Hamiltonian for states is called "free Hamiltonian" and the Hamiltonian for observables is called "interaction Hamiltonian". In symbols:

The Heisenberg picture is the closest to classical mechanics, but the Schrödinger picture is universally considered easiest to understand, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory.

3 Other formulations

3.1 C*-algebraic formulation

This approach is related to the heisenberg formulation, and the mathematical theory of C*-algebras has its roots in the work of von Neumann on quantum mechanics.

In this formulation, the basic structure describing a quantum system is a C*-algebra, whose intended interpretation is the associative algebra of bounded observables of the system. A state in this formulation is a complex linear functional f such that f(x* x) is non-negative for any observable x and f(1)=1. States in this formulation generalize density matrices in the von Neumann formulation.

Given a state, we can construct a unitary representation of it using the Gelfand-Naimark-Segal construction. Two unitarily inequivalent representations are said to belong to different superselection sectors. Relative phase s between superselection sectors are not observable.

3.2 Quantum logic formulation

There is an approach similar to the C*-algebraic formulation, but which instead of an algebra of observables, has as a starting point an orthocomplemented lattice of yes-no questions regarding a quantum mechanical system. The formal rules which govern this lattice can be viewed as a quantum analogue of propositional logic. This approach, referred to as quantum logic, was originated by J. von Neumann and G. Birkhoff and further pursued by G. Mackey. Although it is to some extent superseded by the C*-algebraic formulation, it is more suitable for considering foundational issues such as measurement and decoherence.

4 The problem of measurement

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is the effects of measurement.

4.1 Wavefunction collapse

When von Neumann proposes his mathematical postulate for quantum mechanics he included the following "measurement postulate".

Carrying out a measurement of an observable A with discrete spectrum on a system in the state represented by will cause the system state to collapse into an eigenstate (i.e. eigenvector), of the operator; the observed value corresponds to the eigenvalue a of the eigenstate:

Mathematically, the collapse corresponds to orthogonal projection of onto the eigenspace of A corresponding to the abserved value a

Von Neumann's postulate regarding the effects of measurement has always been a source of confusion and speculation. Fortunately, there is a general mathematical theory of such irreversible operations (see quantum operation) and various physical interpretations of the mathematics.

4.2 The relative state interpretation

An alternative interpretation of measurement is Everett's relative state interpretation, which was later dubbed " many-worlds interpretation" of quantum mechanics.

5 List of mathematical tools

Part of the folklore of the subject concerns the mathematical physics textbook Courant-Hilbert, put together by Richard Courant from David Hilbert's Göttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, up to the Schrödinger equation's advent. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg consulted Hilbert about his matrix mechanics, and Hilbert had observed that his own experience with infinite-dimensional matrices had derived from differential equations; advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of theory was conventional at the time, where the physics was radically new.

The main tools included:

Quantum mechanics

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