Mathematical formulation of quantum mechanics

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One of the remarkable facts of quantum mechanics in its current formulations is its abstractness. Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of differential geometry and partial differential equations and to a lesser extent, probability theory. The first two clearly had a strong visual flavor. Even theories of relativity were still formulated in terms of spatial concepts. During the first 10 to 15 years after the emergence of quantum theory (up to about 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.

Around 1925 that situation changed radically with the appearance of Schrödinger's wave mechanics and Heisenberg's matrix mechanics. Heisenberg's formulation, based on algebras of infinite matrices was certainly very radical in light of the mathematics of classical physics; Schrödinger's also had nonconvential ingredients, particularly in its probabilistic view of the main concepts of the theory. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which generally underlies all approaches. 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.

This framework can be regarded as an identification of the bra-ket notation of Dirac, with the abstract notion of Hilbert space used in functional analysis. The first complete formulation of this approach is attributed to von Neumann, although Weyl in his 1927 classic book already referred to Hilbert spaces (which he called unitary spaces).

Table of contents

1 Basic structure of quantum mechanics

1.1 Other formulations

2 C*-algebraic formulation
3 Quantum logic formulation
4 List of mathematical tools

Basic structure of quantum mechanics

A quantum mechanical system is described by three basic ingredients: states, observables and dynamics. For classical systems these ingredients can be described in fairly direct ways by a phase space model of mechanics: states are points in phase space, observables are real-valued functions on phase space and the dynamics is given by a one-parameter group of transformations of the phase space. To describe these ingredients for a quantum system in the so-called Schrödinger picture of quantum mechanics, we first postulate that such a system is associated with a separable Hilbert space H. Moreover,

Each possible quantum state is represented by a unit vector of H which is unique up to phase.
Any physical observable is represented by a densely-defined self-adjoint operator on H. In quantum physics the association between the value of an observable and the system state is much less direct than in classical mechanics. In fact the only physically meaningful structure associated to a state and an observable is a probability distribution of real values.
The dynamics is given as follows: If denotes the state ket of the system at any one time t, the following Schrödinger equation holds:

where H is a densely-defined self-adjoint (partial differential) operator, called the system Hamiltonian, i is the root of negative unity and is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.

Now this picture is sufficient for description of a completely isolated system. However, it still fails to account for one of the main differences between quantum mechanics and classical mechanics, which is how to account for effects of measurement. In this article we give a somewhat limited description of this process, that is measurement of observables A which have a complete set of eigenvectors:

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

Since there is generally more than one eigenstate for the particular observable, , it can collapse into any one of the set of eigenstates, given by . The probability that a system represented by collapses into eigenstate is given by BornÒs statistical interpretation:

Finally we need some notion of how a the description of a composite system is related to that of its components:

The Hilbert space of a composite system is the Hilbert space tensor product of those 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.

Other formulations

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. In this approach, both continuous and discrete observables may be accommodated; in the former case, the Hilbert space is a space of square-integrable wavefunctions. This approach is close to the approaches based on C*-algebras.

In both the Schrödinger and the Heisenberg framework, one can formulate and prove the uncertainty principle, although the exact sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.

The postulate regarding the effects of measurement has always been a source of confusion and spurious speculation. Fortunately, there is a general mathematical theory of such irreversible operations (see quantum operation) and various physical interpretations of the mathematics. One of the more commonly accepted interpretations is the relative state interpretation which is equivalent to the Everett many-worlds interpretation of quantum mechanics.

C*-algebraic formulation

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 phases between superselection sectors are not observable.

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.