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.
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 spacemodel 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 separableHilbert spaceH. Moreover,
Any physical observable is represented by a densely-defined self-adjointoperator 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:
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 Born’s 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.
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.
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.