3.4.6  Mathematical Formulation

In addition to the fundamental works of Bohr, Heisenberg, Schroedinger, and other scientists in the 1920s, the mathematical formulation of Quantum Mechanics was developed further by various famous physicists, such as Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl. Heisenberg’s matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra, followed by Schroedinger’s wave mechanics, although he himself initially did not understand its fundamental probabilistic nature, until Max Born introduced the interpretation of the absolute square of the wave-function as the probability distribution of the position of a quantum object.

Schroedinger’s wave-function can be seen to be closely related to the classical Hamilton-Jacobi equation. The correspondence to classical mechanics was even more explicit in Heisenberg’s matrix mechanics. Paul Dirac then discovered that the equation for the operators in the Heisenberg representation 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 Schroedinger himself 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 Dirac (1981).

Dirac also discovered a relativistic generalization of the quantum theory, and he introduced the bra-ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis. He showed that Schroedinger’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. His work was particularly fruitful in all kinds of generalizations of the field, as we shall discuss further when we review his contribution in the Quantum Field Theory in section 5.

In the formalism of Quantum Mechanics, the state of a system at a given time is described by a complex wave-function, also referred to as state vector in a complex vector space. This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, to arbitrary precision.

When Quantum Mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

Early attempts to merge Quantum Mechanics with Special Relativity involved the replacement of the Schroedinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation, described in section 5.2. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of Quantum Field Theory, which applies quantization to a field, rather than a fixed set of particles. The first complete Quantum Field Theory, called Quantum Electrodynamics, described in section 5.3, provides a fully quantum description of the electromagnetic interaction. The full apparatus of Quantum Field Theory is often unnecessary for describing electrodynamic systems.