The 6 Formulations of Quantum Mechanics

[Note: This page is in dire need of a re-write, and should really only be considered as a placeholder. The list below doesn’t even mention C*-algebraic QM, or von Neumann’s AQM. Distinctions really should be made between “formulations,” “representations,” and “pictures”. On top of all of that, “interpretations” must be taken into account….]
  1. Matrix Mechanics
    Formulated in 1925 by  Werner Heisenberg, Max Born, and Pascual Jordan (and later extended by P.A.M. Dirac), matrix mechanics was the first mathematically rigorous formulation of QM, treating the physical properties of particles (observables) as matrices (operators) evolving over time.

    • W. Heisenberg, Über quantentheoretische Umdeutung kinematischer und mechanischer BeziehungenZeitschrift für Physik33, 879-893, 1925 (received July 29, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations).]
    • M. Born and P. Jordan, Zur QuantenmechanikZeitschrift für Physik34, 858-888, 1925 (received September 27, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: On Quantum Mechanics).]
    • M. Born, W. Heisenberg, and P. Jordan, Zur Quantenmechanik IIZeitschrift für Physik35, 557-615, 1926 (received November 16, 1925). [English translation in: B. L. van der Waerden, editor, Sources of Quantum Mechanics (Dover Publications, 1968) ISBN 0-486-61881-1 (English title: On Quantum Mechanics II).]
  2. The Schrödinger Picture (wave mechanics)
    In the Schrödinger Picture, the state of a system evolves over time, with this evolution represented by a unitary operator known as the time evolution operator. Alternatively stated, the state vector of the system moves through its Hilbert state space, while the reference frame in which the Hilbert space is observed remains fixed (active transformation). It is via this formulation that undergraduate students are generally first introduced to quantum mechanics.
  3. The Heisenberg Picture
    In the Heisenberg Pictrue, the state vectors of a system are time-independent, but the operators corresponding to observables have a time-dependance. The Heisenberg Picture is simply a formulation of matrix mechanics with an arbitrary basis. Alternatively, the state vector remains fixed in Hilbert space, whereas the reference frame is rotated via the time propagator (passive transformation).
  4. The Interaction Picture (“Dirac Picture”)
    The Interaction Picture is a hybrid of the Schrödinger Picture and the Heisenberg picture in which both state vectors and operators carry part of the time dependence.
  5. Quantum Field Theory: Second Quantization Formulation
    In Second Quantization, fields are treated as operators upon which commutation rules are imposed, enabling the description of systems composed of varying numbers of particles.
  6. Quantum Field Theory: Path Integral Formulation (“sum-over-histories”)
    The Path Integral Formulation, based upon the Calculus of Variations, is a generalization of the Principle of Least Action found in classical Lagrangian mechanics to quantum theory. In this formulation, the integration of the Lagrangian over all possible paths forms the action. The actual evolution of the system is one which minimizes this action. Early efforts at this formulation were stymied by divergences, an issue which was solved by Richard Feynman’s introduction of renormalization. This, combined with the introduction of Feynman diagrams as a calculation aid, made possible quantum electrodynamicsquantum chromodynamics, and electroweak theory as we now know them. Considered by many physicists to be in some fashion more fundamental than the other formulations of quantum mechanics, this approach is the workhorse of physicists working with the Standard Modeland beyond. That having been said, it is not generally introduced to students until well into graduate school.


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