Mathematical Formulation of Quantum Mechanics - List of Mathematical Tools

List of Mathematical Tools

Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics 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, until the advent of Schrödinger's equation. 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 had consulted Hilbert about his matrix mechanics, and Hilbert 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 the theory was conventional at the time, whereas the physics was radically new.

The main tools include:

  • linear algebra: complex numbers, eigenvectors, eigenvalues
  • functional analysis: Hilbert spaces, linear operators, spectral theory
  • differential equations: partial differential equations, separation of variables, ordinary differential equations, Sturm–Liouville theory, eigenfunctions
  • harmonic analysis: Fourier transforms
See also: list of mathematical topics in quantum theory

Read more about this topic:  Mathematical Formulation Of Quantum Mechanics

Famous quotes containing the words list of, list, mathematical and/or tools:

    Religious literature has eminent examples, and if we run over our private list of poets, critics, philanthropists and philosophers, we shall find them infected with this dropsy and elephantiasis, which we ought to have tapped.
    Ralph Waldo Emerson (1803–1882)

    Shea—they call him Scholar Jack—
    Went down the list of the dead.
    Officers, seamen, gunners, marines,
    The crews of the gig and yawl,
    The bearded man and the lad in his teens,
    Carpenters, coal-passers—all.
    Joseph I. C. Clarke (1846–1925)

    As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.
    Blaise Pascal (1623–1662)

    The besetting sin of able men is impatience of contradiction and of criticism. Even those who do their best to resist the temptation, yield to it almost unconsciously and become the tools of toadies and flatterers. “Authorities,” “disciples,” and “schools” are the curse of science and do more to interfere with the work of the scientific spirit than all its enemies.
    Thomas Henry Huxley (1825–95)