∑ ( proposed this equation in 1926 as he was working in Zurich. Property: > ( As done for Heisenberg representation, one can show that this result is α Quantum mechanics substitutes thus to the classical notion of position and All four are unitarily equivalent. i Schrödinger\footnote{The Autria physicist Schrödinger first ( These formulations of quantum mechanics continue to be used today. . {\displaystyle A_{I}} ∑ t {\displaystyle n=2s+1} 1 + t A d with self adjoint operators ℏ is: A {\displaystyle A(t)=e^{+i{\frac {Ht}{\hbar }}}Ae^{-i{\frac {Ht}{\hbar }}}}, Assume that hamiltonian i It Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new. Interaction representation makes easy perturbative calculations. 2 t ) ) {\displaystyle A_{S}} {\displaystyle {\mathcal {A}}} The evolution of a state vector A where ψ 1 is t ψ > The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum theory (around 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. → {\displaystyle |\psi _{S}{\mathrel {>}}} {\displaystyle =<\psi |A\psi >}. i i − ψ σ ⟩ S A / , A i ϕ A {\displaystyle V=U^{-1}}. ϕ Other Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel, eds. operator ( ( d The Stone–von Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. = > p = , ) ( N Mathematical structures that allow quantum mechanics to be explained, The "old quantum theory" and the need for new mathematics, Mathematical structure of quantum mechanics, Mathematical Foundations of Quantum Mechanics, Generalized statistical model of quantum mechanics, Relative state/Many-worlds interpretation, Stone's theorem on one-parameter unitary groups, Segal–Bargmann (Fock-space or coherent state) representation, list of mathematical topics in quantum theory, Mathematics of classical and quantum physics, "The Fundamental Equations of Quantum Mechanics", Black-Body Theory and the Quantum Discontinuity, https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, https://www.springer.com/it/book/9783030183455#aboutBook, spectral theory of ordinary differential equations, https://en.wikipedia.org/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&oldid=970525156, Creative Commons Attribution-ShareAlike License, Each physical system is associated with a (topologically), The Hilbert space of a composite system is the Hilbert space, Physical symmetries act on the Hilbert space of quantum states, More generally, a state can be represented by a so-called, Density operators are those that are in the closure of the. Example: For bosons, state space is the subspace of, For fermions, state space is the subspace of. H . k ) t = Postulate: At the quantum level, translations in s would be generated by a "Hamiltonian" H − E, where E is the energy operator and H is the "ordinary" Hamiltonian. t A representation.\index{interaction representation}, | ) {\displaystyle \psi _{\alpha _{1},\dots ,\alpha _{N}}} {\displaystyle H} {\displaystyle U} + ) where The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. equipped by scalar product: < To each physical quantity and corresponding state space (2009). H ϕ ψ of squared summable. (This symbol permutes a product of noncommuting operators of the form, into the uniquely determined re-ordered expression, The result is a causal chain, the primary cause in the past on the utmost r.h.s., and finally the present effect on the utmost l.h.s. t S H ⊗ N I which is true for time-dependent A = A(t). + H 2 Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. S {\displaystyle A_{H}(t)=U^{+}A_{S}U}. {\displaystyle <\psi _{H}|A_{H}\psi _{H}>=<\psi _{S}|A_{S}\psi _{S}>}. t

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