Quantum theory

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• ConceptSchrödinger equation

  The Schrödinger equation $i\hbar\, \partial_t \psi = -\frac{\hbar^2}{2m} \Delta \psi + V \psi$ describes how the wave function $\psi$ of a quantum particle of mass $m$ in a potential $V$ evolves in time. It is a linear dispersive partial differential equation with complex-valued solutions. Its right-hand side is a Schrödinger operator applied to $\psi$. The solutions are given by a unitary group of operators, so the total probability $\int \abs{\psi}^2 \, dx$ is conserved. It is the basic equation of non-relativistic quantum mechanics and a model problem for dispersive equations.

• ConceptWKB approximation

  The WKB approximation gives approximate solutions of linear differential equations whose highest derivative is multiplied by a small parameter, such as $\varepsilon^2 y'' = Q(x) y$. It writes the solution as the exponential of a series in $\varepsilon$ and keeps the first terms. It is also called the Liouville–Green method, and the name WKB refers to Wentzel, Kramers and Brillouin. It is the basis of semiclassical approximations in quantum mechanics and of the asymptotics of special functions.

• ConjectureQuantum unique ergodicity conjecture

  Let $M$ be a compact connected Riemannian manifold without boundary and with negative sectional curvature, with volume measure $dv$ and total volume $V$. Let $(\varphi_j)$ be any sequence of $L^2$-normalised eigenfunctions of the Laplace–Beltrami operator with eigenvalues $\lambda_j \to \infty$. The conjecture says that for every continuous function $a$ on $M$ $$\lim_{j \to \infty} \int_M a \, \abs{\varphi_j}^2 \, dv = \frac{1}{V} \int_M a \, dv.$$ In its full form it says that the microlocal lifts of the measures $\abs{\varphi_j}^2 \, dv$ converge to the normalised Liouville measure on the unit cotangent bundle.

• ObjectFock space

  Let $H$ be a complex Hilbert space. For $n \ge 1$, let $H^{\otimes n}$ be the tensor product of $n$ copies of $H$, and let $H^{\otimes 0} = \C$. The full Fock space over $H$ is the Hilbert sum $$\mathcal{F}(H) = \widehat{\bigoplus}_{n \ge 0} H^{\otimes n}.$$ Let $S_+$ and $S_-$ be the orthogonal projections of $H^{\otimes n}$ onto its symmetric and antisymmetric tensors. The bosonic Fock space is $\mathcal{F}_+(H) = \widehat{\bigoplus}_{n \ge 0} S_+ H^{\otimes n}$, and the fermionic Fock space is $\mathcal{F}_-(H) = \widehat{\bigoplus}_{n \ge 0} S_- H^{\otimes n}$. Each of them is a Hilbert space. Indeed, each summand is a closed subspace of $H^{\otimes n}$, and a Hilbert sum of Hilbert spaces is a Hilbert space.

• TheoremStone–von Neumann theorem

  Let $n \ge 1$ and let $H$ be a Hilbert space. Let $U$ and $V$ be strongly continuous unitary representations of $\R^n$ on $H$ that satisfy the Weyl relations $$U(s) V(t) = e^{i s \cdot t} \, V(t) U(s) \qquad (s, t \in \R^n).$$ Then $H$ is an orthogonal direct sum of closed subspaces, each invariant under all $U(s)$ and $V(t)$. On each of them, the pair $(U, V)$ is unitarily equivalent to the Schrödinger pair on $L^2(\R^n)$: $$(U_0(s) \psi)(x) = \psi(x + s), \qquad (V_0(t) \psi)(x) = e^{i t \cdot x} \psi(x).$$ In particular, every irreducible pair satisfying the Weyl relations is unitarily equivalent to $(U_0, V_0)$.

• ConceptDeformation quantization

  Deformation quantization equips the smooth functions on a Poisson manifold with a noncommutative star product. The star product is a formal power series in a parameter $\hbar$. The commutator of the star product reproduces the Poisson bracket at first order in $\hbar$. The Moyal product on $\R^{2n}$ is the first example. Kontsevich's formality theorem shows that every Poisson manifold admits a star product.