Technopedia

The technologist’s encyclopedia

Applications

Applications is where the machinery of the other three pages pays off. The same Hilbert-space and operator language that abstracts Euclidean geometry is exactly the language of quantum mechanics, and the same weak/compact-operator ideas give existence theorems for partial differential equations. This page covers the two flagship applications: quantum mechanics (states, observables, time evolution) and PDEs (weak solutions and existence by the direct method).

Authoritative sources

Quantum mechanics

Definition. In quantum mechanics the state of a system is a unit vector (equivalently a ray) in a Hilbert space H\mathcal{H}; an observable is a self-adjoint operator on H\mathcal{H}; and the possible measured values of an observable are its spectrum.

Details

States and expectation values

A pure state is a unit vector ψ∈H\psi \in \mathcal{H} (vectors differing by a phase represent the same physical state, so really a “ray”). If AA is an observable, the expected value of measuring AA in state ψ\psi is the number ⟨Aψ,ψ⟩\langle A\psi, \psi \rangle. Because AA is self-adjoint, this number is real — see Operators.

Example. A spin-1/21/2 particle has state space H=C2\mathcal{H} = \mathbb{C}^2. The spin in the zz-direction is the observable Sz=ℏ2(100−1)S_z = \frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, whose spectrum {ℏ/2,−ℏ/2}\{\hbar/2, -\hbar/2\} is exactly the two outcomes of a Stern–Gerlach measurement.

Time evolution is a one-parameter unitary group

The time evolution of a state with Hamiltonian HH (a self-adjoint operator) is given by the Schrödinger equation

iℏ ddtψ(t)=H ψ(t),ψ(t)=e−iHt/ℏ ψ(0).i\hbar\, \frac{d}{dt}\psi(t) = H\,\psi(t), \qquad \psi(t) = e^{-iHt/\hbar}\,\psi(0).

The family U(t)=e−iHt/ℏU(t) = e^{-iHt/\hbar} is a one-parameter group of unitary operators — this is Stone’s theorem in action, and it is why probability is conserved. The operator e−iHt/ℏe^{-iHt/\hbar} is defined by the functional calculus of the spectral theorem.

psi (state) eigenvector of A
Example problem

Problem. Let AA be a self-adjoint operator and ψ\psi a unit vector. Show that the expectation value ⟨Aψ,ψ⟩\langle A\psi, \psi \rangle is real.

Solution

⟨Aψ,ψ⟩=⟨ψ,A∗ψ⟩=⟨ψ,Aψ⟩=⟨Aψ,ψ⟩‾\langle A\psi, \psi \rangle = \langle \psi, A^* \psi \rangle = \langle \psi, A \psi \rangle = \overline{\langle A\psi, \psi \rangle}, using A=A∗A = A^*. A complex number equal to its own conjugate is real. ∎

PDEs and weak solutions

Definition. A weak solution of a PDE is a function in a Sobolev space that satisfies the equation in the distributional sense (after integrating by parts to move derivatives onto smooth test functions). Existence is often proved by the direct method: a minimizing sequence has a weakly convergent subsequence whose limit is the solution.

Details

Why weak solutions

Classical (smooth) solutions to PDEs often do not exist, especially for nonsmooth data or nonlinear equations. Weak solutions relax the requirement: instead of the equation holding pointwise, it holds after pairing with test functions. This is exactly the language of Distributions — a weak solution is a function whose weak derivatives satisfy the PDE.

Sobolev spaces. The natural home for weak solutions is a Sobolev space H1H^1, the space of L2L^2 functions whose weak first derivatives are also in L2L^2. These are Banach (indeed Hilbert) spaces, so all the machinery of Spaces applies.

The direct method of the calculus of variations

To solve −Δu=f-\Delta u = f with zero boundary data, one minimizes the energy

J(v)=12∫∣∇v∣2−∫f vJ(v) = \frac{1}{2}\int |\nabla v|^2 - \int f\,v

over the affine space of admissible functions. The argument has three steps, each a theorem from the other pages:

  1. Boundedness: a minimizing sequence is bounded in H1H^1.
  2. Weak compactness: because H1H^1 is reflexive, a subsequence converges weakly to some uu (see Spaces).
  3. Lower semicontinuity: the energy satisfies J(u)≤lim inf⁡J(un)J(u) \leq \liminf J(u_n), so uu is the minimizer, and the Euler–Lagrange equation gives −Δu=f-\Delta u = f in the weak sense.
u (weak limit) bounded minimizing sequence -> weakly convergent subsequence
Example problem

Problem. Explain why the Poisson equation −Δu=f-\Delta u = f on a bounded domain with f∈L2f \in L^2 and zero boundary conditions has a weak solution in H01H_0^1.

Solution

Minimize J(v)=12∫∣∇v∣2−∫fvJ(v) = \frac{1}{2}\int |\nabla v|^2 - \int f v over H01H_0^1. By the Poincaré inequality, JJ is coercive and bounded below, so a minimizing sequence (vn)(v_n) is bounded in H01H_0^1. Since H01H_0^1 is a Hilbert (hence reflexive) space, a subsequence converges weakly to some u∈H01u \in H_0^1. The norm v↦∫∣∇v∣2v \mapsto \int |\nabla v|^2 is convex and continuous, hence weakly lower semicontinuous, so uu minimizes JJ. Taking the first variation of JJ at uu gives ∫∇u⋅∇φ=∫fφ\int \nabla u \cdot \nabla \varphi = \int f \varphi for all test functions φ\varphi, which is precisely the weak form of −Δu=f-\Delta u = f. ∎

Quiz

Example problem

Problem. In quantum mechanics, show that if HH is a self-adjoint Hamiltonian, then the time-evolution operator U(t)=e−iHt/ℏU(t) = e^{-iHt/\hbar} is unitary for every real tt.

Solution

By the functional calculus, U(t)∗=eiHt/ℏU(t)^* = e^{iHt/\hbar} (the adjoint conjugates the exponent). Then U(t)∗U(t)=eiHt/ℏe−iHt/ℏ=e0=IU(t)^* U(t) = e^{iHt/\hbar} e^{-iHt/\hbar} = e^{0} = I, since the two exponents are functions of the same self-adjoint operator HH and hence commute. So U(t)U(t) is unitary, which is why it preserves the norm (and hence total probability) of a state. ∎