Mathematics · Advanced Topics

Green's Functions

The Green's function is the response of a linear system to a point source — an impulse. Once known, the response to any source is a superposition. Green's functions unify potential theory, wave propagation, heat conduction, and quantum propagators.

PrerequisitesDifferentialequations(Ch.DE)⋅Complexanalysis(Ch.CA)⋅Fourieranalysis(Ch.F)⋅EDifferential equations (Ch. DE) \cdot Complex analysis (Ch. CA) \cdot Fourier analysis (Ch. F) \cdot Electrostatics (Ch. ES)
Learning Goals
  • DefinetheGreen′sfunctionGviaLG=δ(r−r′)andusesuperpositiontowritethegeneraDefine the Green's function G via LG = \delta(r - r') and use superposition to write the general solution as a convolution with the source.
  • Derivethe3Dfree−spaceLaplacianGreen′sfunctionG=1/(4π∣r−r′∣)andconnectittoCDerive the 3D free-space Laplacian Green's function G = 1/(4\pi|r - r'|) and connect it to Coulomb's law and the method of images.
  • WritetheoutgoingHelmholtzGreen′sfunctionG+=eik∣r−r′∣/(4π∣r−r′∣)anduseitsfarWrite the outgoing Helmholtz Green's function G^{+} = e^{ik|r-r'|}/(4\pi|r-r'|) and use its far-field expansion to derive the Born scattering amplitude.
  • Find the retarded Green's function for a driven harmonicoscillatorandexpressthesolutionasaconvolutionofthedrivingforcewithsin(ωmonic oscillator and express the solution as a convolution of the driving force with sin(\omega
  • Statethespectral(Lehmann)representationG+(E)=∑∣n⟩⟨n∣/(E−En+iε)andidentifypolState the spectral (Lehmann) representation G^{+}(E) = \sum|n⟩⟨n|/(E - E_{n} + i\varepsilon) and identify poles as energy levels and Im G as the density of states.

GF.1 Motivation — The Impulse Response

Consider a linear differential operator L acting on functions of position (or time). The Green's functionG(r, r') is defined by:

LG(r,r′)=δ(r−r′)(definingequationfortheGreen′sfunction)L G(r, r') = \delta(r - r') \qquad (defining equation for the Green's function)(GF.1)

Once G is known, the solution to L u = f(r) with appropriate boundary conditions is:

u(r)=∫G(r,r′)f(r′)d3r′(superpositionprinciple)u(r) = \int G(r, r') f(r') d^{3}r' \qquad (superposition principle)(GF.2)

Physical interpretation: f(r) is a source distribution. G(r, r') is the response at r due to a point source at r'. The total response is the sum (integral) over all sources. This is the mathematical expression of the superposition principle for linear systems.

Reciprocity:for self-adjoint operators, G(r, r') = G(r', r) — the response at r due to a source at r' equals the response at r' due to a source at r.

GF.2 Poisson's Equation — Electrostatics

The electrostatic potential satisfies ∇²φ = −ρ/ε₀. The Green's function for the Laplacian in 3D free space (L = −∇²):

G(r,r′)=1/(4π∣r−r′∣)(free−spaceGreen′sfunction,3D)G(r, r') = 1/(4\pi|r - r'|) \qquad (free-space Green's function, 3D)(GF.3)

This is just Coulomb's law in disguise. The potential from an arbitrary charge distribution:

ϕ(r)=(1/4πε0)∫ρ(r′)/(∣r−r′∣)d3r′(Coulombintegral)\phi(r) = (1/4\pi\varepsilon_{0}) \int \rho(r')/(|r - r'|) d^{3}r' \qquad (Coulomb integral)(GF.4)

For problems with boundaries, G must satisfy the boundary conditions. For a point charge q above a grounded plane (z = 0): the image charge at the mirror position gives the correct G with G = 0 on the plane — method of images is just finding the boundary-corrected Green's function.

In 2D: G = −(1/2π) ln|r − r'|. In 1D: G = −|x − x'|/2. Note that dimensionality changes the form of G.

GF.3 The Helmholtz Equation

The Helmholtz equation (∇² + k²)u = −f arises in acoustics, EM waves, and quantum mechanics (time-independent Schrödinger). The outgoing Green's function (satisfying the Sommerfeld radiation condition — waves going out, not in):

G+(r,r′)=eik∣r−r′∣/(4π∣r−r′∣)(retarded/outgoingHelmholtzGreen′sfunction)G+(r, r') = e^{ik|r-r'|}/(4\pi|r-r'|) \qquad (retarded/outgoing Helmholtz Green's function)(GF.5)

At large distance r ≫ r': G+ ≈ (e^(ikr)/4πr) e^(−ik̂·r'). This factor e^(−ik̂·r') is precisely the far-field factor in the Born approximation for quantum scattering:

f(k,k′)=−(m/2πℏ2)∫e−ik′\cdotr′V(r′)ψ(r′)d3r′(Bornscatteringamplitude)f(k, k') = -(m/2\pi\hbar^{2}) \int e^{-ik'\cdotr'} V(r') \psi(r') d^{3}r' \qquad (Born scattering amplitude)(GF.6)

For weak potentials (Born approx): ψ ≈ e^(ik·r), giving f ∝ Ṽ(q) (the Fourier transform of V at momentum transfer q = k − k'). The Rutherford cross section follows from V = Ze²/r and its Fourier transform ∝ 1/q² → dσ/dΩ ∝ 1/sin⁴(θ/2).

Example GF.1 — Driven Harmonic Oscillator — Retarded Green's Function

FindtheGreen′sfunctionforx¨+ω02x=f(t)/m(drivenharmonicoscillator)satisfyingGFind the Green's function for ẍ + \omega_{0}^{2} x = f(t)/m (driven harmonic oscillator) satisfying G = 0 for t < t' (causality).

Define:G(t,t′)satisfiesG˙¨+ω02G=δ(t−t′)/m.Fort>t′:Gobeysthehomogeneousequation→G(t, t') satisfies Ġ̈ + \omega_{0}^{2} G = \delta(t-t')/m. For t > t': G obeys the homogeneous equation \to G=Asin(ω0(t−t′))+Bcos(ω0(t−t′G = A sin(\omega_{0}(t-t')) + B cos(\omega_{0}(t-t'
Boundary conditions:Causality(G=0,t<t′):Gmustvanishfort<t′.Jumpconditions:att=t′+,integratCausality (G = 0, t < t'): G must vanish for t < t'. Jump conditions: at t = t'+, integrateonce→G˙jumpsby1/m.SoG(t′+,t′)=0andG˙(t′+,t′)=1/me once \to Ġ jumps by 1/m. So G(t'+,t') = 0 and Ġ(t'+,t') = 1/m.
Solve:G(t′+)=Asin(0)+Bcos(0)=B=0.G˙(t′+)=Aω0cos(0)=Aω0=1/m→A=1/(mω0G(t'+) = A sin(0) + B cos(0) = B = 0. Ġ(t'+) = A\omega_{0} cos(0) = A\omega_{0} = 1/m \to A = 1/(m\omega_{0}.
Result:G(t,t′)=θ(t−t′)/(mω0)×sin(ω0(t−t′))whereθistheHeavisidestepfunctionG(t,t') = \theta(t-t')/(m\omega_{0}) \times sin(\omega_{0}(t-t')) where \theta is the Heaviside step function.
Solution:x(t)=\intG(t,t′)f(t′)dt′=∫(−∞tot)f(t′)/(mω0)sin(ω0(t−t′))dt′.Thisistheconvolutx(t) = \intG(t,t') f(t') dt' = \int(-\infty to t) f(t')/(m\omega_{0}) sin(\omega_{0}(t-t')) dt'. This is the convolutionformula—Ftimestheimpulseresponsesin(ω0t)/(mω0ion formula — F times the impulse response sin(\omega_{0}t)/(m\omega_{0}.

GF.4 Diffusion — Heat Equation Green's Function

For the heat equation ∂u/∂t − D∇²u = f(r,t), the causal Green's function is the Gaussian spreading kernel (already seen in diffusion):

G(r,t;r′,t′)=θ(t−t′)/(4\piD(t−t′))3/2×exp(−∣r−r′∣2/(4D(t−t′)))G(r, t; r', t') = \theta(t-t')/(4\piD(t-t'))^{3/2} \times exp(-|r-r'|^{2}/(4D(t-t')))(GF.7)

For an initial condition u(r, 0) = u₀(r) with no source f = 0:

u(r,t)=∫G(r,t;r′,0)u0(r′)d3r′(propagateinitialdata)u(r, t) = \int G(r, t; r', 0) u_{0}(r') d^{3}r' \qquad (propagate initial data)(GF.8)

This is the complete solution to the heat equation from any initial condition. The same Green's function appears in the path integral for quantum mechanics (imaginary time τ = it, D = ℏ/(2m)):

K(r,t;r′,0)=(m/(2\pii\hbart))3/2×exp(im∣r−r′∣2/(2\hbart))(free−particlepropagator)K(r, t; r', 0) = (m/(2\pii\hbart))^{3/2} \times exp(im|r-r'|^{2}/(2\hbart)) \qquad (free-particle propagator)(GF.9)

GF.5 Spectral Representation

For a self-adjoint operator L with eigenfunctions φₙ and eigenvalues λₙ (L φₙ = λₙ φₙ):

G(r,r′)=∑nϕn(r)ϕn∗(r′)/(λn−λ)(spectral/Lehmannrepresentation)G(r, r') = \sum_{n} \phi_{n}(r) \phi_{n}*(r') / (\lambda_{n} - \lambda) \qquad (spectral/Lehmann representation)(GF.10)

where λ is the eigenvalue of the source equation (Lu = λu + f, i.e., (L−λ)G = δ). This representation connects Green's functions to quantum mechanics: the retarded Green's function of the Schrödinger equation is:

G+(E)=∑n∣n⟩⟨n∣/(E−En+iε)(quantumretardedpropagator)G+(E) = \sum_{n} |n⟩⟨n| / (E - E_{n} + i\varepsilon) \qquad (quantum retarded propagator)(GF.11)

Poles of G+(E) at E = Eₙ − iε give the energy levels. The imaginary part (spectral function) −(1/π) Im G+(E) = Σₙ |n⟩⟨n| δ(E − Eₙ) — the density of states. In condensed matter, the interacting Green's function's poles give quasiparticle energies; its spectral function is measured by ARPES (angle-resolved photoemission spectroscopy).

Definition GF.1 — Common Traps
  • A Green's function depends on boundary conditions: the same operator can have different Green's functions.
  • The delta function is a distribution: use it under integrals, not as an ordinary function.
  • Convolution solves linear problems: nonlinear equations do not superpose this way.
  • Retarded and advanced choices encode causality: choose the one matching the physical problem.
Exercises — GF.1–GF.5 Green's Functions
1.
FindtheGreen′sfunctionfor−d2G/dx2=δ(x−x′)on[0,L]withG(0)=G(L)=0.UseittoFind the Green's function for -d^{2}G/dx^{2} = \delta(x-x') on [0,L] with G(0) = G(L) = 0. Use it to solvetheequationwithuniformsourcef=1solve the equation with uniform source f = 1
Straightforward
2.FindtheretardedGreen′sfunctionforthe3Dwaveequation(∂2/\partialt2−c2∇2)G=δ3(r)δ(tFind the retarded Green's function for the 3D wave equation (\partial^{2}/\partialt^{2} - c^{2}\nabla^{2})G = \delta^{3}(r)\delta(t. Interpret the result physically.
Intermediate
3.UsetheBornapproximationtofinddσ/dΩforscatteringfromaYukawapotentialV(r)=V0Use the Born approximation to find d\sigma/d\Omega for scattering from a Yukawa potential V(r) = V_{0} e−\mur/r.ShowthatRutherfordscatteringistheμ→0limite^{-\mur}/r. Show that Rutherford scattering is the \mu \to 0 limit
Intermediate
4.Write down the exact propagator K(x,t; x', 0) for the harmonic oscillator. Show it encodes both the energy spectrum (as poles) and the partition function (Euclidean rotation).
Challenging
Key Takeaways
  • GdefinedbyLG=δ(r−r′).SolutiontoLu=fisu(r)=\intG(r,r′)f(r′)dr′(superpositionG defined by LG = \delta(r-r'). Solution to Lu = f is u(r) = \intG(r,r')f(r')dr' (superposition.
  • 3DLaplacian:G=1/(4π∣r−r′∣)—Coulomb′slaw.Methodofimages=boundary−correctedG3D Laplacian: G = 1/(4\pi|r-r'|) — Coulomb's law. Method of images = boundary-corrected G.
  • Helmholtz:G+=eik∣r−r′∣/(4π∣r−r′∣).Bornapproxfollowsfromfar−fieldexpansionHelmholtz: G+ = e^{ik|r-r'|}/(4\pi|r-r'|). Born approx follows from far-field expansion.
  • Harmonicoscillator:GR=θ(t−t′)sin(ω0(t−t′))/(mω0).ConvolutiongivesdrivenresponseHarmonic oscillator: G_{R} = \theta(t-t') sin(\omega_{0}(t-t'))/(m\omega_{0}). Convolution gives driven response.
  • Heat/Schro¨dinger:sameGaussiankernel,realvsimaginarytime.Euclidean→partitionfuncHeat/Schrödinger: same Gaussian kernel, real vs imaginary time. Euclidean \to partition function.
  • Spectralform:G+(E)=∑∣n⟩⟨n∣/(E−En+iε).Poles=energies;ImG=densityofstatesSpectral form: G+(E) = \sum|n⟩⟨n|/(E-E_{n}+i\varepsilon). Poles = energies; Im G = density of states.