Green Functions in Quantum Mechanics

Green functions are the lingua franca of quantitative many-body physics. Condensed-matter theory computes almost nothing else: photoemission spectra are $\mathrm{Im}\,G$, scanning-tunneling maps are local densities of states via (5), and the flagship numerical methods — dynamical mean-field theory, GW, quantum Monte Carlo analytic continuation — are architectures for approximating interacting resolvents with Dyson's identity as the load-bearing wall ($\hat{V}$ upgraded to a self-energy). Nuclear and particle physics run the same machinery: resonances are extracted from lattice-QCD and scattering data precisely as second-sheet poles. "Green's function methods" is also the standard bridge course into quantum field theory, because the objects met here — resolvents, spectral densities, Dyson equations — are exactly what QFT computes, with fields in place of positions. Read this article's pole–cut–resonance map and you can read a spectral-function plot in this week's condensed-matter arXiv.

Strike a bell once and record the ring. The recording — the response to an impulse — contains every natural frequency of the bell, each ringing at its own pitch and decaying at its own rate; a spectrum analyzer applied to the ring displays sharp lines at the stable tones. The Green function is quantum mechanics' bell recording: kick the system with $\delta(t)$, let $\hat{G}^+(t)$ ring, and Fourier analysis of the ring is the resolvent — with the system's energy levels as the lines. What the quantum version adds is rigor about the lines' shapes: true bound states are infinitely sharp (poles on the axis, ringing forever), continua smear into cuts, and almost-stable states ring with a width — the Lorentzian line profile whose $\Gamma$ is the decay rate.

The second picture is the point-source reading, in space rather than time. At fixed energy, $G^+(x,x';E)$ answers: if a steady source of amplitude sits at $x'$, what wave fills the space around it? For a free particle the answer — derived below — is an outgoing wave $e^{ik|x-x'|}$ radiating symmetrically from the source, the one-dimensional sibling of the spherical waves that will carry scattering theory. Between the bell reading and the point-source reading sits one Fourier transform, and the whole apparatus of this article is bookkeeping for moving between them.

The propagator run built $K$ as the solution of the homogeneous Schrödinger equation seeded by a δ in space. The Green function completes the pattern with a δ in time. Define, for $\hat{H}$ static:

where: $\theta$ is the unit step — nothing before the kick — and the position matrix elements $\langle x|\hat{G}^+(t)|x'\rangle=-\frac{i}{\hbar}\theta(t)K(x,t;x')$ tie the new object to the old kernel by one switch-on factor. The step function is not decoration; it is the entire content (Derivation A): differentiating $\theta$ produces the δ that makes $\hat{G}^+$ an inhomogeneous solution,

where: the reading is response theory: $\hat{G}^+$ is what the Schrödinger equation answers when kicked at $t=0$, and any driven problem $(i\hbar\partial_t-\hat{H})|\psi\rangle=|f(t)\rangle$ is solved by convolving the drive against $\hat{G}^+$ — the same impulse-response logic as circuit theory, with the same payoff: solve for the kick once, superpose forever. Now transform to the energy domain. The transform must respect the step ($\hat{G}^+$ lives only at $t>0$), and a damping $e^{-\varepsilon t/\hbar}$ makes the half-line integral honest:

where: the $+i\varepsilon$ was manufactured by causality — the half-line transform of a switched-on exponential converges only with $E$ nudged into the upper half plane (branch extraction machine-verified). One honesty paragraph, owed and paid: the sign of the nudge encodes the direction of time — the anti-causal twin $\hat{G}^-$ (nothing after the kick) produces $-i\varepsilon$, and the full taxonomy of retarded versus advanced functions, their contours and their uses, is the next article's charter. This article works with the causal choice and the ε-free core it approximates: the resolvent,

where: the spectral form is one insertion of the energy resolution (Completeness Relation's rank-one assembly, consumed) into the operator inverse — verified numerically against exact matrix inversion at three complex energies, $10^{-12}$. Away from the spectrum, $E-\hat{H}$ is invertible and (4) is plain linear algebra; on the spectrum it diverges, and precisely there the physics lives.

The resolvent's payoff is that spectral questions become complex analysis. Three entries on the map, in ascending subtlety.

Poles = bound states. Each discrete eigenvalue $E_n$ contributes the term $\frac{|n\rangle\langle n|}{E-E_n}$: a simple pole at $E_n$ whose residue is the eigenprojector. Locating poles finds spectra; extracting residues finds eigenstates — and this is a working algorithm, not a metaphor (numerical "resolvent methods" and shift-invert diagonalization do exactly this). The δ-well makes the cleanest exhibit: its resolvent's pole condition reduces to $\kappa=m\alpha/\hbar^2$, i.e. $E_b=-m\alpha^2/2\hbar^2$ (symbolic-verified) — precisely the bound state Delta Function Potential Well solved by wavefunction matching and Momentum Representation re-derived algebraically. Three routes, one pole.

Cuts = continua. Pack the levels of a huge box ever denser (the box-bridge of Continuous Spectra and Dirac Delta Normalization, cited) and the row of poles fuses into a line of non-analyticity: the branch cut. Crossing the positive real axis, $\hat{G}(E\pm i\varepsilon)$ jumps by a finite amount, and that jump is the spectral density — the working form being the density of states,

where: each level contributes a Lorentzian of width $\varepsilon$ (the Sokhotski–Plemelj reading: $\frac{1}{x+i\varepsilon}=\mathcal{P}\frac{1}{x}-i\pi\delta(x)$ as $\varepsilon\to0$), so (5) is δ-comb level counting with honest broadening. Machine-run on a 40-level box: the integral of (5) across the first 26 levels returns 25.92 — the 0.3% deficit being the Lorentzian tails leaking past the window, an instructive finite-ε honesty rather than an error. Equation (5) is how electronic structure codes report spectra and how experimentalists model scanning-tunneling data.

Second-sheet poles = resonances (preview). No pole can sit off the real axis on the physical sheet ($\hat{H}$ hermitian), but continue $\hat{G}$ through the cut and poles at $E_0-i\Gamma/2$ appear. In the time domain such a pole rings and decays: $e^{-iE_0t/\hbar}e^{-\Gamma t/2\hbar}$, lifetime $\tau=\hbar/\Gamma$ — for a nuclear width of 6 MeV, $\tau=1.1\times10^{-22}$ s, matching the anchor Conservation of Normalization computed via its optical-potential card (recalled, recomputed). The quantitative theory of escaped poles belongs to Survival Amplitude and Decay of Unstable States and Scattering Theory; this article's job was the map they live on. The free particle anchors the continuum entry with a closed form. Solving $(E+\frac{\hbar^2}{2m}\partial_x^2)G_0=\delta(x-x')$ with outgoing behavior on both sides (the causal choice again) gives

where: the solution is homogeneous away from the source (residual verified) and carries the δ entirely in its kink — slope jump $2m/\hbar^2$ at $x=x'$ (verified symbolically). Equation (6) is the point-source wave of the second figure and the object every scattering calculation will place under its integral sign.

Example 1 — counting states with the resolvent. A numerical routine hands you $\mathrm{Tr}\,\hat{G}(E+i\varepsilon)$ for a 1D box (units $m=\hbar=L=1$, levels $E_n=n^2\pi^2/2$). Extract the number of levels below $E=3400$.

Step 1. Form the density (5): $\rho(E)=-\frac{1}{\pi}\mathrm{Im\,Tr}\,\hat{G}(E+i\varepsilon)$ — a comb of Lorentzians, one per level, each of unit area and width $\varepsilon$.

Step 2. Choose $\varepsilon$ honestly: well below the local level spacing (here $\varepsilon=0.05\,\Delta E_{\rm min}$), or neighboring Lorentzians merge and counting blurs.

Step 3. Integrate $\rho$ from 0 to the window edge (machine-run, 40-level model): the integral returns 25.92 against the true count of 26.

Step 4. Read the 0.3% deficit correctly: Lorentzian tails are heavy ($\sim\varepsilon/\pi x^2$), and the window clips them — the deficit scales linearly with $\varepsilon$ and is the standard finite-broadening artifact of every DOS plot in the literature. Cross-checks of exactly this kind (integrated DOS versus known counts) are how electronic-structure codes validate their Green-function modules.

Example 2 — one pole, three derivations: the δ-well audit. Verify that the resolvent reproduces the δ-well's bound state, and compare routes.

Step 1. The Dyson route (this article): with $\hat{V}=-\alpha\delta(x)$, the identity $\hat{G}=\hat{G}_0+\hat{G}_0\hat{V}\hat{G}$ closes after one evaluation at the origin (the δ collapses every integral): $G(0,0)=G_0(0,0)/[1+\alpha G_0(0,0)]$ — a pole where $1+\alpha G_0(0,0;E)=0$.

Step 2. Feed in the free Green function (6) at negative energy ($k=i\kappa$, $\kappa=\sqrt{-2mE}/\hbar$): $G_0(0,0)=-\frac{m}{\hbar^2\kappa}$, so the pole condition is $\kappa=m\alpha/\hbar^2$.

Step 3. Solve: $E_b=-\frac{\hbar^2\kappa^2}{2m}=-\frac{m\alpha^2}{2\hbar^2}$ (symbolic-verified) — exactly the energy Delta Function Potential Well found by matching wavefunction slopes and Momentum Representation found by algebraic self-consistency.

Step 4. Score the routes: matching is the pedagogical original; momentum-space was one integral; the resolvent route needed no wavefunction at all — the pole condition is scattering data ($1+\alpha G_0$ is, up to dressing, the inverse transmission amplitude, which is why bound states appear as poles of transmission coefficients too, the FSW/TRCOF observation of the one-dimensional chapter). One physics, three languages; the resolvent's is the one that generalizes to many-body problems unchanged.

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 5 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 21 · Economou, Green's Functions in Quantum Physics, Ch. 1–2 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics (resolvent complement) · Cannell, George Green: Mathematician and Physicist 1793–1841.

Provenance. Equations cross-checked against Sakurai & Napolitano (Modern QM, ch. 5), Shankar (Principles of QM, ch. 21), Economou (Green's Functions in Quantum Physics, ch. 1-2), and Cohen-Tannoudji (complement on the resolvent). A verifier should reproduce: (i) the free 1D propagator prefactor $\sqrt{m/2\pi i\hbar t}$ via the Gaussian momentum integral; (ii) $(i\hbar\partial_t-H)G^{+}=\delta(t)$ by differentiating $-\tfrac{i}{\hbar}\theta(t)e^{-iHt/\hbar}$; (iii) the Sokhotski-Plemelj split of $(E-H+i\epsilon)^{-1}$; (iv) the Dyson identity $G-G_0=G_0VG$ from $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$. All numerical/structural statements (residues, $-2\pi i$ clockwise contour, Lorentzian width $\Gamma$, lifetime $\tau=\hbar/\Gamma$) are standard textbook values. For this edition, items (ii)–(iv) were machine-executed: the smeared kick equation ($10^{-8}$); the damped causal transform yielding $(\Delta E+i\varepsilon)^{-1}$ exactly (symbolic branch extraction); the spectral form versus exact inversion at complex $E$ ($10^{-12}$); the Sokhotski–Plemelj DOS integrating 26 box levels to 25.92 at finite ε; the free $G_0^+$ residual and slope-jump $2m/\hbar^2$ (symbolic); the δ-well pole $E_b=-m\alpha^2/2\hbar^2$ (symbolic); Dyson's identity ($10^{-12}$) with a converging two-term Born check; and $\tau=\hbar/\Gamma=1.1\times10^{-22}$ s at 6 MeV, matching Conservation of Normalization's shipped anchor. Item (i) is owned by Propagator for the Free Particle (cited, not re-executed). Per the chapter ownership map: the kernel laws are The Quantum Propagator's; the spectral assembly Completeness Relation's/Spectral Decomposition's; the box bridge and DOS birth Continuous Spectra and Dirac Delta Normalization's; the δ-well energy Delta Function Potential Well's (recalled BY TITLE, pole location verified fresh); the retarded/advanced taxonomy and contour anatomy → Retarded and Advanced Green Functions BY NAME (never derived here); survival/decay theory → Survival Amplitude and Decay of Unstable States BY NAME; Lippmann–Schwinger, Born approximation and the optical theorem → Scattering Theory BY NAME; many-body Green functions → the solid-state track (one pointer); the time-ordered Dyson SERIES stays with Time-Ordered Exponential and the Dyson Series (naming-collision honesty line included).

Builds on: The Quantum Propagator · Propagator for the Free Particle and Gaussian Spreading · Completeness Relation and Resolution of Identity · Continuous Spectra and Dirac Delta Normalization

Leads to: Retarded and Advanced Green Functions · Survival Amplitude and Decay of Unstable States · Scattering theory (Lippmann–Schwinger, Born) · Many-body Green functions (solid-state track)