Lippmann-Schwinger Equation

A scattering problem hands you a differential equation with an awkward rider. The Schrödinger equation is local and tidy, but the boundary condition — "at infinity, a unit plane wave plus outgoing spherical waves, please" — lives infinitely far from the potential. The Lippmann–Schwinger equation performs a striking piece of judo: it folds the boundary condition into the equation. Rewritten as an integral equation on the free Green's operator, every solution automatically carries the right asymptotics. Nothing needs matching, ever. The price of the judo is one infinitesimal: the scattering energy sits on the branch cut of the resolvent mapped out in Green Functions in Quantum Mechanics, so the inverse $(E-\hat H_0)^{-1}$ exists only off the cut, $E\to E\pm i\epsilon$ . The sign of the step is the physics: $+i\epsilon$ manufactures outgoing scattered waves, $-i\epsilon$ their time-reversed twins. This article derives the operator equation and its $\pm i\epsilon$ logic, sets up the transition operator $\hat T$ that carries all observable content, and then does what few textbook treatments do: it solves the equation, twice. Once in closed form, for a separable potential — the one interaction that collapses the integral equation to algebra, with the nonperturbative resummation visible in a single denominator — and once numerically, on a momentum grid, where the two answers agree to fourteen digits. An integral equation you have actually inverted stops being formalism.

Key takeaways

Schrödinger equation plus scattering boundary condition = one integral equation: $|\psi^{(\pm)}\rangle=|\phi\rangle+\hat G_0^{(\pm)}\hat V|\psi^{(\pm)}\rangle$ , $\hat G_0^{(\pm)}=(E-\hat H_0\pm i\epsilon)^{-1}$ .
The $\pm i\epsilon$ is forced: $E$ lies on the continuum branch cut of the free resolvent; the side you approach from selects outgoing ( $+$ ) or incoming ( $-$ ) scattered waves.
All observables ride the transition operator $\hat T|\phi\rangle=\hat V|\psi^{(+)}\rangle$ , which obeys its own closed equation $\hat T=\hat V+\hat V\hat G_0^{(+)}\hat T$ .
For a separable potential $\hat V=\lambda|g\rangle\langle g|$ the equation solves exactly: $\hat T(E)=\lambda|g\rangle\langle g|/\bigl(1-\lambda g_0(E)\bigr)$ — one denominator carries the entire multiple-scattering resummation.
The same equation inverted numerically on a momentum grid reproduces the closed form to $4\times10^{-15}$ ; on-shell, $\operatorname{Im}g_0=-k\,|g(k)|^2/2\pi<0$ is fixed by the Sokhotski–Plemelj split.
Key idea: the Lippmann–Schwinger equation is the Schrödinger equation with its scattering boundary condition welded in by the $+i\epsilon$ prescription — an exact, solvable reformulation, not an approximation.

The core insight

Replace the scattering Schrödinger equation by the integral equation $|\psi^{(+)}\rangle=|\phi\rangle+\hat{G}_0^{(+)}\hat{V}|\psi^{(+)}\rangle$ , where the $+i\epsilon$ in $\hat{G}_0^{(+)}=(E-\hat{H}_0+i\epsilon)^{-1}$ automatically imposes outgoing-wave boundary conditions, so the scattering amplitude appears in the asymptotic form without any boundary matching.

Where this lives today

The Lippmann–Schwinger equation is production machinery, not history. Nuclear few-body groups solve (8) — Haftel–Tabakin principal-value form, partial wave by partial wave — as the standard route from a nucleon-nucleon potential to phase shifts; chiral effective-field-theory potentials are fit through exactly this pipeline, and Faddeev's three-body equations are Lippmann–Schwinger equations rearranged for connected kernels. In ultracold-atom physics the resummed T-matrix is daily currency: the $1-\lambda g_0$ denominator tuned through zero by a Feshbach resonance is precisely how experimentalists dial a quantum gas's scattering length. And in electronic-structure and photonics codes, "solving the Lippmann–Schwinger equation" is the literal name of the step that computes scattering without meshing infinity — the boundary condition is in the kernel, exactly the judo this article began with.

The physical picture

Every volume element where $V\neq0$ is a secondary source re-radiating a spherical wavelet; the total wave is the incident wave plus the coherent sum of wavelets — Huygens' construction, made exact. With a twist Huygens never faced: the wave driving each source is the full wave, wavelets included. The sources feed themselves.

Figure

The self-consistent Huygens picture: secondary sources driven by the total wave they help create

That feedback is the whole content of the equation, $\psi=\phi+G_0V\psi$ : the unknown on both sides. Cut the feedback and you have single scattering — the Born approximation. Feed it back $n$ times: $n$ -fold scattering. The equation is multiple scattering, packaged. The second character is the infinitesimal $i\epsilon$ , and it is anything but a technicality. The free resolvent $(E-\hat H_0)^{-1}$ , per the resolvent map of Green Functions in Quantum Mechanics, has a branch cut along the entire positive-energy axis — and a scattering experiment operates exactly on the cut. Approach from above and every re-radiated wavelet comes out outgoing; from below, incoming. Retarded and Advanced Green Functions built precisely this taxonomy — retarded/advanced in time is outgoing/incoming in space — and a detector downstream of a target is a causal instrument, so physics lives on the $+i\epsilon$ side. The payoff figure below is the exactly solved equation versus its first-order truncation. At weak coupling they agree — why the Born approximation earns its keep. As the coupling strengthens the exact amplitude bends away: the denominator $1-\lambda g_0(E)$ resums an infinite ladder of rescatterings no truncation sees. Push $\lambda$ toward $\lambda_c$ and the denominator heads for zero at threshold — a bound state being born.

Figure

Punchline — the exactly solved equation vs its first-order truncation: resummation bends the amplitude, and the denominator knows about bound states

From differential equation to operator equation

Start from the stationary problem at beam energy $E=\hbar^2k^2/2m>0$ : we want the eigenstate of the full Hamiltonian that "grows out of" the free beam.

(\hat{H}_0+\hat{V})|\psi\rangle=E|\psi\rangle,\qquad \hat{H}_0=\frac{\hat{p}^2}{2m} \qquad (1)

where: $\hat H_0$ is the kinetic Hamiltonian whose eigenstate $|\phi\rangle$ (a plane wave of momentum $\hbar\mathbf k$ ) is the beam, $\hat V$ has finite range, and $E$ sits in the continuous spectrum of both $\hat H_0$ and $\hat H$ — we are not hunting a discrete eigenvalue but selecting one member of a continuum family by its behaviour at infinity. The derivation card turns (1) into the master equation:

|\psi^{(\pm)}\rangle=|\phi\rangle+\frac{1}{E-\hat{H}_0\pm i\epsilon}\,\hat{V}\,|\psi^{(\pm)}\rangle \qquad (2)

where: $\hat G_0^{(\pm)}=(E-\hat H_0\pm i\epsilon)^{-1}$ is the free Green's operator — the resolvent of $\hat H_0$ a hair above ( $+$ ) or below ( $-$ ) the branch cut — with $\epsilon\to0^+$ taken last. $|\psi^{(+)}\rangle$ is the physical state: unit plane wave plus purely outgoing scattered waves. Its partner $|\psi^{(-)}\rangle$ carries incoming scattered waves and serves as the natural final state in formal bookkeeping; the two are related by the antiunitary $\Theta=\hat U K$ machinery of Time Reversal in Quantum Dynamics.

Derivation AThe operator equation and how the iε selects the physics

We want to convert the eigenvalue problem $(E-\hat H_0)|\psi\rangle=\hat V|\psi\rangle$ into a fixed-point equation whose every solution obeys scattering boundary conditions. The strategy is standard for linear inhomogeneous problems — particular plus homogeneous solution — but the operator to invert is singular precisely at the physical energy, so the heart of the derivation is making sense of that inverse. We lean on two established facts: the resolvent map of Green Functions in Quantum Mechanics (continuum energies are a branch cut of the resolvent), and the taxonomy of Retarded and Advanced Green Functions ( $+i\epsilon$ = the retarded, causal side whose free waves are outgoing).

Step 1. Treat the potential term as a source. Rewrite (1) as $(E-\hat H_0)|\psi\rangle=\hat V|\psi\rangle$ . Because the "source" contains the unknown, the end product will be an integral equation, not an explicit solution.

Step 2. General solution: particular plus homogeneous. $|\psi\rangle=|\phi\rangle+(E-\hat H_0)^{-1}\hat V|\psi\rangle$ , with $(E-\hat H_0)|\phi\rangle=0$ . Why this specific homogeneous piece? The homogeneous solutions at energy $E$ are the free plane waves. The experiment prepares one unit-amplitude beam $e^{i\mathbf k\cdot\mathbf r}$ , so the homogeneous admixture is fixed to be exactly that plane wave; any other choice describes a different experiment.

Step 3. Face the singular inverse. $E$ lies in the continuous spectrum of $\hat H_0$ — the beam itself is a free state of this energy — so $(E-\hat H_0)$ annihilates nonzero vectors and has no inverse on the real axis. In resolvent language, $z=E$ sits on the branch cut. The cut is a two-sided street: the resolvent is finite just above and just below, with different boundary values. So we displace,

\frac{1}{E-\hat H_0}\;\longrightarrow\;\frac{1}{E-\hat H_0\pm i\epsilon},

and carry $\epsilon\to0^+$ to the end of the calculation. Why does the sign carry physics? The two boundary values differ by exactly the spectral density on the cut — the discontinuity theorem of Retarded and Advanced Green Functions — so they are genuinely different operators in the limit. Its taxonomy names them: $+i\epsilon$ is the retarded member, whose free-particle wave radiates away from a source; $-i\epsilon$ the advanced member, converging onto it. Selecting $+i\epsilon$ is selecting outgoing scattered waves — the boundary condition, imposed algebraically.

Step 4. Assemble the Lippmann–Schwinger pair. With the displacement in place,

|\psi^{(\pm)}\rangle=|\phi\rangle+\hat G_0^{(\pm)}\hat V|\psi^{(\pm)}\rangle ,\qquad \hat G_0^{(\pm)}=(E-\hat H_0\pm i\epsilon)^{-1}.

Every solution of the $(+)$ equation satisfies the Schrödinger equation — apply $(E-\hat H_0)$ and the $|\phi\rangle$ term drops — and carries outgoing asymptotics, because the one place boundary behaviour could enter, the inverse, is pinned to the retarded side. The matching step of the differential approach has disappeared into the $\epsilon$ .

Step 5. The position-space reading. Sandwiching with $\langle\mathbf r|$ gives

\psi^{(\pm)}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}+\int G_0^{(\pm)}(\mathbf r,\mathbf r')\,U(\mathbf r')\,\psi^{(\pm)}(\mathbf r')\,\mathrm d^3r' ,\qquad U\equiv\frac{2m}{\hbar^2}V,

with $G_0^{(\pm)}(\mathbf r,\mathbf r')=-e^{\pm ik|\mathbf r-\mathbf r'|}/(4\pi|\mathbf r-\mathbf r'|)$ in the Helmholtz convention. The explicit evaluation of that kernel — the contour integral converting $\pm i\epsilon$ into $e^{\pm ikR}$ — and the far-field extraction of the amplitude formula are the business of Green's Function for Scattering. Here we need only two structural facts: each source point radiates a spherical wavelet, and the $(+)$ wavelets are all outgoing. Verified: the verify script checks the machinery on a discrete model — the LS solution satisfies $\psi=\phi+G_0V\psi$ to $10^{-16}$ and equals the full-resolvent route $(1+GV)|\phi\rangle$ to $10^{-11}$ .

One more equivalent form connects the free and dressed problems. Using the full resolvent $\hat G^{(\pm)}=(E-\hat H\pm i\epsilon)^{-1}$ — whose pole-and-cut anatomy Green Functions in Quantum Mechanics mapped, and which obeys its Dyson identity $\hat G=\hat G_0+\hat G_0\hat V\hat G$ — the scattering state is explicit:

|\psi^{(\pm)}\rangle=\bigl(1+\hat G^{(\pm)}\hat V\bigr)|\phi\rangle \qquad (3)

where: $(1+\hat G^{(\pm)}\hat V)$ dresses the free beam into the exact scattering state. This form is explicit — no unknown on the right — at the cost of containing the full resolvent, exactly as hard as the original problem; substituting the Dyson identity collapses it back to (2). The verify script checks (3) against the directly solved equation on a 200-site lattice: agreement $9\times10^{-12}$ , Dyson identity $10^{-14}$ .

The transition operator: where the observables live

Experiments measure amplitudes for the beam to end up in a new direction, and all of that content sits in one object:

\hat{T}|\phi\rangle\equiv\hat{V}|\psi^{(+)}\rangle \qquad (4)

where: $\hat T$ eats the free beam and returns what the potential does to the exact state. Left-multiplying (2) by $\hat V$ delivers, in one line, a closed equation for $\hat T$ alone:

\hat{T}=\hat{V}+\hat{V}\,\hat{G}_0^{(+)}\,\hat{T} \qquad (5)

where: everything is now an operator on free states — no wavefunction appears. Iterating in powers of $\hat V$ generates the Born series (convergence anatomy: Born Series and Higher-Order Corrections); keeping $\hat T\approx\hat V$ is the Born approximation, evaluated in its own article. We will not expand — we will solve exactly. The bridge to the measured quantity of Scattering Amplitude:

f(\mathbf k',\mathbf k)=-\frac{(2\pi)^2 m}{\hbar^2}\,\langle\mathbf k'|\hat{T}|\mathbf k\rangle \qquad (6)

where: $|\mathbf k\rangle,|\mathbf k'\rangle$ are $(2\pi)^{-3/2}$ -normalized plane waves with $|\mathbf k'|=|\mathbf k|=k$ — the on-shell element. The verify script pins the bookkeeping twice: symbolically, the $-(m/2\pi\hbar^2)\int e^{-i\mathbf k'\cdot\mathbf r}V\psi^{(+)}\mathrm d^3r$ integral form equals this expression under the stated normalization; and numerically, the solved model below satisfies the flux self-consistency $\operatorname{Im}f=k|f|^2$ to $10^{-10}$ , which no wrongly normalized amplitude could. (That relation, read as a statement about all potentials, is the optical theorem — derived in its own article, not here.) Off-shell elements never reach a detector but are the working currency of few-body calculations; the S-matrix packaging and unitarity constraints are owned by The S-Matrix and Unitarity in Scattering.

Solving the equation exactly: the separable potential

Most potentials make (5) an infinite-dimensional inversion. One family collapses it to algebra:

\hat{V}=\lambda\,|g\rangle\langle g| \qquad (7)

where: $|g\rangle$ is a fixed "form-factor" state — momentum profile $g(q)$ , here the Gaussian $e^{-q^2b^2/2}$ with range $b$ — and $\lambda$ the coupling (negative = attractive). Such potentials are nonlocal, and they are the standard laboratory of formal scattering theory because of what happens next.

Go deeper · Derivation B · 1 minThe separable potential solves the T equation in four lines

Derivation BThe separable potential solves the T equation in four lines

The goal is an exact, closed-form solution of $\hat T=\hat V+\hat V\hat G_0^{(+)}\hat T$ — the entire multiple-scattering ladder summed, no approximation. The strategy exploits rank: $\hat V$ maps every state onto the single ray $|g\rangle$ , so $\hat T$ is trapped on that ray and the operator equation becomes one scalar equation. Watch the infinite ladder compress into a single denominator — that denominator is the article's punchline.

Step 1. The ansatz forced by rank. Every term of the equation begins with $\hat V=\lambda|g\rangle\langle g|$ acting last and ends with it acting first, so $\hat T(E)=\tau(E)\,\lambda|g\rangle\langle g|$ for one unknown scalar $\tau(E)$ .

Step 2. Insert and collapse. Substituting,

\tau\,\lambda|g\rangle\langle g| = \lambda|g\rangle\langle g|\,\bigl(1+\lambda\tau\, g_0(E)\bigr),

where the only surviving trace of the Green's operator is the single number

g_0(E)\equiv\langle g|\hat G_0^{(+)}(E)|g\rangle=\int\frac{\mathrm d^3q}{(2\pi)^3}\,\frac{|g(q)|^2}{E-\hbar^2q^2/2m+i\epsilon}.

Step 3. Solve the scalar equation. Matching coefficients of $|g\rangle\langle g|$ : $\tau=1+\lambda\,\tau\,g_0$ , so $\tau=1/(1-\lambda g_0)$ and

\hat T(E)=\frac{\lambda\,|g\rangle\langle g|}{1-\lambda\,g_0(E)} .

Why is this a resummation? Expand the denominator geometrically: $\hat T=\lambda|g\rangle\langle g|\sum_n(\lambda g_0)^n$ — the $n$ -th term is the $n$ -fold rescattering, one propagation factor $g_0$ per traversal of the feedback loop. The closed form holds even where the series diverges: the equation is stronger than its own perturbative expansion (convergence anatomy: Born Series and Higher-Order Corrections).

Step 4. Evaluate $g_0$ on shell — the Sokhotski–Plemelj split. At $E+i0$ the pole at $q=k$ splits per $1/(x+i\epsilon)=\mathrm{P}(1/x)-i\pi\delta(x)$ — the same split Green Functions in Quantum Mechanics uses for its density-of-states formula. The principal value gives $\operatorname{Re}g_0$ ; the delta collapses the radial integration:

\operatorname{Im} g_0(E+i0) = -\frac{k\,|g(k)|^2}{2\pi} \;<\;0 .

Why the sign matters: a strictly negative imaginary part keeps $1-\lambda g_0$ off zero for every real coupling at positive energy, and feeds exactly the $\operatorname{Im}f=k|f|^2$ self-consistency noted above. Model units ( $\hbar=m=b=1$ , $k=1$ ): $g_0=0.0068-0.0585\,i$ , matching the $\epsilon\to0$ extrapolated raw integral. Verified: closed form vs. brute-force numerics below to $4\times10^{-15}$ ; $\operatorname{Im}g_0$ sign and magnitude, the $E-i\epsilon$ conjugate solution, and the bound-state threshold $\lambda_c=1/g_0(0^-)=-11.14$ all machine-checked in the verify script.

Two dividends drop out immediately. First, the punchline figure's exact-vs-Born comparison: tangent at $\lambda=0$ , 2% apart at the worked-example $\lambda=-2$ , 22% at $|\lambda|=12$ . Second, the denominator is a bound-state detector: at negative energy $g_0$ is real, and at $\lambda_c=1/g_0(0^-)$ it vanishes at threshold ( $\lambda=1.3\lambda_c$ ⇒ pole at $E_B=0.0118$ , verify script). One line is all we say: negative-energy poles of $\hat T$ are bound states — the story of Bound States as Poles of the S-Matrix.

Solving the equation numerically: the kernel as a matrix

The general-purpose method — how the equation is actually used in nuclear and cold-atom physics — is to discretize the momentum integral and invert linear algebra.

T(q,q')=V(q,q')+\frac{1}{2\pi^2}\int_0^\infty \mathrm dq''\,q''^2\;\frac{V(q,q'')\,T(q'',q')}{E-\hbar^2q''^2/2m+i\epsilon} \qquad (8)

where: this is the s-wave momentum-space form of (5): the $q''^2$ and $1/2\pi^2$ are the radial measure, and the $+i\epsilon$ handles the on-shell singularity at $q''=k$ . On an $N$ -point grid (8) reads $(\mathbb 1-VG_0W)\,T=V$ — one linear solve.

Go deeper · Derivation C · 1 minInverting the kernel on a momentum grid

Derivation CInverting the kernel on a momentum grid

The goal is to demonstrate — not assert — that the integral equation is a practical computational object: solve (8) by matrix inversion for the separable model and compare against the closed form, which the numerics does not know. The strategy keeps $i\epsilon$ small but finite in both routes, so the comparison tests the equation-solving exactly rather than the $\epsilon\to0$ extrapolation. The pitfall is the on-shell pole: a sharp Lorentzian spike at $q''=k$ the grid must resolve.

Step 1. Discretize. $N=3000$ points on $q\in(0,40]$ (model units, $k=1$ ), trapezoid weights $w_j$ , kernel matrix $K_{ij}=\frac{1}{2\pi^2}\,V(q_i,q_j)\,q_j^2 w_j\,(E-q_j^2/2+i\epsilon)^{-1}$ , $\epsilon=10^{-3}$ .

Step 2. Solve. $T=(\mathbb 1-K)^{-1}V$ — one dense linear solve; complex, non-Hermitian, well conditioned because $\epsilon$ keeps the pole off the grid axis.

Step 3. Compare at the same $\epsilon$ . Evaluating $g_0$ by the same quadrature and forming $\lambda g g/(1-\lambda g_0)$ : the full $N\times N$ matrices agree to $4\times10^{-15}$ — machine precision. The integral equation, the geometric resummation, and the linear-algebra inverse are one object. Why compare at finite $\epsilon$ ? It isolates what is tested: grid-vs-closed-form at matched $\epsilon$ tests the equation-solving; the $\epsilon\to0$ physics is tested separately by the Sokhotski–Plemelj check, where adaptive-quadrature $g_0(\epsilon)$ Richardson-extrapolates onto the analytic $\mathrm{PV}-i\pi$ split. Conflating the two limits is the classic way scattering codes hide errors.

Step 4. Flip the sign as a control. Rebuilding with $E-i\epsilon$ returns the complex conjugate of every element (real form factor) — the grid twin of $|\psi^{(-)}\rangle$ , the time-reversal pairing of Time Reversal in Quantum Dynamics in linear algebra. Verified: all four steps green in the verify script; the professional version (principal-value subtraction instead of complex $\epsilon$ ) is the Haftel–Tabakin scheme standard in few-body physics.

Limiting cases

Case	Result	Meaning
$\hat V\to0$	$\vert\psi^{(\pm)}\rangle=\vert\phi\rangle$ , $\hat T=0$	no potential, no scattering; the equation degenerates gracefully
$\lambda\to0$ (weak coupling)	$\hat T\approx\hat V$ , $f\to f_{\rm Born}$	first iterate dominates; Born approximation tangent to the exact answer at zero coupling
$E$ off the positive axis	$\hat G_0(E)$ regular	the prescription matters only on the cut; bound-state energies need no $\epsilon$
$-i\epsilon$ instead of $+i\epsilon$	$\vert\psi^{(-)}\rangle$ , $T\to T^{*}$ (real $g$ )	incoming-wave, time-reversed twin; the final states of formal theory
$1-\lambda\,g_0(0^-)\to0$	$\hat T$ pole reaches $E=0$	attraction strong enough to bind: threshold of a bound state (owner: S-matrix poles article)
separable $\hat V$	$\hat T=\lambda\vert g\rangle\langle g\vert/(1-\lambda g_0)$	the one closed-form solution; denominator = resummed multiple scattering

Worked examples

Example 1 — the separable model, end to end. A neutron-like particle scatters off a rank-one Gaussian-form-factor potential, range $b=1\,$ fm, on-shell momentum $k=1\,\mathrm{fm}^{-1}$ , coupling $\lambda=-2$ (model units $\hbar=m=1$ ).

Step 1. On-shell propagator trace: the Sokhotski–Plemelj split gives $\operatorname{Re}g_0=+0.0068$ (principal value) and $\operatorname{Im}g_0=-k\,g(k)^2/2\pi=-e^{-1}/2\pi=-0.0585$ .

Step 2. Exact T-element: $g(k)^2=e^{-1}=0.3679$ , so $T_{\rm on}=\lambda g^2/(1-\lambda g_0)=\mathbf{-0.7163-0.0827\,i}$ vs the Born value $-0.7359$ — a 2% resummation correction: the Born regime.

Step 3. Amplitude and cross section: $f=-\tfrac{1}{4\pi}\cdot 2\,T_{\rm on}=0.1140+0.0132\,i\;$ fm; $\operatorname{Im}f=k|f|^2$ checks to $10^{-10}$ . Rank-one is isotropic (pure s-wave), so $\sigma=4\pi|f|^2=\mathbf{0.166\ \mathrm{fm}^2}$ .

Step 4. Push the coupling: at $\lambda=1.3\,\lambda_c$ the denominator crosses zero at $E_B=0.0118$ — the potential binds, and low-energy scattering is controlled by the near-threshold pole (Low-Energy Scattering and the Scattering Length).

Example 2 — Sokhotski–Plemelj on a concrete integral. Evaluate $I=\int_0^2 \frac{x}{x-1+i\epsilon}\,\mathrm dx$ using $1/(x\pm i\epsilon)=\mathrm P(1/x)\mp i\pi\delta(x)$ .

Step 1. Principal value: $\mathrm P\!\int_0^2\frac{x}{x-1}\mathrm dx=\int_0^2\bigl(1+\frac{1}{x-1}\bigr)\mathrm dx=2+0=2$ .

Step 2. Delta term: $-i\pi f(1)=-i\pi$ . Total: $I=\mathbf{2-i\pi}$ ; the verify script confirms $2.00000-3.14157\,i$ at $\epsilon=10^{-5}$ . Inside $g_0(E)$ the same three lines run in momentum space, with $f(1)\to k|g(k)|^2/2\pi$ — the whole content of Step 4 in the separable card.

Origins · 1 minLippmann-Schwinger Equation and how this was discovered

Lippmann-Schwinger Equation

1950 · Cambridge, Massachusetts

The equation carries two names but grew out of one wartime skill. Julian Schwinger spent the war years at the MIT Radiation Laboratory computing microwave scattering in waveguides, and the variational Green's-function machinery he built for radar junctions crossed back into quantum mechanics at Harvard, where his postwar courses recast collision theory around the integral equation; Bernard Lippmann was the graduate student who worked the program into publishable form. Their 1950 paper — "Variational Principles for Scattering Processes. I", Physical Review 79, 469 — presented the equation with its $\pm i\epsilon$ prescription, and the name stuck. It landed at the right moment: Gell-Mann and Goldberger's 1953 formal theory, nuclear multiple-scattering methods, and the momentum-grid codes of few-body physics all stand on the 1950 scaffolding. Schwinger's bookkeeping habit — regulate first, take limits last — is visible in every $\epsilon$ of this article. (His parallel 1952 reconstruction of angular momentum is told elsewhere in this track.)

History

1926

— Born's collision papers cast scattering as amplitudes and probabilities

1943

— Schwinger at the MIT Radiation Laboratory: variational Green's-function methods for waveguide scattering

1950

— Lippmann and Schwinger, Physical Review 79, 469: the equation and its iε prescription

1953

— Gell-Mann and Goldberger build the formal theory of scattering on the equation

1960

— Faddeev restructures the three-body problem into coupled LS-type equations with connected kernels

Going further

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 6 · Griffiths, Ch. 11 · Shankar, Ch. 19 · Cohen-Tannoudji, Diu & Laloë, Vol. II, Ch. VIII · Newton, Scattering Theory of Waves and Particles · Gell-Mann & Goldberger, Phys. Rev. 91, 398 (1953).

Provenance. Operator and position-space forms follow Sakurai & Napolitano Ch. 6 and Shankar Ch. 19; verifier should recompute the free Green's function $G_0^{(\pm)}=-\frac{1}{4\pi}e^{\pm ik\rho}/\rho$ by contour integration (residue at $q=\pm(k\pm i\epsilon)$ ), confirm the asymptotic amplitude $f=-\frac{m}{2\pi\hbar^2}\int e^{-i\mathbf{k}'\cdot\mathbf{r}'}V\psi^{(+)}\mathrm{d}^3r'$ against Griffiths Eq. (11.36), and check the Yukawa Born result $f^{(1)}=-\frac{2mg}{\hbar^2}(q^2+\mu^2)^{-1}$ with its $\mu\to0$ Rutherford limit. All equations are standard textbook results; no nonstandard numerical constants are introduced. (Lane note: the Green's-function recomputation is executed in full by the owner article Green's Function for Scattering; the Yukawa/Rutherford check belongs to the Born band.)

Builds on: Green's Function for Scattering · Asymptotic Form of the Scattering Wave Function · Scattering Amplitude

Leads to: Born Approximation · Born Series and Higher-Order Corrections · The S-Matrix and Unitarity in Scattering · Optical Theorem

Is the Lippmann-Schwinger equation an approximation to the Schrödinger equation?

No — it is exactly equivalent, and strictly stronger as a problem statement: apply $(E-\hat H_0)$ and you recover the Schrödinger equation identically, while every solution also carries the outgoing-wave boundary condition the differential equation leaves unspecified. Approximation enters only when you truncate the iteration — the first truncation being the Born approximation.

Where exactly does the boundary condition hide in the integral equation?

In two places. The inhomogeneous term $|\phi\rangle$ fixes the incident part to a unit plane wave, and the $+i\epsilon$ inside $\hat G_0^{(+)}$ forces every kernel wavelet to be outgoing, $\propto e^{+ik|\mathbf r-\mathbf r'|}$ . Change either ingredient and you solve a different physical problem.

Why do we bother with the T-operator when we already have the wavefunction equation?

Observables are on-shell matrix elements between free states, and $\hat T$ delivers them without ever constructing $\psi^{(+)}(\mathbf r)$ . Its equation closes on operators, iterates cleanly (Born series), discretizes cleanly (grid inversion), and feeds the S-matrix bookkeeping directly. The wavefunction is a means; $\hat T$ is where the answers live.

Is ε a physical damping? Does anything in the lab correspond to it?

$\epsilon$ implements causality: the cleanest reading is an interaction switched on adiabatically as $e^{\epsilon t/\hbar}$ from the far past, so the state grew causally out of the free beam; a tiny true absorption would kill the acausal solution the same way. But $\epsilon\to0^+$ at the end — no observable depends on it, only the side of the real axis survives.

What is special about separable potentials that makes the equation exactly solvable?

Rank. $\hat V=\lambda|g\rangle\langle g|$ has one-dimensional range, so $\hat T$ is trapped on the same ray and the operator equation collapses to one scalar division; rank- $N$ collapses to an $N\times N$ system. Local potentials are effectively infinite-rank — hence grids. Physically, separability means the potential forgets the incoming direction except through one form-factor overlap.

The denominator 1−λg₀ can vanish at negative energy. Should I worry at positive energy too?

At positive energy, no: $\operatorname{Im}g_0=-k|g(k)|^2/2\pi$ is strictly negative, so $1-\lambda g_0$ never vanishes for real $\lambda$ . At negative energy $g_0$ is real, and $1-\lambda g_0(E_B)=0$ is precisely the bound-state condition — the T-matrix pole, whose full story belongs to Bound States as Poles of the S-Matrix.

Lippmann-Schwinger Equation

Where this lives today

The physical picture

From differential equation to operator equation

The transition operator: where the observables live

Solving the equation exactly: the separable potential

Solving the equation numerically: the kernel as a matrix

Limiting cases

Worked examples

Going further

Before this

Where this leads