The Lippmann-Schwinger Equation

Derivation

The Lippmann-Schwinger Equation

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Statement

For a particle of mass \( m \) and energy \( E = \dfrac{\hbar^2 k^2}{2m} > 0 \) scattering off a localized potential \( V(\vec{r}\,) \), the time-independent Schrödinger equation \( \left( -\dfrac{\hbar^2}{2m}\nabla^2 + V \right)\psi = E\psi \) is exactly equivalent to the integral (Lippmann-Schwinger) equation \( \psi_{\vec{k}}(\vec{r}\,) = \phi_{\vec{k}}(\vec{r}\,) + \dfrac{2m}{\hbar^2}\displaystyle\int G_0^{+}(\vec{r},\vec{r}\,')\,V(\vec{r}\,')\,\psi_{\vec{k}}(\vec{r}\,')\,d^3r' \), where \( \phi_{\vec{k}} \) solves the free equation and \( G_0^{+} \) is the outgoing free Green's function. This builds the scattering boundary condition into the equation itself.

Why it matters

The differential Schrödinger equation for scattering is underdetermined: the same operator equation admits incoming, outgoing, and standing-wave solutions, and one must impose boundary conditions by hand. The Lippmann-Schwinger form packages the correct outgoing-wave boundary condition directly into the equation through the choice of Green's function, so that any solution automatically has the physically required asymptotic structure \( e^{i\vec{k}\cdot\vec{r}} + f(\theta,\phi)\,e^{ikr}/r \).

It is the launching point for the entire machinery of scattering theory: iterating it generates the Born series, its formal solution defines the T-matrix and the scattering amplitude, and its operator form \( |\psi^{+}\rangle = |\phi\rangle + G_0^{+}V|\psi^{+}\rangle \) underlies the Green's-function methods used across nuclear, atomic, condensed-matter and particle physics.

Assumptions

The potential is short-rangedIf \( V \) does not fall faster than \( 1/r \) (e.g. the Coulomb potential), the scattering states are not asymptotically free plane waves; the integral does not converge in the ordinary sense and logarithmic phase distortions appear, requiring Coulomb-modified (Dollard) treatment. The energy is positive, \( E>0 \)For \( E<0 \) (bound states) the free term \( \phi_{\vec{k}} \) is absent and the Green's function is exponentially decaying rather than oscillatory; the homogeneous equation \( \psi = \frac{2m}{\hbar^2}\int G_0 V\psi \) becomes an eigenvalue problem, not a scattering problem. The outgoing (retarded) contour \( E\to E+i\epsilon \) is chosenIf dropped, \( G_0 \) is ambiguous: the operator \( (E-H_0)^{-1} \) has a pole on the real axis. Choosing \( -i\epsilon \) gives incoming spherical waves \( \psi^{-} \); choosing the principal value gives standing waves. The physical scattering solution requires the \( +i\epsilon \) prescription. The interacting and free spectra overlap at \( E \)The manipulation \( (E-H_0)^{-1} \) presumes \( E \) lies in the continuous spectrum of \( H_0 \). If it did not, the resolvent would be bounded and no scattering (no on-shell singularity) would occur.

Derivation

\[ \left( -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{r}\,) \right)\psi_{\vec{k}}(\vec{r}\,) = E\,\psi_{\vec{k}}(\vec{r}\,) \]

Time-independent Schrödinger equation for a stationary scattering state of definite energy \( E \). A

\[ \left( \nabla^2 + k^2 \right)\psi_{\vec{k}}(\vec{r}\,) = \frac{2m}{\hbar^2}\,V(\vec{r}\,)\,\psi_{\vec{k}}(\vec{r}\,) \]

Move \( V\psi \) to the right, multiply through by \( -2m/\hbar^2 \), and define \( k^2 \equiv 2mE/\hbar^2 \). This isolates the free Helmholtz operator on the left with a source on the right. A

\[ \left( \nabla^2 + k^2 \right)\phi_{\vec{k}}(\vec{r}\,) = 0, \qquad \phi_{\vec{k}}(\vec{r}\,) = e^{i\vec{k}\cdot\vec{r}} \]

The homogeneous (source-free) equation is the free-particle Schrödinger/Helmholtz equation; its physical solution is the incident plane wave with wavevector \( \vec{k} \). Any particular solution of Step 2 plus any homogeneous solution is again a solution. A

\[ \left( \nabla^2 + k^2 \right) G_0(\vec{r},\vec{r}\,') = \delta^{3}(\vec{r}-\vec{r}\,') \]

Introduce the Green's function of the Helmholtz operator: the response to a unit point source. Its existence lets us invert the differential operator by convolution (result greens-function-method). B

\[ \psi_{\vec{k}}(\vec{r}\,) = \phi_{\vec{k}}(\vec{r}\,) + \frac{2m}{\hbar^2}\int G_0(\vec{r},\vec{r}\,')\,V(\vec{r}\,')\,\psi_{\vec{k}}(\vec{r}\,')\,d^3r' \]

Superpose the homogeneous solution with the convolution of \( G_0 \) against the source \( \frac{2m}{\hbar^2}V\psi \). Applying \( (\nabla^2+k^2) \) to both sides and using Steps 3-4 reproduces Step 2 identically, so this is an exact restatement — still implicit, since \( \psi \) appears under the integral. B

\[ G_0^{\pm}(\vec{r},\vec{r}\,') = -\frac{1}{(2\pi)^3}\int \frac{e^{i\vec{q}\cdot(\vec{r}-\vec{r}\,')}}{q^2 - k^2 \mp i\epsilon}\,d^3q \]

Solve Step 4 by Fourier transform. The integrand has poles at \( q = \pm k \) on the real axis; the operator \( (E - H_0)^{-1} \) is singular. Displace the energy \( E \to E \pm i\epsilon \) (equivalently \( k^2 \to k^2 \pm i\epsilon \)) to define the resolvent unambiguously and fix the boundary condition. C

\[ G_0^{+}(\vec{r},\vec{r}\,') = -\frac{1}{4\pi}\,\frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|} \]

Evaluate the angular integral in Step 6 and close the \( q \)-contour in the upper half-plane (Jordan's lemma). The \( +i\epsilon \) prescription selects the pole giving \( e^{+ik|\vec{r}-\vec{r}\,'|} \): an outgoing spherical wave, the physical retarded Green's function. The \( -i\epsilon \) choice would give \( e^{-ik|\vec{r}-\vec{r}\,'|} \), incoming. C

\[ \psi_{\vec{k}}^{+}(\vec{r}\,) = e^{i\vec{k}\cdot\vec{r}} - \frac{2m}{\hbar^2}\,\frac{1}{4\pi}\int \frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|}\,V(\vec{r}\,')\,\psi_{\vec{k}}^{+}(\vec{r}\,')\,d^3r' \]

Insert the outgoing Green's function (Step 7) into the integral equation (Step 5). The result is the Lippmann-Schwinger equation with the outgoing-wave boundary condition built in. B

Result

\[ \psi_{\vec{k}}^{+}(\vec{r}\,) = e^{i\vec{k}\cdot\vec{r}} - \frac{m}{2\pi\hbar^2}\int \frac{e^{ik|\vec{r}-\vec{r}\,'|}}{|\vec{r}-\vec{r}\,'|}\,V(\vec{r}\,')\,\psi_{\vec{k}}^{+}(\vec{r}\,')\,d^3r' \]

Reading. The full scattering wavefunction equals the incident plane wave plus a coherent superposition of outgoing spherical wavelets \( e^{ikR}/R \), each emitted from a volume element at \( \vec{r}\,' \) with strength proportional to the local potential \( V(\vec{r}\,') \) and the local wavefunction \( \psi^{+}(\vec{r}\,') \) there. It is Huygens' principle for quantum scattering. The equation is implicit — \( \psi^{+} \) appears on both sides — which is exactly what encodes multiple scattering.

Units check. The prefactor \( m/(2\pi\hbar^2) \) has units \( \mathrm{kg}/(\mathrm{J^2\,s^2}) = \mathrm{kg}/(\mathrm{kg^2\,m^4\,s^{-2}}\cdot \mathrm{s^2}) = \mathrm{kg^{-1}\,m^{-4}\,s^{2}} \). The integrand \( \frac{1}{R}V\psi\,d^3r' \) carries \( \mathrm{m^{-1}}\cdot \mathrm{J}\cdot[\psi]\cdot \mathrm{m^3} = \mathrm{kg\,m^{4}\,s^{-2}}\cdot[\psi] \). The product yields \( [\psi] \), matching the left side. Dimensionally consistent.

Limiting cases

Weak potential (first Born approximation). Replace \( \psi^{+}\to\phi \) inside the integral: \( \psi^{+}\approx e^{i\vec{k}\cdot\vec{r}} - \frac{m}{2\pi\hbar^2}\int \frac{e^{ikR}}{R}V e^{i\vec{k}\cdot\vec{r}\,'}d^3r' \), giving \( f_B(\vec{q}\,) = -\frac{m}{2\pi\hbar^2}\tilde V(\vec{q}\,) \).
Vanishing potential, \( V\to 0 \). The integral term drops and \( \psi^{+}\to e^{i\vec{k}\cdot\vec{r}} \): pure incident wave, no scattering, \( f\to 0 \).
Asymptotic region, \( r\to\infty \). Using \( |\vec{r}-\vec{r}\,'|\approx r - \hat r\cdot\vec{r}\,' \) gives \( \psi^{+}\to e^{i\vec{k}\cdot\vec{r}} + f(\theta,\phi)\frac{e^{ikr}}{r} \) with \( f = -\frac{m}{2\pi\hbar^2}\int e^{-i\vec{k}'\cdot\vec{r}\,'}V\psi^{+}d^3r' \), \( \vec{k}'=k\hat r \).
Low energy, \( k\to 0 \). \( e^{ikR}\to 1 \) and \( f\to -a_s \) (constant), the s-wave scattering length; cross section \( \sigma\to 4\pi a_s^2 \).

Breaks when

Long-range (Coulomb) potentials. For \( V\sim 1/r \) the asymptotic states are never free plane waves; the Green's-function convolution diverges and the scattering amplitude acquires a divergent Coulomb phase. One must use parabolic-coordinate Coulomb wavefunctions or split off the long-range part explicitly.
Bound-state / negative energy regime. For \( E<0 \) there is no incident free wave (\( \phi=0 \)) and \( k \) is imaginary; the equation degenerates into a homogeneous eigenvalue problem whose nontrivial solutions are the bound states, not scattering states.
Resonances and near-threshold poles. When \( E \) sits near a pole of the T-matrix the Born series diverges and naive iteration fails; the full implicit equation must be solved (e.g. by matrix inversion on a momentum grid), not expanded.
Strong/singular potentials. Potentials more singular than \( 1/r^2 \) at the origin make the integral kernel non-compact; existence and uniqueness of \( \psi^{+} \) are no longer guaranteed by Fredholm theory.

Failure modes

Sign/prescription confusion. Using \( G_0^{-} \) (the \( -i\epsilon \), incoming) Green's function for a standard scattering problem — this yields \( \psi^{-} \), appropriate for time-reversed final states, and gives the wrong sign of the spherical phase.
Treating it as explicit. Forgetting that \( \psi^{+} \) appears inside the integral and reading the equation as a closed-form solution rather than an implicit (integral) equation requiring iteration or inversion.
Born used out of range. Applying the first Born approximation \( \psi^{+}\to\phi \) to a strong or resonant potential where the series does not converge.
Dropping the \( 2m/\hbar^2 \). Confusing the "reduced potential" \( U=2mV/\hbar^2 \) (units of \( \mathrm{m^{-2}} \)) with \( V \) itself, giving an amplitude wrong by that factor.
Wrong Green's-function normalization. Omitting the \( -1/4\pi \) or the \( \delta^3 \) normalization of \( G_0 \), which propagates into a wrong prefactor \( m/2\pi\hbar^2 \) in the amplitude.
Coulomb by force. Plugging a bare \( 1/r \) potential into the plane-wave Born formula and quoting the resulting (finite) Rutherford amplitude as if the derivation were valid — it accidentally gives the right \( |f|^2 \) but the phase and convergence are ill-defined.

Discussion

The deep content of the Lippmann-Schwinger equation is that it converts a boundary-value problem into a convolution. In the differential form, the outgoing-wave condition is an external constraint applied at infinity; in the integral form it has been absorbed into the kernel through the choice \( G_0^{+} \). This is why the equation is often written compactly in operator form, \( |\psi^{+}\rangle = |\phi\rangle + \frac{1}{E-H_0+i\epsilon}V|\psi^{+}\rangle \), where the \( +i\epsilon \) is the entire physical content of "outgoing." The resolvent \( (E-H_0+i\epsilon)^{-1} \) is the free propagator; the interacting propagator \( (E-H+i\epsilon)^{-1} \) obeys the analogous Dyson equation.

Iterating the equation generates the Born series \( \psi^{+} = \phi + G_0^{+}V\phi + G_0^{+}VG_0^{+}V\phi + \cdots \), which admits a transparent physical reading: a term with \( n \) factors of \( V \) is the amplitude for \( n \)-fold scattering off the potential, each propagation between events carried by \( G_0^{+} \). This is the quantum-mechanical seed of Feynman diagrams. Resumming the series formally defines the transition operator \( T = V + VG_0^{+}T \), whose on-shell matrix element \( \langle \vec{k}'|T|\vec{k}\rangle \) is proportional to the scattering amplitude \( f(\theta,\phi) \).

The equation also makes the connection to the optical theorem and unitarity manifest. Because \( \mathrm{Im}\,G_0^{+} \) is proportional to an on-shell delta function \( \delta(E-H_0) \), the imaginary part of the forward amplitude is tied to the total flux removed from the beam, \( \mathrm{Im}\,f(0) = \frac{k}{4\pi}\sigma_{\text{tot}} \). Conservation of probability is thus encoded in the analytic structure of the Green's function, not imposed separately.

At the most abstract level, the free and full states are related by the Møller wave operators \( \Omega^{\pm} = 1 + G^{\pm}V \), with \( |\psi^{\pm}\rangle = \Omega^{\pm}|\phi\rangle \), and the S-matrix is \( S = (\Omega^{-})^{\dagger}\Omega^{+} \). The two Lippmann-Schwinger solutions \( \psi^{+} \) and \( \psi^{-} \) form complete but distinct bases of the continuous spectrum, related by time reversal; their overlap is precisely the S-matrix. The \( \pm i\epsilon \) prescriptions are the operator statement of the arrow of time in scattering — \( \psi^{+} \) coincides with a plane wave in the far past, \( \psi^{-} \) in the far future.

Common misconceptions. The equation is not an approximation — it is exact and equivalent to the Schrödinger equation; the approximation enters only when one truncates the Born series. And \( G_0^{+} \) is not "the" Green's function but a specific boundary-condition-selecting choice among several; the physics of "outgoing waves" lives entirely in the infinitesimal \( +i\epsilon \).

Worked examples

Born amplitude and cross section for a Yukawa potential

Take \( V(r) = V_0 \dfrac{e^{-\mu r}}{\mu r} \), a screened potential of range \( 1/\mu \). Compute the first-Born scattering amplitude and differential cross section. B

\[ f_B(\vec{q}\,) = -\frac{m}{2\pi\hbar^2}\int e^{-i\vec{q}\cdot\vec{r}\,'}V(r')\,d^3r', \qquad \vec{q} = \vec{k}-\vec{k}', \; q = 2k\sin\tfrac{\theta}{2} \]

First Born approximation: replace \( \psi^{+}\to e^{i\vec{k}\cdot\vec{r}} \) in the Result. \( \vec q \) is the momentum transfer. B

\[ \int e^{-i\vec{q}\cdot\vec{r}}\,\frac{e^{-\mu r}}{r}\,d^3r = \frac{4\pi}{q^2+\mu^2} \quad\Rightarrow\quad f_B(q) = -\frac{2mV_0}{\hbar^2\mu}\,\frac{1}{q^2+\mu^2} \]

Standard Fourier transform of the Yukawa kernel (the same integral, run backwards, that produced \( G_0 \)). Insert the \( V_0/\mu \) prefactor. B

\[ \frac{d\sigma}{d\Omega} = |f_B(q)|^2 = \left(\frac{2mV_0}{\hbar^2\mu}\right)^2 \frac{1}{\left(4k^2\sin^2\frac{\theta}{2}+\mu^2\right)^2} \]

Differential cross section is \( |f|^2 \); substitute \( q^2 = 4k^2\sin^2(\theta/2) \). A

Numbers: \( m = 9.11\times10^{-31}\,\mathrm{kg} \), \( \mu^{-1} = 1\,\text{\AA} = 10^{-10}\,\mathrm{m} \) so \( \mu = 10^{10}\,\mathrm{m^{-1}} \), \( V_0 = 5\,\mathrm{eV} = 8.0\times10^{-19}\,\mathrm{J} \), electron energy \( E = 10\,\mathrm{eV} \Rightarrow k = \frac{\sqrt{2mE}}{\hbar} = 1.62\times10^{10}\,\mathrm{m^{-1}} \), at \( \theta = 90^\circ \).

Insert values. \( \sin^2 45^\circ = 0.5 \), so \( q^2 = 4k^2(0.5) = 2k^2 = 5.25\times10^{20}\,\mathrm{m^{-2}} \); \( \mu^2 = 1.0\times10^{20}\,\mathrm{m^{-2}} \). A

\[ \frac{2mV_0}{\hbar^2\mu} = \frac{2(9.11\times10^{-31})(8.0\times10^{-19})}{(1.055\times10^{-34})^2(10^{10})} = 1.31\times10^{10}\,\mathrm{m}\;;\quad \frac{d\sigma}{d\Omega} = \frac{(1.31\times10^{10})^2}{(6.25\times10^{20})^2} = 4.4\times10^{-22}\,\mathrm{m^2/sr} \]

Reading. The differential cross section at 90° is about \( 4.4\times10^{-22}\,\mathrm{m^2/sr} = 0.044\,\text{\AA}^2/\mathrm{sr} \). The amplitude peaks sharply forward (small \( q \)) and the screening length \( 1/\mu \) sets the angular width, as expected for a range-\( 1/\mu \) potential.

Coulomb limit: recovering Rutherford scattering

Take the Yukawa result and let the screening vanish, \( \mu\to 0 \), with \( V_0/\mu \to Z_1Z_2 e^2/4\pi\varepsilon_0 \) held fixed, to recover Rutherford scattering of a charge \( Z_1 e \) off \( Z_2 e \). B

\[ V(r) = \frac{Z_1 Z_2 e^2}{4\pi\varepsilon_0 r} = \lim_{\mu\to0}\left(\frac{Z_1Z_2e^2}{4\pi\varepsilon_0}\mu\right)\frac{e^{-\mu r}}{\mu r} \]

Identify the Yukawa strength: \( V_0 = \dfrac{Z_1Z_2e^2}{4\pi\varepsilon_0}\mu \), so \( V_0/\mu = \dfrac{Z_1Z_2e^2}{4\pi\varepsilon_0} \) is finite as \( \mu\to0 \). B

\[ f(q) = -\frac{2m}{\hbar^2 q^2}\cdot\frac{Z_1Z_2e^2}{4\pi\varepsilon_0} \quad\Rightarrow\quad \frac{d\sigma}{d\Omega} = \left(\frac{2m}{\hbar^2 q^2}\frac{Z_1Z_2e^2}{4\pi\varepsilon_0}\right)^2 \]

Take \( \mu\to0 \) in \( f_B(q) = -\frac{2m(V_0/\mu)}{\hbar^2(q^2+\mu^2)} \). The angular part survives cleanly even though the amplitude's phase is (correctly) ill-defined. B

\[ q^2 = 4k^2\sin^2\tfrac{\theta}{2}, \quad E = \frac{\hbar^2k^2}{2m} \;\Rightarrow\; \frac{d\sigma}{d\Omega} = \left(\frac{Z_1Z_2e^2}{4\pi\varepsilon_0}\right)^2\frac{1}{16E^2\sin^4\frac{\theta}{2}} \]

Substitute \( q^2 \) and \( \hbar^2 k^2 = 2mE \); the \( m \)'s and \( \hbar \)'s combine into \( (4E)^2 \). This is exactly the Rutherford formula. A

Numbers: \( \alpha \)-particle on gold, \( Z_1=2 \), \( Z_2=79 \), \( E = 5\,\mathrm{MeV} = 8.0\times10^{-13}\,\mathrm{J} \), \( \theta = 60^\circ \). \( \dfrac{e^2}{4\pi\varepsilon_0} = 2.31\times10^{-28}\,\mathrm{J\,m} \).

\( \sin(30^\circ)=0.5 \Rightarrow \sin^4 30^\circ = 0.0625 \). Prefactor \( \dfrac{Z_1Z_2e^2}{4\pi\varepsilon_0} = 2\cdot79\cdot2.31\times10^{-28} = 3.65\times10^{-26}\,\mathrm{J\,m} \). A

\[ \frac{d\sigma}{d\Omega} = \frac{(3.65\times10^{-26})^2}{16\,(8.0\times10^{-13})^2\,(0.0625)} = \frac{1.33\times10^{-51}}{6.4\times10^{-25}} = 2.1\times10^{-27}\,\mathrm{m^2/sr} \]

Reading. About \( 2.1\times10^{-27}\,\mathrm{m^2/sr} = 21\,\mathrm{mb/sr} \) at 60°. Remarkably, the Born approximation reproduces the exact Rutherford cross section for the Coulomb potential — a famous coincidence: the \( 1/q^4 \) angular law is exact, even though the derivation's convergence assumptions fail for \( 1/r \). Only the (unobservable) phase of \( f \) is wrong.

Problems

Green's function verification. Show by direct differentiation that \( G_0^{+}(\vec r,\vec r\,') = -\frac{1}{4\pi}\frac{e^{ik|\vec r-\vec r\,'|}}{|\vec r-\vec r\,'|} \) satisfies \( (\nabla^2+k^2)G_0^{+}=\delta^3(\vec r-\vec r\,') \).
Solution
Let \( R=|\vec r-\vec r\,'| \). Away from \( R=0 \), \( \nabla^2\frac{e^{ikR}}{R} = \frac{1}{R}\frac{d^2}{dR^2}\!\left(R\cdot\frac{e^{ikR}}{R}\right) = \frac{1}{R}\frac{d^2}{dR^2}e^{ikR} = -k^2\frac{e^{ikR}}{R} \), so \( (\nabla^2+k^2)\frac{e^{ikR}}{R}=0 \) for \( R\neq0 \). Near \( R=0 \), \( e^{ikR}\to1 \) and \( \nabla^2\frac{1}{R} = -4\pi\delta^3(\vec r-\vec r\,') \). Hence \( (\nabla^2+k^2)\left(-\frac{1}{4\pi}\frac{e^{ikR}}{R}\right) = -\frac{1}{4\pi}(-4\pi\delta^3) = \delta^3(\vec r-\vec r\,') \). QED.
Optical theorem. Using \( f(\theta) = -\frac{m}{2\pi\hbar^2}\int e^{-i\vec k'\cdot\vec r\,'}V\psi^{+}d^3r' \) and the first Born amplitude \( f_B(q) = -\frac{2mV_0}{\hbar^2\mu(q^2+\mu^2)} \) for the Yukawa potential, verify that \( f_B \) is real and hence that the first Born approximation violates the optical theorem \( \mathrm{Im}f(0)=\frac{k}{4\pi}\sigma_{\text{tot}} \). Explain physically.
Solution
\( f_B(q) \) is manifestly real for all \( q \), so \( \mathrm{Im}f_B(0)=0 \). But \( \sigma_{\text{tot}}=\int|f_B|^2 d\Omega > 0 \). The optical theorem \( \mathrm{Im}f(0)=\frac{k}{4\pi}\sigma_{\text{tot}} \) is therefore violated at first order. Reason: unitarity relates \( \mathrm{Im}f \) at order \( V \) to \( |f|^2 \) at order \( V^2 \). The first Born term is \( O(V) \); the required imaginary part first appears in the second Born term. The optical theorem is only satisfied order-by-order when the full (or consistently truncated) series is kept.
Total Yukawa cross section. Integrate the Born differential cross section \( \frac{d\sigma}{d\Omega}=\left(\frac{2mV_0}{\hbar^2\mu}\right)^2\frac{1}{(4k^2\sin^2\frac{\theta}{2}+\mu^2)^2} \) over solid angle to obtain \( \sigma_{\text{tot}}(k) \).
Solution
With \( u=\sin^2\frac{\theta}{2} \), \( d\Omega = 2\pi\sin\theta\,d\theta = 4\pi\,du \) over \( u\in[0,1] \). Let \( A=\frac{2mV_0}{\hbar^2\mu} \). Then \( \sigma = 4\pi A^2\int_0^1\frac{du}{(4k^2u+\mu^2)^2} = 4\pi A^2\left[-\frac{1}{4k^2(4k^2u+\mu^2)}\right]_0^1 = \frac{4\pi A^2}{4k^2}\left(\frac{1}{\mu^2}-\frac{1}{4k^2+\mu^2}\right) = \frac{\pi A^2}{k^2}\cdot\frac{4k^2}{\mu^2(4k^2+\mu^2)} = \frac{4\pi A^2}{\mu^2(4k^2+\mu^2)} \). So \( \sigma_{\text{tot}} = \left(\frac{2mV_0}{\hbar^2\mu}\right)^2\frac{4\pi}{\mu^2(4k^2+\mu^2)} \). At low energy \( k\to0 \): \( \sigma\to \frac{16\pi m^2V_0^2}{\hbar^4\mu^6} \), energy-independent, as expected for s-wave.
Validity of the Born approximation. The first Born approximation requires the correction term at the origin to be small: \( \left|\frac{m}{2\pi\hbar^2}\int\frac{e^{ikr'}}{r'}V(r')e^{ikr'}d^3r'\right|\ll1 \). For the Yukawa potential in the low-energy limit \( k\to0 \), estimate the condition on \( V_0 \).
Solution
At \( k=0 \), the condition is \( \frac{m}{2\pi\hbar^2}\left|\int\frac{V_0 e^{-\mu r'}/(\mu r')}{r'}d^3r'\right|\ll1 \). Compute \( \int\frac{e^{-\mu r'}}{\mu r'^2}\,4\pi r'^2\,dr' = \frac{4\pi}{\mu}\int_0^\infty e^{-\mu r'}dr' = \frac{4\pi}{\mu^2} \). So condition is \( \frac{m}{2\pi\hbar^2}\cdot\frac{4\pi V_0}{\mu^2} = \frac{2mV_0}{\hbar^2\mu^2}\ll1 \), i.e. \( V_0 \ll \frac{\hbar^2\mu^2}{2m} \). Physically: the potential strength must be much less than the kinetic energy \( \hbar^2\mu^2/2m \) associated with localization to the potential's range \( 1/\mu \). For an electron with \( 1/\mu = 1\,\text{\AA} \): \( \hbar^2\mu^2/2m = \frac{(1.055\times10^{-34})^2(10^{10})^2}{2(9.11\times10^{-31})} = 6.1\times10^{-19}\,\mathrm{J}\approx3.8\,\mathrm{eV} \), so Born requires \( V_0\ll3.8\,\mathrm{eV} \) — our worked example with \( V_0=5\,\mathrm{eV} \) is only marginally in this regime.
Incoming-wave solution. Write the Lippmann-Schwinger equation obtained from the \( -i\epsilon \) prescription, \( G_0^{-}=-\frac{1}{4\pi}\frac{e^{-ikR}}{R} \), and state its asymptotic form. What is the physical role of \( \psi^{-} \)?
Solution
The equation is \( \psi_{\vec k}^{-}(\vec r) = e^{i\vec k\cdot\vec r} - \frac{m}{2\pi\hbar^2}\int\frac{e^{-ik|\vec r-\vec r\,'|}}{|\vec r-\vec r\,'|}V(\vec r\,')\psi_{\vec k}^{-}(\vec r\,')d^3r' \). Asymptotically \( \psi^{-}\to e^{i\vec k\cdot\vec r} + f^{*}(\theta)\frac{e^{-ikr}}{r} \): a plane wave plus an incoming spherical wave. Physically, \( \psi^{-}_{\vec k} \) is the state that behaves as a pure plane wave of momentum \( \hbar\vec k \) in the distant future (after collision), i.e. it is the appropriate scattering state for a specified final momentum. It is the time-reverse of \( \psi^{+} \), and the two are related by \( \psi^{-}_{\vec k}=(\psi^{+}_{-\vec k})^{*} \). The overlap \( \langle\psi^{-}_{\vec k'}|\psi^{+}_{\vec k}\rangle \) yields the S-matrix element.