Born Approximation and Partial Waves

Derivation

Born Approximation and Partial Waves

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Statement

For a particle of mass \(m\) and energy \(E=\hbar^2 k^2/2m\) scattering off a localised potential \(V(\mathbf r)\), we derive the exact stationary scattering state from the Lippmann–Schwinger equation, extract the scattering amplitude \(f(\theta,\phi)\) whose modulus squared is the differential cross-section \(d\sigma/d\Omega=|f|^2\), reduce it in the first Born approximation to the Fourier transform of the potential, \(f_{\mathrm B}(\mathbf q)=-\dfrac{m}{2\pi\hbar^2}\displaystyle\int e^{i\mathbf q\cdot\mathbf r'}V(\mathbf r')\,d^3r'\) with momentum transfer \(\mathbf q=\mathbf k-\mathbf k'\), and independently expand \(f\) in partial waves \(f(\theta)=\dfrac1k\sum_\ell(2\ell+1)e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta)\), from which the optical theorem \(\sigma_{\text{tot}}=\dfrac{4\pi}{k}\,\mathrm{Im}\,f(0)\) follows.

Why it matters

Scattering is the primary experimental probe of the interactions we cannot see directly: the size of the nucleus, the shape of an atomic potential, the coupling in a Feynman diagram — all are read off from angular distributions of scattered flux. The Born series and the partial-wave decomposition are the two complementary languages in which every such measurement is interpreted, one perturbative and momentum-space, the other exact and angular-momentum-space.

The optical theorem ties the two together and enforces conservation of probability: the total loss of flux from the forward beam is fixed by the imaginary part of the forward amplitude. This single identity underpins the analysis of everything from neutron transmission to hadron total cross-sections, and it is the cleanest place a wrong sign or a dropped factor announces itself.

Assumptions

The potential is short-ranged, \(rV(r)\to 0\) faster than \(1/r\).If dropped (e.g. bare Coulomb), the asymptotic scattered wave is not a clean spherical wave \(e^{ikr}/r\); logarithmic phase distortions appear and \(f(\theta)\) as defined does not exist without screening/regularisation.

The scattering is elastic and time-independent; a single fixed energy \(E>0\).If dropped, flux is lost to inelastic channels, \(S\)-matrix eigenvalues have modulus \(<1\), and the real phase shift \(\delta_\ell\) must be promoted to a complex quantity.

The incident state is a plane wave \(e^{i\mathbf k\cdot\mathbf r}\) of sharp momentum.If dropped (finite wave-packet), one must convolve \(f\) with the momentum spread; for a well-collimated monoenergetic beam this is a negligible correction.

Born: the scattered wave is a small correction to the incident wave inside the interaction region.If dropped, replacing \(\psi\to e^{i\mathbf k\cdot\mathbf r'}\) in the integral is illegitimate; the first Born term can be off by orders of magnitude near resonances or for deep/strong potentials, and the Born series may not converge.

Central potential \(V=V(r)\) (for the partial-wave section only).If dropped, angular momentum \(\ell\) is not conserved and the \(S\)-matrix is no longer diagonal in \(\ell\); a single real \(\delta_\ell\) per channel no longer suffices and off-diagonal coupling \(S_{\ell'\ell}\) appears.

Derivation

\[ \left(\nabla^2+k^2\right)\psi(\mathbf r)=U(\mathbf r)\,\psi(\mathbf r),\qquad U(\mathbf r)\equiv\frac{2m}{\hbar^2}V(\mathbf r),\quad k^2=\frac{2mE}{\hbar^2} \]

Rewrite the time-independent Schrödinger equation \(\left(-\tfrac{\hbar^2}{2m}\nabla^2+V\right)\psi=E\psi\) as an inhomogeneous Helmholtz equation, collecting the potential into the reduced potential \(U\). A

\[ \left(\nabla^2+k^2\right)G_0^{+}(\mathbf r-\mathbf r')=\delta^{3}(\mathbf r-\mathbf r'),\qquad G_0^{+}(\mathbf R)=-\frac{e^{ikR}}{4\pi R} \]

Introduce the outgoing Green's function of the Helmholtz operator (prior result: Green's function of the Helmholtz equation); the \(+\) branch enforces purely outgoing scattered waves. A

\[ \psi(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}+\int G_0^{+}(\mathbf r-\mathbf r')\,U(\mathbf r')\,\psi(\mathbf r')\,d^3r' \]

General solution = any solution of the homogeneous equation (the incident plane wave, satisfying \((\nabla^2+k^2)e^{i\mathbf k\cdot\mathbf r}=0\)) plus the particular solution built from \(G_0^{+}\). This is the Lippmann–Schwinger equation. B

\[ |\mathbf r-\mathbf r'|=r\sqrt{1-\frac{2\hat{\mathbf r}\cdot\mathbf r'}{r}+\frac{r'^2}{r^2}}\;\xrightarrow{r\to\infty}\;r-\hat{\mathbf r}\cdot\mathbf r'+O(1/r) \]

Take the detector far outside the range of \(V\): expand \(|\mathbf r-\mathbf r'|\) for \(r\gg r'\), keeping the linear term in the phase (which survives \(\times k r\)) and only the leading term \(1/r\) in the amplitude. B

\[ \psi(\mathbf r)\;\xrightarrow{r\to\infty}\;e^{i\mathbf k\cdot\mathbf r}+f(\theta,\phi)\,\frac{e^{ikr}}{r},\qquad f(\theta,\phi)=-\frac{1}{4\pi}\int e^{-i\mathbf k'\cdot\mathbf r'}\,U(\mathbf r')\,\psi(\mathbf r')\,d^3r' \]

Substitute the asymptotics of Step 4 into Step 3 with \(\mathbf k'\equiv k\hat{\mathbf r}\) the outgoing wavevector; this defines the scattering amplitude \(f\) and gives the canonical scattering boundary form. B

\[ \mathbf j=\frac{\hbar}{m}\,\mathrm{Im}\!\left(\psi^{*}\nabla\psi\right)\;\Rightarrow\;\frac{d\sigma}{d\Omega}=\frac{r^2 j_{\text{sc}}}{j_{\text{inc}}}=|f(\theta,\phi)|^2 \]

The differential cross-section is (radial scattered flux through \(r^2 d\Omega\))/(incident flux). The incident \(j_{\text{inc}}=\hbar k/m\) and scattered \(j_{\text{sc}}=(\hbar k/m)|f|^2/r^2\) cancel all prefactors. A

\[ f_{\mathrm B}(\theta,\phi)=-\frac{1}{4\pi}\int e^{-i\mathbf k'\cdot\mathbf r'}\,U(\mathbf r')\,e^{i\mathbf k\cdot\mathbf r'}\,d^3r' =-\frac{m}{2\pi\hbar^2}\int e^{i\mathbf q\cdot\mathbf r'}\,V(\mathbf r')\,d^3r',\quad \mathbf q\equiv\mathbf k-\mathbf k' \]

First Born approximation: under the weak-scattering assumption replace the exact \(\psi\) in Step 5 by the incident plane wave. The amplitude becomes the Fourier transform of \(V\) at momentum transfer \(\mathbf q\). B

\[ |\mathbf q|=|\mathbf k-\mathbf k'|=\sqrt{k^2+k^2-2k^2\cos\theta}=2k\sin\!\frac{\theta}{2} \]

Elastic scattering means \(|\mathbf k|=|\mathbf k'|=k\); the momentum-transfer magnitude follows from the law of cosines with scattering angle \(\theta\) between \(\mathbf k\) and \(\mathbf k'\). A

\[ f_{\mathrm B}(q)=-\frac{2m}{\hbar^2 q}\int_0^{\infty} r\,V(r)\,\sin(qr)\,dr \]

For a central potential do the angular integral in Step 7: \(\int e^{i\mathbf q\cdot\mathbf r'}d\Omega'=4\pi\,\dfrac{\sin(qr')}{qr'}\). The amplitude depends only on \(q\), hence only on \(\theta\). B

\[ e^{i\mathbf k\cdot\mathbf r}=\sum_{\ell=0}^{\infty}(2\ell+1)\,i^{\ell}\,j_\ell(kr)\,P_\ell(\cos\theta),\qquad \psi=\sum_\ell \frac{u_\ell(r)}{r}P_\ell(\cos\theta) \]

Switch to the exact, non-perturbative route: for a central potential expand both incident and full wavefunctions in Legendre polynomials (prior result: spherical harmonics / orbital angular momentum). Each \(\ell\) decouples. B

\[ \frac{u_\ell(r)}{r}\;\xrightarrow{r\to\infty}\;\frac{1}{kr}\sin\!\left(kr-\frac{\ell\pi}{2}+\delta_\ell\right) \]

Outside the range of \(V\) each radial equation is free, so \(u_\ell/r\) is a combination of \(j_\ell\) and \(n_\ell\); the sole effect of the potential is a shift \(\delta_\ell\) of the asymptotic phase. This defines the phase shift. C

\[ f(\theta)=\frac{1}{2ik}\sum_{\ell=0}^{\infty}(2\ell+1)\left(e^{2i\delta_\ell}-1\right)P_\ell(\cos\theta) =\frac{1}{k}\sum_{\ell}(2\ell+1)\,e^{i\delta_\ell}\sin\delta_\ell\,P_\ell(\cos\theta) \]

Match the partial-wave expansion of \(\psi=e^{i\mathbf k\cdot\mathbf r}+f\,e^{ikr}/r\): the coefficient of the outgoing wave \(e^{ikr}/r\) in channel \(\ell\) is \(S_\ell=e^{2i\delta_\ell}\), and subtracting the incident part isolates \(f\). C

\[ \sigma_{\text{tot}}=\int|f|^2\,d\Omega=\frac{4\pi}{k^2}\sum_{\ell=0}^{\infty}(2\ell+1)\sin^2\delta_\ell \]

Integrate \(|f|^2\) using orthogonality \(\int P_\ell P_{\ell'}d\Omega=\dfrac{4\pi}{2\ell+1}\delta_{\ell\ell'}\); cross terms vanish, leaving one term per partial wave. B

\[ \mathrm{Im}\,f(0)=\frac{1}{k}\sum_\ell(2\ell+1)\sin^2\delta_\ell\,P_\ell(1)=\frac{1}{k}\sum_\ell(2\ell+1)\sin^2\delta_\ell \;\Rightarrow\; \boxed{\;\sigma_{\text{tot}}=\frac{4\pi}{k}\,\mathrm{Im}\,f(0)\;} \]

Set \(\theta=0\) so \(P_\ell(1)=1\), take the imaginary part of Step 12, and compare with Step 13. The optical theorem is a direct consequence of unitarity of the \(S\)-matrix. C

Result

\[ \frac{d\sigma}{d\Omega}=|f(\theta,\phi)|^2,\qquad f_{\mathrm B}(\mathbf q)=-\frac{m}{2\pi\hbar^2}\int e^{i\mathbf q\cdot\mathbf r'}V(\mathbf r')\,d^3r',\qquad \sigma_{\text{tot}}=\frac{4\pi}{k^2}\sum_\ell(2\ell+1)\sin^2\delta_\ell=\frac{4\pi}{k}\,\mathrm{Im}\,f(0) \]

Reading. The scattered intensity in a direction is the squared amplitude \(f\). In the Born (weak, fast) regime that amplitude is simply the Fourier transform of the potential evaluated at the momentum transferred to the particle: forward scattering (\(q\to0\)) probes the volume integral of \(V\), while large-angle scattering (\(q\) large) probes short-distance structure. The exact, complementary statement is the partial-wave sum, where all the physics of channel \(\ell\) is compressed into one real number \(\delta_\ell\); the optical theorem then fixes the total cross-section by the forward amplitude alone, a bookkeeping of the flux removed from the incident beam.

Units check. \(f\) carries dimensions of length: in \(f_{\mathrm B}\), \([m/\hbar^2]=\mathrm{kg\,J^{-2}s^{-2}}=\mathrm{kg\,(kg\,m^2 s^{-2})^{-2}s^{-2}}=\mathrm{kg^{-1}m^{-4}s^{2}}\); times \([V\,d^3r]=\mathrm{J\,m^3}=\mathrm{kg\,m^{5}s^{-2}}\) gives \(\mathrm{m}\). Hence \(d\sigma/d\Omega=|f|^2\) has units \(\mathrm{m^2\,sr^{-1}}\) and \(\sigma\) units \(\mathrm{m^2}\). In the partial-wave form \([1/k^2]=\mathrm{m^2}\) and \(\sin^2\delta_\ell\) is dimensionless, consistent. Optical theorem: \([1/k]\cdot[\mathrm{Im}\,f]=\mathrm{m\cdot m}=\mathrm{m^2}\). ✓

Limiting cases

Low energy, \(k\to0\): only \(\ell=0\) survives (centrifugal barrier freezes higher \(\ell\)); \(f\to -a_s\), a single scattering length, and \(\sigma_{\text{tot}}\to4\pi a_s^2\), isotropic.
Screened Coulomb (Yukawa) \(V=\beta e^{-\mu r}/r\): \(f_{\mathrm B}=-\dfrac{2m\beta}{\hbar^2(\mu^2+q^2)}\); as \(\mu\to0\) this reproduces the exact Rutherford result \(d\sigma/d\Omega=(\beta/4E)^2\csc^4(\theta/2)\).
High energy / small angle: \(q=2k\sin(\theta/2)\) is small only near \(\theta=0\), so fast particles scatter predominantly forward into a cone of width \(\sim1/(k\,R)\) set by the potential range \(R\).
Weak potential: the Born phase shift \(\tan\delta_\ell\approx-\dfrac{2mk}{\hbar^2}\displaystyle\int_0^\infty [j_\ell(kr)]^2 V(r)\,r^2\,dr\) links the two formalisms; attractive \(V<0\) gives \(\delta_\ell>0\).
Resonance, \(\delta_\ell\to\pi/2\): that partial wave saturates its unitarity bound \(\sigma_\ell^{\max}=\dfrac{4\pi}{k^2}(2\ell+1)\); the cross-section peaks sharply — inaccessible to first Born.

Breaks when

Strong or resonant potentials. The first Born approximation replaces \(\psi\) by the incident wave; near a bound state at threshold or a shape resonance the true wavefunction is enormously enhanced inside the potential, and \(f_{\mathrm B}\) can be wrong by orders of magnitude. It also cannot produce \(\delta_\ell=\pi/2\).
Long-range Coulomb. With \(V\sim1/r\) the assumption \(rV\to0\) fails; the scattered wave carries a logarithmic phase \(e^{i(kr-\eta\ln 2kr)}/r\), the partial-wave sum diverges, and \(f(\theta)\) is not defined without screening. The Born series for pure Coulomb is only accidentally (numerically) correct at leading order.
Inelastic thresholds. Above the energy where new channels open, probability leaks out; \(|S_\ell|<1\), \(\delta_\ell\) becomes complex, and the elastic optical theorem in the form \(\sigma_{\text{el}}=\tfrac{4\pi}{k}\mathrm{Im}f(0)\) must be replaced by \(\sigma_{\text{tot}}=\sigma_{\text{el}}+\sigma_{\text{inel}}\).
Slowly convergent partial-wave sum. When \(kR\gg1\) (short wavelength, wide potential) thousands of \(\ell\) contribute; truncating the sum, or using it at all, becomes impractical and a semiclassical/eikonal treatment is preferred.

Failure modes

Dropping the \(2\ell+1\) weight in either the sum for \(\sigma\) or the amplitude — the degeneracy of the \(\ell\)-multiplet is not optional.
Using \(k\) where \(k^2\) belongs in \(\sigma_{\text{tot}}=\tfrac{4\pi}{k^2}\sum(2\ell+1)\sin^2\delta_\ell\) but \(\tfrac{4\pi}{k}\mathrm{Im}f(0)\) — one power of \(k\) differs precisely because \(\mathrm{Im}f\) already carries a \(1/k\).
Confusing momentum transfer \(q=2k\sin(\theta/2)\) with the wavenumber \(k\), giving the wrong argument in the Fourier transform.
Squaring before summing: writing \(\sigma=\tfrac{4\pi}{k^2}\big(\sum\ldots\big)^2\) instead of summing \(\sin^2\delta_\ell\) — orthogonality kills the cross terms only after integration.
Sign of the phase shift: asserting attractive \(\Rightarrow\delta_\ell<0\). Attractive potentials pull the wave in, advancing the phase, so \(\delta_\ell>0\).
Forgetting the \(-m/2\pi\hbar^2\) versus \(-1/4\pi\) bookkeeping, i.e. mixing up \(U=2mV/\hbar^2\) with \(V\) in the amplitude, producing a factor \(2m/\hbar^2\) error.
Applying first Born to Coulomb without screening and being surprised the integral diverges at \(q\to0\).

Discussion

The two derivations tell the same story in dual bases. The Born series is an expansion in powers of the potential — physically, in the number of times the particle interacts: \(f=f^{(1)}+f^{(2)}+\dots\), where the second term \(\sim\int U G_0^+ U\) is a double scattering, and so on. It is natural when the potential is weak or the energy high, and it lives in momentum space, where \(f_{\mathrm B}(\mathbf q)\) is literally the Fourier transform of \(V\). This is why electron and neutron diffraction directly image charge and nuclear density distributions: the measured angular pattern is the transform.

The partial-wave expansion, by contrast, is exact and organises the problem by angular momentum. Each channel contributes an independent unitary phase \(e^{2i\delta_\ell}\); the entire content of the interaction, for a central potential, is the set \(\{\delta_\ell(E)\}\). At low energy the centrifugal barrier \(\hbar^2\ell(\ell+1)/2mr^2\) suppresses all but \(\ell=0\), and scattering collapses to a single number, the scattering length — the organising quantity of cold-atom and nuclear low-energy physics.

The optical theorem is the hinge. It is not a dynamical statement but a consequence of unitarity: the forward amplitude must have exactly the imaginary part needed to account, by interference of the scattered wave with the incident beam, for every particle removed from the forward direction. That is why a purely real first-Born amplitude — which has \(\mathrm{Im}\,f_{\mathrm B}(0)=0\) — appears to violate it: the missing imaginary part is precisely the second-order Born term, restoring the balance order by order. The theorem thereby diagnoses the internal consistency of any approximate amplitude.

Analytic structure ties everything together at the deepest level. Continued to complex energy, the \(S\)-matrix element \(S_\ell(E)=e^{2i\delta_\ell(E)}\) has poles on the positive-imaginary \(k\)-axis at bound states and, just below the real axis, at resonances; Levinson's theorem \(\delta_\ell(0)-\delta_\ell(\infty)=n_\ell\pi\) counts the bound states of channel \(\ell\) by the total phase travelled. The Born amplitude is the first term of a series whose convergence is governed by the nearest such singularity, which is why proximity to a bound state or resonance is exactly where first Born fails.

Common misconceptions. The Born approximation is not a low-energy approximation — it is a weak-scattering approximation, and in fact improves at high energy where the particle spends less time in the potential. The phase shift is not the phase of \(f\); it is the phase advance of the radial wave far from the target. And "cross-section" is an area only dimensionally: it is the effective target size the flux argument assigns to the interaction, not a geometric footprint.

Worked examples

Example 1 — Born cross-section for a screened Coulomb (Yukawa) potential. An electron of energy \(E=100\ \mathrm{eV}\) scatters off a screened nuclear charge \(V(r)=\dfrac{e^2}{4\pi\varepsilon_0}\dfrac{e^{-\mu r}}{r}\) with screening length \(1/\mu=a_0\) (Bohr radius). Find \(d\sigma/d\Omega\) at \(\theta=60^\circ\).

\[ f_{\mathrm B}(q)=-\frac{2m}{\hbar^2 q}\int_0^\infty r\,\frac{\beta e^{-\mu r}}{r}\sin(qr)\,dr =-\frac{2m\beta}{\hbar^2 q}\cdot\frac{q}{\mu^2+q^2}=-\frac{2m\beta}{\hbar^2(\mu^2+q^2)},\quad\beta\equiv\frac{e^2}{4\pi\varepsilon_0} \]

Insert the Yukawa form into Step 9 and use \(\int_0^\infty e^{-\mu r}\sin(qr)\,dr=q/(\mu^2+q^2)\). Symbols first. B

\[ \frac{2m\beta}{\hbar^2}=\frac{2m e^2}{4\pi\varepsilon_0\hbar^2}=\frac{2}{a_0},\qquad a_0=\frac{4\pi\varepsilon_0\hbar^2}{me^2}=5.29\times10^{-11}\ \mathrm{m} \]

Recognise the Bohr radius, so \(f_{\mathrm B}=-\dfrac{2/a_0}{\mu^2+q^2}\) with \(\mu=1/a_0\). Clean grouping before numbers. A

\[ k=\frac{\sqrt{2mE}}{\hbar}=\frac{\sqrt{2(9.11\times10^{-31})(100\cdot1.602\times10^{-19})}}{1.055\times10^{-34}}=5.12\times10^{10}\ \mathrm{m^{-1}} \]

Convert energy to wavenumber; \(E=100\ \mathrm{eV}=1.602\times10^{-17}\ \mathrm J\). A

\[ q=2k\sin\frac\theta2=2(5.12\times10^{10})\sin30^\circ=5.12\times10^{10}\ \mathrm{m^{-1}},\quad q^2=2.62\times10^{21}\ \mathrm{m^{-2}} \]

Momentum transfer from Step 8 at \(\theta=60^\circ\). A

\[ \mu^2=\frac{1}{a_0^2}=3.57\times10^{20}\ \mathrm{m^{-2}},\qquad \mu^2+q^2=2.98\times10^{21}\ \mathrm{m^{-2}},\qquad \frac{2}{a_0}=3.78\times10^{10}\ \mathrm{m^{-1}} \]

Assemble the pieces. A

\[ f_{\mathrm B}=-\frac{3.78\times10^{10}}{2.98\times10^{21}}=-1.27\times10^{-11}\ \mathrm m \]

Numerical amplitude; length dimension confirmed. A

\[ \frac{d\sigma}{d\Omega}\bigg|_{60^\circ}=|f_{\mathrm B}|^2=1.61\times10^{-22}\ \mathrm{m^2\,sr^{-1}}\approx1.6\times10^{6}\ \mathrm{barn/sr} \]

Reading. A large forward-peaked cross-section, characteristic of the Coulomb-like \(1/(\mu^2+q^2)^2\) fall-off; screening tames the \(q\to0\) divergence that bare Coulomb would have.

Example 2 — Hard-sphere partial waves at low energy. A neutron scatters off an impenetrable sphere of radius \(a=2.0\ \mathrm{fm}\), with \(V=\infty\) for \(r<a\) and \(0\) outside, in the low-energy limit \(ka=0.10\). Find the \(s\)-wave phase shift and the total cross-section.

\[ \frac{u_0(a)}{a}=0\;\Rightarrow\;\sin(ka+\delta_0)=0\;\Rightarrow\;\delta_0=-ka \]

The wavefunction must vanish at the impenetrable surface; the \(\ell=0\) asymptotic form \(\propto\sin(kr+\delta_0)\) is exact everywhere outside, so the boundary condition fixes \(\delta_0\) directly. B

\[ \delta_0=-ka=-0.10\ \mathrm{rad},\qquad \sin^2\delta_0=\sin^2(0.10)=9.97\times10^{-3} \]

Numbers in. Higher \(\ell\) give \(\delta_\ell\sim(ka)^{2\ell+1}\), negligible here. A

\[ k=\frac{ka}{a}=\frac{0.10}{2.0\times10^{-15}\ \mathrm m}=5.0\times10^{13}\ \mathrm{m^{-1}},\qquad k^2=2.5\times10^{27}\ \mathrm{m^{-2}} \]

Extract \(k\) from the dimensionless \(ka\). A

\[ \sigma_{\text{tot}}\approx\frac{4\pi}{k^2}\sin^2\delta_0=\frac{4\pi}{2.5\times10^{27}}(9.97\times10^{-3})=5.01\times10^{-29}\ \mathrm{m^2} \]

Keep only \(\ell=0\) in Step 13. A

\[ 4\pi a^2=4\pi(2.0\times10^{-15})^2=5.03\times10^{-29}\ \mathrm{m^2} \]

Cross-check against the exact low-energy hard-sphere result \(\sigma\to4\pi a^2\) — four times the geometric area \(\pi a^2\). B

\[ \delta_0=-0.10\ \mathrm{rad},\qquad \sigma_{\text{tot}}=5.0\times10^{-29}\ \mathrm{m^2}=0.50\ \mathrm{barn} \]

Reading. At low energy the sphere looks four times its geometric size to the wave; the negative phase shift signals the repulsive (excluding) core, which pushes the wave outward.

Problems

Born amplitude for a square well. For \(V(r)=-V_0\) (\(r<a\)), \(0\) otherwise, compute \(f_{\mathrm B}(q)\) and its forward value \(f_{\mathrm B}(0)\).
Solution
From Step 9, \(f_{\mathrm B}=\dfrac{2mV_0}{\hbar^2 q}\displaystyle\int_0^a r\sin(qr)\,dr\). Using \(\int_0^a r\sin(qr)\,dr=\dfrac{\sin(qa)}{q^2}-\dfrac{a\cos(qa)}{q}\), \[ f_{\mathrm B}(q)=\frac{2mV_0}{\hbar^2}\,\frac{\sin(qa)-qa\cos(qa)}{q^3}. \] Forward limit: expand \(\sin(qa)\to qa-\tfrac{(qa)^3}{6}\) and \(qa\cos(qa)\to qa-\tfrac{(qa)^3}{2}\); their difference is \(\tfrac{(qa)^3}{3}\), so \(f_{\mathrm B}(0)=\dfrac{2mV_0}{\hbar^2}\cdot\dfrac{a^3}{3}=\dfrac{2mV_0 a^3}{3\hbar^2}\), i.e. proportional to the volume integral \(\int V\,d^3r=-\tfrac43\pi a^3 V_0\) as expected (\(f_{\mathrm B}(0)=-\tfrac{m}{2\pi\hbar^2}\int V\,d^3r\)).
Optical theorem as a check. Verify explicitly that \(\sigma_{\text{tot}}=\tfrac{4\pi}{k^2}\sum_\ell(2\ell+1)\sin^2\delta_\ell\) and \(\tfrac{4\pi}{k}\mathrm{Im}\,f(0)\) agree, and explain why first-Born \(f_{\mathrm B}\) (real) seems to violate it.
Solution
Take \(\mathrm{Im}\) of \(f(\theta)=\tfrac1k\sum(2\ell+1)e^{i\delta_\ell}\sin\delta_\ell P_\ell(\cos\theta)\): since \(\mathrm{Im}(e^{i\delta_\ell}\sin\delta_\ell)=\sin^2\delta_\ell\), at \(\theta=0\) (\(P_\ell(1)=1\)) we get \(\mathrm{Im}f(0)=\tfrac1k\sum(2\ell+1)\sin^2\delta_\ell\). Multiplying by \(4\pi/k\) gives \(\tfrac{4\pi}{k^2}\sum(2\ell+1)\sin^2\delta_\ell=\sigma_{\text{tot}}\). ✓ First Born gives a real \(f_{\mathrm B}\) (the Fourier transform of a real, even-in-\(\mathbf r\) potential is real), so \(\mathrm{Im}f_{\mathrm B}(0)=0\); the required imaginary part first appears at second order, \(\mathrm{Im}f^{(2)}(0)=\tfrac{k}{4\pi}\int|f^{(1)}|^2 d\Omega\), which restores the theorem order-by-order.
Rutherford from Born. Starting from the Yukawa result \(f_{\mathrm B}=-\dfrac{2m\beta}{\hbar^2(\mu^2+q^2)}\), take \(\mu\to0\) and show \(\dfrac{d\sigma}{d\Omega}=\left(\dfrac{\beta}{4E}\right)^2\csc^4(\theta/2)\).
Solution
As \(\mu\to0\), \(f_{\mathrm B}\to-\dfrac{2m\beta}{\hbar^2 q^2}\). With \(q^2=4k^2\sin^2(\theta/2)\) and \(\hbar^2k^2=2mE\): \[ f_{\mathrm B}=-\frac{2m\beta}{\hbar^2\cdot4k^2\sin^2(\theta/2)}=-\frac{\beta}{4E\sin^2(\theta/2)}. \] Therefore \(\dfrac{d\sigma}{d\Omega}=|f_{\mathrm B}|^2=\dfrac{\beta^2}{16E^2\sin^4(\theta/2)}=\left(\dfrac{\beta}{4E}\right)^2\csc^4\dfrac\theta2\), the exact Rutherford formula — a famous accident of the Coulomb potential.
Unitarity bound and resonance. Show the maximum contribution of one partial wave to \(\sigma_{\text{tot}}\) is \(\tfrac{4\pi}{k^2}(2\ell+1)\), attained at \(\delta_\ell=\pi/2\). Evaluate for \(\ell=0\) at neutron energy \(E=1.0\ \mathrm{eV}\) (mass \(1.675\times10^{-27}\ \mathrm{kg}\)).
Solution
\(\sigma_\ell=\tfrac{4\pi}{k^2}(2\ell+1)\sin^2\delta_\ell\) is maximised when \(\sin^2\delta_\ell=1\), i.e. \(\delta_\ell=\pi/2\), giving \(\sigma_\ell^{\max}=\tfrac{4\pi}{k^2}(2\ell+1)\). For \(\ell=0\): \(k=\sqrt{2mE}/\hbar=\sqrt{2(1.675\times10^{-27})(1.602\times10^{-19})}/(1.055\times10^{-34})=2.20\times10^{11}\ \mathrm{m^{-1}}\). Then \(\sigma_0^{\max}=\tfrac{4\pi}{(2.20\times10^{11})^2}=2.6\times10^{-22}\ \mathrm{m^2}=2.6\times10^{6}\ \mathrm{barn}\). Resonances approach this ceiling; first Born, which never produces \(\delta_\ell=\pi/2\), cannot describe them.
Validity of first Born. For a well of depth \(V_0\) and range \(a\), estimate the low-energy validity condition on \(V_0\) and evaluate the threshold depth for \(a=2.0\ \mathrm{fm}\), nucleon mass \(1.67\times10^{-27}\ \mathrm{kg}\).
Solution
First Born requires the scattered wave to be small inside the potential: from the Lippmann–Schwinger equation the correction at the origin is \(\sim\left|\int G_0^+ U\,d^3r\right|\). At low energy (\(k\to0\), \(G_0^+\to-1/4\pi r\)) this is \(\sim\dfrac{2m}{\hbar^2}\dfrac{1}{4\pi}\displaystyle\int\dfrac{V_0}{r}d^3r\sim\dfrac{mV_0 a^2}{\hbar^2}\). Validity: \(\dfrac{2m|V_0|a^2}{\hbar^2}\ll1\). Threshold \(V_0^\ast\) where this is \(\sim1\): \(V_0^\ast\approx\dfrac{\hbar^2}{2ma^2}=\dfrac{(1.055\times10^{-34})^2}{2(1.67\times10^{-27})(2.0\times10^{-15})^2}=8.3\times10^{-13}\ \mathrm J=5.2\ \mathrm{MeV}\). Real nuclear wells (\(\sim35\ \mathrm{MeV}\)) far exceed this, which is exactly why the Born approximation is inadequate for nuclear bound-state problems and partial-wave/variational methods are used instead.