Lippmann–Schwinger Equation

The Lippmann–Schwinger equation rewrites the stationary scattering problem as a free incoming state plus the wave generated by the potential. It is the formal starting point for scattering Green functions and the Born expansion. Scattering States and Boundary Conditions owns the incoming and outgoing prescriptions; S-Matrix owns the asymptotic operator; T-Matrix owns the transition operator; Green Function for Scattering owns the outgoing-kernel and far-field extraction; Born Series owns repeated iteration and convergence; and Lippmann–Schwinger Equation Preview owns the general resolvent map.

Let

$$H=H_0+V,$$

where $H_0$ is the free Hamiltonian and $V$ is a short-range scattering potential. A scattering state at energy $E$ satisfies

$$H\lvert\psi^{(\pm)}\rangle = E\lvert\psi^{(\pm)}\rangle.$$

The Lippmann–Schwinger equation is

$$\lvert\psi^{(\pm)}\rangle = \lvert\phi\rangle + G_0^{(\pm)}(E) V \lvert\psi^{(\pm)}\rangle,$$

where $\lvert\phi\rangle$ is a free solution at the same energy and

$$G_0^{(\pm)}(E) = \frac{1}{E-H_0\pm i0}$$

is the free resolvent with outgoing or incoming boundary conditions. Its coordinate kernel and normalization are derived in Energy Green Function.

Resolvent Setup

Start from

$$(E-H_0)\lvert\psi\rangle = V\lvert\psi\rangle.$$

If $E-H_0$ were invertible in the ordinary sense, one could multiply by $(E-H_0)^{-1}$. But $E$ lies in the continuous spectrum of $H_0$, so the inverse needs a boundary-condition prescription. This is why the resolvents are written as

$$G_0^{(+)}(E) = \lim_{\epsilon\to0^+} \frac{1}{E-H_0+i\epsilon},$$

and

$$G_0^{(-)}(E) = \lim_{\epsilon\to0^+} \frac{1}{E-H_0-i\epsilon}.$$

The sign selects outgoing or incoming spherical waves at infinity.

Incoming and Outgoing States

The state $\lvert\psi^{(+)}\rangle$ is the physical scattering state built from a specified incoming free state and outgoing scattered radiation:

$$\lvert\psi^{(+)}\rangle = \lvert\phi\rangle + G_0^{(+)}V\lvert\psi^{(+)}\rangle.$$

The state $\lvert\psi^{(-)}\rangle$ has the opposite boundary condition and is useful for formal scattering theory and matrix elements:

$$\lvert\psi^{(-)}\rangle = \lvert\phi\rangle + G_0^{(-)}V\lvert\psi^{(-)}\rangle.$$

The superscripts do not mean positive and negative energy. They label boundary conditions.

Coordinate-Space Form

For a particle in three dimensions with incoming plane wave $e^{i\mathbf k\cdot\mathbf r}$,

$$E = \frac{\hbar^2k^2}{2m}.$$

The outgoing free Green function gives

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

This is an integral equation: the unknown wavefunction appears on both sides.

The sign, source normalization, current direction, and controlled far-field expansion of this kernel are developed in Green Function for Scattering.

Scattering Amplitude

At large $r$ with $\mathbf r'$ in the finite range of the potential,

$$|\mathbf r-\mathbf r'| \approx r-\hat{\mathbf r}\cdot\mathbf r'.$$

Then

$$\frac{ e^{ik|\mathbf r-\mathbf r'|} }{ |\mathbf r-\mathbf r'| } \approx \frac{e^{ikr}}{r} e^{-i\mathbf k'\cdot\mathbf r'}, \qquad \mathbf k'=k\hat{\mathbf r}.$$

Comparing with

$$\psi_{\mathbf k}^{(+)}(\mathbf r) \sim e^{i\mathbf k\cdot\mathbf r} + f(\theta,\phi) \frac{e^{ikr}}{r},$$

one obtains

$$f(\theta,\phi) = - \frac{m}{2\pi\hbar^2} \int d^3r'\, e^{-i\mathbf k'\cdot\mathbf r'} V(\mathbf r') \psi_{\mathbf k}^{(+)}(\mathbf r').$$

This formula is exact within the stated potential-scattering assumptions. It becomes a practical approximation when $\psi_{\mathbf k}^{(+)}$ under the integral is approximated.

Born-Series Handoff

Repeated substitution formally gives

$$\lvert\psi^{(+)}\rangle = \lvert\phi\rangle + G_0^{(+)}V\lvert\phi\rangle + G_0^{(+)}VG_0^{(+)}V\lvert\phi\rangle + \cdots.$$

This equation records where the expansion comes from. Born Series owns its interaction-order interpretation, second-order momentum and coordinate forms, convergence conditions, unitarity bookkeeping, and failure modes.

Rigorous Caveats

Plane waves are not normalizable Hilbert-space vectors, and scattering states live naturally in a continuum-normalized or rigged-Hilbert-space setting. The $i0$ prescription is also a distributional boundary condition, not an ordinary finite matrix inverse.

For practical quantum mechanics, the equation is still extremely useful: it packages boundary conditions, Green functions, and scattering amplitudes into one compact formalism. For mathematical scattering theory, existence and completeness of wave operators require additional hypotheses on $V$.

Common Mistakes

• Interpreting the $+i0$ sign as a small physical absorption rather than an outgoing boundary condition.
• Forgetting that $\lvert\psi^{(+)}\rangle$ appears on both sides of the equation.
• Replacing $\psi^{(+)}$ by the incoming plane wave without recognizing that this is the first Born approximation.
• Applying the short-range asymptotic form without checking long-range Coulomb behavior.
• Treating plane-wave states as square-normalizable bound states.

Exercises

1. Explain why $G_0^{(+)}$ and $G_0^{(-)}$ are different even though both contain the same free Hamiltonian.

Solution

The difference is the boundary-condition prescription:

$$G_0^{(\pm)}(E) = \frac{1}{E-H_0\pm i0}.$$

Because $E$ lies in the continuous spectrum, the limiting prescription matters. The plus sign selects outgoing-wave behavior, while the minus sign selects incoming-wave behavior.

2. Starting from the coordinate-space Lippmann–Schwinger equation, identify the term that produces the outgoing spherical wave.

Solution

The integral term contains

$$\frac{ e^{ik|\mathbf r-\mathbf r'|} }{ |\mathbf r-\mathbf r'| }.$$

At large $r$ this behaves as

$$\frac{e^{ikr}}{r} e^{-i\mathbf k'\cdot\mathbf r'},$$

which has the outgoing spherical factor $e^{ikr}/r$. Its coefficient is the scattering amplitude.

3. What approximation is made when $\psi_{\mathbf k}^{(+)}(\mathbf r')$ is replaced by $e^{i\mathbf k\cdot\mathbf r'}$ inside the amplitude integral?

Solution

That replacement keeps only the incident free wave inside the potential region and neglects rescattering. It is the first Born approximation. It is valid only when the scattered wave generated by the potential is small enough compared with the incident wave in the region that contributes to the integral.

Cross-Links

• Lippmann–Schwinger Equation Preview
• Energy Green Function
• Green Function for Scattering
• Born Series
• Scattering Amplitude
• First Born Approximation
• Cross Sections
• Coulomb Scattering

References

• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.