Scattering Operators and Unitarity

The operator formulation puts boundary conditions, repeated interactions, and probability conservation into one framework. Start from an exact scattering equation, choose an approximation to solve it, and then check the resulting amplitudes against unitarity. Keeping these steps distinct explains why an exact identity can be useful even when a truncated approximation does not satisfy it exactly.

Helpful background. Scattering states and observables introduces amplitudes and flux. Projectors and the energy Green function help with operator decompositions and resolvents.

An equation that includes the boundary condition

For $H=H_0+V$, the outgoing Lippmann–Schwinger equation is

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

where $|\phi\rangle$ is the incident free state and $G_0^{(+)}(E)=(E-H_0+i0)^{-1}$ is the outgoing resolvent boundary value. The $i0$ specifies a limiting prescription; it is not an arbitrary physical absorption rate.

The equation says that the total wave is the incident wave plus the wave radiated by the interaction acting on that same total wave. The scattering Green function makes this source interpretation explicit in position space and shows how its far-field behavior gives the amplitude.

From repeated interactions to the Born approximation

The transition operator satisfies $T=V+VG_0^{(+)}T$. Iteration gives the formal Born series,

$$T={}V+VG_0^{(+)}V +VG_0^{(+)}VG_0^{(+)}V+\cdots.$$

Each extra factor of $V$ represents another interaction, with free propagation between interactions. The series needs convergence or a justified finite-order approximation; merely writing the iteration does not establish either.

First Born theory keeps $T\approx V$. For a local potential, its on-shell matrix element becomes a Fourier transform. Born validity explains why weak coupling, range, energy, and nearby poles must all be considered. High energy or a simple potential shape alone is not a universal guarantee.

The $T$ operator is also useful when Born theory is inaccurate: it can be obtained nonperturbatively. Its conversion to a position-space amplitude depends on state normalization, so compare complete conventions rather than isolated coefficients.

What unitarity adds

The S-matrix maps incoming asymptotic states to outgoing ones. For a complete set of open channels in a closed, probability-conserving scattering problem, $S^\dagger S=I$. Unitarity translates this statement into channel sums and partial-wave bounds. An elastic subblock can lose probability to other open channels even though the full matrix is unitary.

The optical theorem expresses the same conservation law through the forward elastic amplitude. In the standard short-range convention for an incident channel of wave number $k$,

$$\sigma_{\mathrm{tot}}=\frac{4\pi}{k}\operatorname{Im}f(0).$$

This does not say that scattering occurs only in the forward direction. Interference between the incident wave and forward scattered wave accounts for flux redirected into all final directions and channels.

For a real potential, the first Born forward amplitude is real, while $|f_{\mathrm B}|^2$ gives a generally nonzero leading cross section. There is no contradiction: that cross section is second order in the interaction, and the corresponding forward imaginary part first appears at second Born order. Check both sides at a consistent perturbative order.

Exercises

Follow the interaction order. Suppose $f=\lambda f_1+\lambda^2f_2+\cdots$, with $\lambda$ real and $f_1(0)$ real. Which term supplies the forward imaginary part needed to balance the leading cross section?

Solution

The leading cross section is $\lambda^2\int |f_1|^2d\Omega$. The matching optical-theorem term is $\lambda^2(4\pi/k)\operatorname{Im}f_2(0)$. Comparing the first-order forward amplitude alone with the second-order cross section mixes orders.

An incomplete channel set. An incoming state has elastic outgoing probability $0.7$ and inelastic outgoing probability $0.3$. Is the full scattering operator nonunitary?

Solution

No violation follows: the total outgoing probability is one. The elastic restriction alone does not preserve norm because it omits the inelastic channel. Full unitarity imposes further orthogonality relations among different incoming-channel columns, not only this one probability sum.

Continue with partial waves and thresholds to use these ideas channel by channel in a central potential.

References

• Lippmann, B. A., and J. Schwinger. “Variational principles for scattering processes. I.” Physical Review 79, 469–480 (1950). doi:10.1103/PhysRev.79.469. The scattering integral-equation framework.
• Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Dover (2006; reprint of the 1972 edition). Resolvents, transition operators, and unitarity.