I'm doing a computational project on neutron scattering and I've found that if you simulate (n,n) collisions on Uranium 238, the program predicts resonances not found in experimental data.

The optical model includes imaginary surface or volume potential terms which supposedly damp out these resonances.

How should an imaginary potential be interpreted in this context?


1 Answer 1


The time-dependent Schrödinger equation (one dimension will do for this discussion) is \begin{align} V(x) \psi - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \psi = i\hbar \frac{\partial}{\partial t} \psi \end{align} Usually you can separate the solution $\psi$ into a spatial part $e^{ikx}$ and a time-varying part $e^{-i\omega t}$, so the right-hand side of the Schrödinger equation gives you $\hbar\omega\psi = E\psi$.

If your problem includes absorption, you want to include that in your wavefunction as well. A term in the wavefunction $\psi \propto e^{-\Gamma t/2\hbar}$ accounts for the fact that the probability of detecting your particle anywhere, \begin{align} P_\text{anywhere} = \int_\infty\!\! dx\ |\psi|^2, \end{align} decreases exponentially, with its time constant $\tau$ inversely proportional to the energy width $\Gamma$.

Following the usual recipe for converting from the time-dependent to time-independent Schödinger equations gives \begin{align} V(x) \psi - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \psi &= \left(\hbar \omega - \frac{i\Gamma}2 \right)\psi \\ \left( V(x) + \frac{i\Gamma}2 \right) \psi - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \psi &= \hbar \omega \psi \\ \end{align} So the effect of a complex potential (at least, as long as the imaginary part is strictly positive) is to describe solutions where absorption plays an important part in the dynamics. This is by far the most common use of complex potentials in the literature, including in neutron optics.

It's not immediately obvious to me how this approach would "damp out resonances," but I can imagine a handwaving argument involving changing narrow, long-lived energy eigenstates from a potential with real $V(x)$ to states with wide intrinsic energy; the probability densities for the short-lived states get smeared over an energy of width $\Gamma$ and therefore don't compete with the short-lived resonances any more.


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