In quantum field theory the LSZ reduction formula gives us a method of calculating S-matrix elements. In order to understand better scattering in QFT, I will study scattering in non-relativistic quantum mechanics and that question ocurred to me.


In QFT, LSZ formula is a tool to obtain S-matrix from correlation function. In QFT correlation function is very easy to calculate(free and interacting theories). When outgoing particles became on-shell then we can relate S-matrix element to correlation function. I QM all the particles are always on-shell. In non-relativistic limit scattering matrix(amplitude) should be compared to Born-approximation. So if we Fourier transforms the $i \cal{M}$ (non-relativistic limit) back into position space, then we can see the behavior of potential.

In QM, if we can calculate correlation function, then we can easily obtain the S-matrix element. Calculation of correlation function is not easy in QM. But on the other hand calculation of S-matrix element is very easy in Born approximation.

But what we can do is to write an action of which Schrodinger equation is just the equation of motion of that action.

$$S = \int_{xt} \psi^\dagger \left(i{\partial \over \partial t} + {\nabla^2\over 2m}\right)\psi - \psi^\dagger(x) \psi(x) V(x).$$ After that we can do the usual trick using generating functional. After taking the functional derivative with respect to current($J$), we can find correlation function. $${\displaystyle Z[J]=\int {\mathcal {D}}\phi e^{i(S[\phi ]+\int d^{d}xJ(x)\phi (x))}~,}$$ Ofcourse, propagator are different in QM as compared to QFT. In this way we can find S-matrix element using correlation function.

So at last, In quantum mechanics, LSZ formula is not much use. But we can take the usual LSZ formula and go into the non-relativistic limit.

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