Is there an analogue of the LSZ reduction formula in quantum mechanics?

Question

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.

Answer 1

By Huygen's principle we can write the wave function at some later time $t_f > t_i$ at a point $\mathbf{x}_f$ in terms of the wave function at some initial time $t_i$ at a point $\mathbf{x}_i$ as

$$\theta(t_f -t_i)\Psi(\mathbf{x}_f,t_f) = i \int d^3 \mathbf{x}_i G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)\psi_I(\mathbf{x}_i,t_i)$$

where $G(x_f:x_i) = 0$ when $t_f < t_i$.

(If this is unfamiliar, a simple way to appreciate it is to consider the stationary state expansion $\Psi(\mathbf{x},t) = \sum_n c_n \psi_n(\mathbf{x}_f) e^{-iE_n t}$, solve for $c_n = \int d \mathbf{x}_f \Psi(\mathbf{x},t) \psi_n^*(\mathbf{x}) e^{iE_n t}$ and then insert $c_n$ at some earlier time $t_1$ integrated over $\mathbf{x}_i$ into $\Psi(\mathbf{x}_f,t_f)$, and multiply both sides by $\theta(t_f - t_i)$ explicitly to ensure both sides vanish for $t_f < t_i$, which gives $G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i) = (-i) \theta(t_f - t_i) \sum_n \psi_n(\mathbf{x}_f) \psi_n^*(\mathbf{x}_i) e^{-iE_n(t_f - t_i)}$).

We can then form

$$ S_{FI} = \int d^3\mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f)\theta(t_f -t_i)\Psi(\mathbf{x}_f,t_f) = \int d^3 \mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f) i \int d^3 \mathbf{x}_i G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)\psi_I(\mathbf{x}_i,t_i) $$

and re-write this in LSZ form using $$ i \frac{\partial }{\partial t} \psi_I(x) = - \frac{\nabla^2}{2m} \psi_I(x) \ \ , \ \ - i \frac{\partial }{\partial t} \psi_I^*(x) = - \frac{\nabla^2}{2m} \psi_I^*(x) $$ by writing (recall $G(x_f:x_i) = 0$ when $t_f < t_i$, thus we get the $-$ sign) \begin{align*} \begin{split} S_{FI} &= \int d^3 \mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f) i \int d^3 \mathbf{x}_i G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)\psi_I(\mathbf{x}_i,t_i) \\ &= \int d^3 \mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f) (-i) \int d^4 x_i \frac{\partial}{\partial t_i} [ G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)\psi_I(\mathbf{x}_i,t_i)] \\ &= \int d^3 \mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f) \int d^4 x_i [(-i) \frac{\partial}{\partial t_i} G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)\psi_I(\mathbf{x}_i,t_i) + G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)(-i)\frac{\partial}{\partial t_i} \psi_I(\mathbf{x}_i,t_i)] \\ &= \int d^3 \mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f) \int d^4 x_i [ (-i) \frac{\partial}{\partial t_i} G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)\psi_I(\mathbf{x}_i,t_i) + G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i) \frac{\nabla_i^2}{2m} \psi_I(\mathbf{x}_i,t_i)] \\ &= \int d^3 \mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f) \int d^4 x_i [ (-i) \frac{\partial}{\partial t_i} G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)\psi_I(\mathbf{x}_i,t_i) + \frac{\nabla_i^2}{2m} G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i) \psi_I(\mathbf{x}_i,t_i)] \\ &= \int d^4 x_i \int d^3 \mathbf{x}_f \psi_F^*(\mathbf{x}_f,t_f) G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i) \overleftarrow{[ (-i) \frac{\partial}{\partial t_i} + \frac{\nabla_i^2}{2m} ]} \psi_I(\mathbf{x}_i,t_i) \\ &= (-i) \int d^4 x_i \int d^4 x_f i \frac{\partial}{\partial t_f} [\psi_F^*(\mathbf{x}_f,t_f) G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i)] \overleftarrow{[ (-i) \frac{\partial}{\partial t_i} + \frac{\nabla_i^2}{2m} ]} \psi_I(\mathbf{x}_i,t_i) \\ &= (-i) \int d^4 x_i \int d^4 x_f i \psi_F^*(\mathbf{x}_f,t_f) \overrightarrow{[ i \frac{\partial}{\partial t_f} + \frac{\nabla_f^2}{2m} ]} G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i) \overleftarrow{[ (-i) \frac{\partial}{\partial t_i} + \frac{\nabla_i^2}{2m} ]} \psi_I(\mathbf{x}_i,t_i) \\ &= (-i) \int d^4 x_i \int d^4 x_f i \psi_F^*(\mathbf{x}_f,t_f) \psi_I(\mathbf{x}_i,t_i) [ i \frac{\partial}{\partial t_f} + \frac{\nabla_f^2}{2m} ] [ (-i) \frac{\partial}{\partial t_i} + \frac{\nabla_i^2}{2m} ] G(\mathbf{x}_f,t_f;\mathbf{x}_i,t_i) . \end{split} \end{align*}

Thus it's basically at the core of propagator theory. This is written in terms of wave functions, obviously it applies to single particle quantum field operators also. Thus it technically applies to creation and annihilation operators also, and so you can see why the usual LSZ formula derivation will end up looking like it does. The link to path integrals can similarly be seen in a few lines [2]

On sending $ [ i \frac{\partial}{\partial t_f} + \frac{\nabla_f^2}{2m} ] \to (\partial^2 + m^2)$ you get the Klein-Gordon LSZ. I haven't seen nor tried to use this in this form in practice non-relativistically, and references that do so would be interesting to read if anybody has any.

References:

1. Bjorken and Drell, Relativistic Quantum Mechanics and Relativistic Quantum Fields, Ch. 6 (Eq. 6.1) and Ch. 16 (Eq. 16.82).
2. Sadovskii, Quantum Field Theory, Ch. 9.

Answer 2

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.