@kangermu is correct. The general principle is this:

You have an initial state $|i\rangle$, which is modeled as a plane wave incident on some slit pattern.

You detect a final state $|f\rangle$, which is also a plane wave, until it splats on your detector screen.

The probability of observing a final state is:

$$ P_{fi} = M_{fi}^*M_{fi} $$

where $M_{fi}$ is total (complex) amplitude for $|i\rangle$ to transition to $|f\rangle$.

Now here is where you need to make some approximations, maybe.

If there are two slits, (L, R), then:

$$ M_{fi} = M^{(L)}_{fi} + M^{(R)}_{fi} $$

Finding the modulus of that leads to $(M^{(L)*}_{fi}M^{(R)}_{fi})$ terms which make the interference pattern.

Meanwhile, to the experiment properly, the slit width (in $x$) need so be chosen so that:

$$ M^{(\alpha)}_{fi} = \int_{x_{\alpha^-}}^{x_{\alpha^+}} [{\rm (plane\ wave\ in\ z)}] dx $$

with $\alpha \in [L, R]$ so that the slits have mutually overlapping diffraction patterns.

So that is a general coherent sum over all possible paths.

Now you want to introduce a busted up middle path were the beam particle scatters of something into the slit.

AFAIL, if the scattering is elastic than we just add that amplitude into sum over all possible paths.

If it is inelastic, then a record of the path taken could exist, and if it does: the sum over paths goes through one slit and there is no interference pattern.

If the scattering is elastic, and there is no way from not Man, but God, to tell us which way the particle went: you need to sum over all possible paths, and there is a interference pattern.

Note: if the scattering leads to a phase shift (this is equivalent to putting a piece of glass over one slit), then that just shifts the pattern left or right (in $x$). [$z$ is direction of propagation, $x$ is the transverse direction, and $y$ translation symmetry is implied in the set up].

tl;dr: always sum amplitudes over all possible paths. That's it.

See also: https://www.science.org/doi/10.1126/sciadv.aav4020

While many people oh and ah over these results, as my thesis advisor put it: "Quantum mechanics works. So what? We've known this for 100 years already!"

@kangermu is correct. The general principle is this:

You have an initial state $|i\rangle$, which is modeled as a plane wave incident on some slit pattern.

You detect a final state $|f\rangle$, which is also a plane wave (or perhaps a spherical wave emanating from the slit), until it splats on your detector screen.

The probability of observing a final state is:

$$ P_{fi} = M_{fi}^*M_{fi} $$

where $M_{fi}$ is total (complex) amplitude for $|i\rangle$ to transition to $|f\rangle$.

Now here is where you need to make some approximations, maybe.

If there are two slits, ($L, R$), then:

$$ M_{fi} = M^{(L)}_{fi} + M^{(R)}_{fi} $$

Finding the modulus of that leads to a $(M^{(L)*}_{fi}M^{(R)}_{fi})$ terms )and it's complex conjugate) which make the interference pattern.

Meanwhile, to do the experiment properly, the slit width (in $x$) need so be chosen so that:

$$ M^{(\alpha)}_{fi} = \int_{x_{\alpha^-}}^{x_{\alpha^+}} [{\rm (plane\ wave\ in\ z)}] dx $$

with $\alpha \in [L, R]$ so that the slits have mutually overlapping diffraction patterns.

So that is a general coherent sum over all possible paths.

Now you want to introduce a busted up middle path where the beam/particle scatters off something into the slit.

AFAIK, if the scattering is elastic then we just add that amplitude into sum over all possible paths.

If it is inelastic, then a record of the path taken could exist, and if it does: the sum over paths goes through one slit and there is no interference pattern.

If the scattering is elastic, and there is no way from not Man, but God, to tell us which way the particle went: you need to sum over all possible paths, and there is a interference pattern.

Note: if the scattering leads to a phase shift (this is equivalent to putting a piece of glass over one slit), then that just shifts the pattern left or right (in $x$). [$z$ is direction of propagation, $x$ is the transverse direction, and $y$ translation symmetry is implied in the set up].

tl;dr: always sum amplitudes over all possible paths. That's it.

See also: https://www.science.org/doi/10.1126/sciadv.aav4020

While many people oh and ah over these results, as my thesis advisor put it: "Quantum mechanics works. So what? We've known this for 100 years already!"