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When I first learned quantum mechanics almost everything was in terms of wave functions or matrix mechanics, not path integrals. Not having learned much about path integrals besides some brief reading I am struggling to see their benefits or the motivation behind them, besides a desire for an approach based on action/lagrangians. It seems that some problems, like the hydrogen atom would even be more difficult in that formulation. However I would expect there has to be some, probably significant, advantages to using path integrals in certain situations.

What are the advantages (and disadvantages) to the path integral formulation compared to other approaches?

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You are quite right. The path integral is (mostly) useless in non-relativistic quantum mechanics. The main use of the path integral comes from quantum field theory. In QFT, the path integral is taken to be an integral over the field variables (i.e., the value of the field at every point in space-time) as opposed to the position variables in non-relativistic quantum mechanics. There are a few tricks you can do in quantum field theory to easily derive the Feynman rules (where you view the entire path integral as a Gaussian integral, and interactions as perturbations of your Gaussian). Historically, I believe, Feynman didn't consider the field path integral in his approach to QFT. He thought the diagrams themselves were the paths (and indeed, they can be written in this way, although it is quite useless for computational purposes).

Nevertheless, there are still uses for the path integral in non-relativistic quantum mechanics, although they are usually "tricks" to get numbers you want, and not for computing amplitudes. One use of the path integral is for studying instantons, which don't have to be in QFT setting. Another use is that you can express the quantum partition function of a statistical mechanical system using a path integral. (Both of these uses involve time being "Wick Rotated.") A third use of the path integral is that it can be used to find the energy spectrum of a system, even though only approximately. This is done by integrating over the expectation value of 1/energy and finding the poles.

In the case of statistical mechanics, because the partition function can be written as a path integral, often times people use computers to evaluate them. Here is one example: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.67.279

This is a benefit of the path integral in general, actually. If you really just integrate over all of the paths, you can get answers to questions that are too difficult to tackle otherwise. This is the entire field of "Lattice QCD."

Also, apparently you can solve for the hydrogen atom if you really want to. http://users.physik.fu-berlin.de/~kleinert/83/83.pdf

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