When we studied classical mechanics on the undergraduate level, on the level of

Question

When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics.

Now when we studied QM, on the level of Griffiths, we always dealt with Hamiltonians not Lagrangians. Why is that? I know that QFT and particle physics rely instead on Lagrangian!!

Could someone provide some insights on why, it seems that, "only" Hamiltonians are used in undergrad QM.

Also it would be great if providing some insights on why "only" Lagrangians appear in high energy physics (KG Lagrangian, Dirac Lagrangian, Standard Model Lagrangian...etc)

Explanation / Answer

In order to use Lagrangians in QM, one has to use the path integral formalism. This is usually not covered in a undergrad QM course and therefore only Hamiltonians are used. In current research, Lagrangians are used a lot in non-relativistic QM.

In relativistic QM, one uses both Hamiltonians and Lagrangians. The reason Lagrangians are more popular is that it sets time and spacial coordinates on the same footing, which makes it possible to write down relativistic theories in a covariant way. Using Hamiltonians, relativistic invariance is not explicit and it can complicate many things.

So both formalism are used in both relativistic and non-relativistic quantum physics. This is the very short answer.

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