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I have been reading Sakurai's Modern Quantum Mechanics and I notice that every proposition that book has depends primarily on the Hamiltonian formalism of classical mechanics. Even the time evolution of a quantum system is determined by the Hamiltonian (since the Hamiltonian is the generator of time). This seems quite odd since even though we are doing new physics (compared to classical mechanics) we can not get rid of the basic tools of old physics.

I want to ask whether the Hamiltonian formalism is the best that we can do? It is pretty obvious for physics that the limitations of our theory is the limitations of our axioms. What is currently the best formalism (the one which can cover as many regions of physics as possible) for physics, and what are some ongoing research on this field and related reading material?

To narrow the question down: Can you give examples of useful formalisms in mathematical physics, which can accomplish what Hamiltonian formalism can't?


This is precisely what I am asking for:

Are you asking if this operator can be something else than 𝐻?

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closed as too broad by ZeroTheHero, Qmechanic May 1 at 11:28

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I really do not understand this question. Some operator has to generate the dynamics. Are you asking if this operator can be something else than $H$? Are you asking if the dynamics can lead to higher order derivatives? Are you asking for quantum systems with no classical analogues? $\endgroup$ – ZeroTheHero May 1 at 11:55
  • $\begingroup$ @ZeroTheHero Yes! I am asking why does this Hamiltonian operator specifically pop-up in every field. $\endgroup$ – user212860 May 1 at 11:56
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/265248/2451 and links therein. $\endgroup$ – Qmechanic May 1 at 12:02
  • $\begingroup$ @ZeroTheHero Can you give answers to the questions that you have indicated. They look really interesting. $\endgroup$ – user212860 May 1 at 13:43
  • $\begingroup$ Comment to the post (v4): Consider to only ask 1 subquestion per post. $\endgroup$ – Qmechanic May 1 at 14:39