Roadmap to understanding the quantum mechanics

Lagrangian mechanics

Action and principle of minimal action: Leibniz, Maupertuis, D’Alembert’s and Hamilton’s

More of a philosophical then practical origin. Leibniz.

“Nature is parsimonious.”

$S = \int \limits_{t_1}^{t_2} L(t) dt$, where $S$ is the total action, and $L(t)$ is the unit of change of action through an infinitesimal period of time.

Another pretentious name for the subject is “principle of stationary action”, which indicates that the first derivative of the action turns 0, but this, generally speaking, does not guarantee that it is a minimum - it could be a saddle point or maximum. Let us not be distracted by this technicality.

TODO

Configuration space and generalized coordinates for constrained optimization

Remember, Lagrange was known for the method of Lagrange multipliers.

Example: Body, attached to a wire.

TODO

Calculus of variations

TODO

Lagrangian

$S = \int \limits_{x_1}^{x_2} L(f(x, \dot{x}, t)) dt$

$L = T - U$

Why Lagrangian was chosen the way it was chosen? 4 criteria.

TODO

Noether’s theorem

Every law of conservation (e.g. conservation of momentum, energy etc.) corresponds to some symmetry (e.g. translational symmetry of time, space etc.).

See: gauge theory (in Russian)

TODO

Hamiltonian mechanics

• Phase space vs configuration space, coordinates and their derivatives vs coordinates and momenta, Legendre transform
• Hamiltonian vs Lagrangian
• n-forms, exterior forms, differential forms, symplectic manifolds

Basics of physics for quantum mechanics

• Planck constant as quantum of action

References

• https://www.motionmountain.net/ - the best book ever in the class of “physics for dummies” books
• https://loshijosdelagrange.files.wordpress.com/2013/04/v-arnold-mathematical-methods-of-classical-mechanics-1989.pdf - classical soviet textbook on mechanics by the great V.Arnold. Don’t you ever try to read this crap directly, soviet and post-soviet mathematicians write books for themselves. Useful as a table of contents, though.
• https://pdfroom.com/books/introduction-to-topology/jb5qOml6gxQ - associated topology questions by V.Vasiliev. Again, don’t read it directly.
• https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%D0%B8%D0%B1%D1%80%D0%BE%D0%B2%D0%BE%D1%87%D0%BD%D0%B0%D1%8F_%D0%B8%D0%BD%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C - gauge theory page contains a table of correspondence between transforms (on which physical laws are invariant) and conservation laws
• https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf - transition from the axioms of classical mechanics to quantum
• https://sbseminar.wordpress.com/2012/01/09/what-is-a-symplectic-manifold-really/ - basic explanation of symplectic manifolds
• https://emilman.net.technion.ac.il/files/2017/02/JoramSeminar-2016-Talk.pdf

Written by Boris Burkov who lives in Moscow, Russia, loves to take part in development of cutting-edge technologies, reflects on how the world works and admires the giants of the past. You can follow me in Telegram