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August 17th, 2018, 04:18 PM  #1 
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics  Math methods from classical mechanics to quantum
Hello! I am curious about what mathematical methods in classical mechanics transfer to quantum mechanics. Classical mechanics is a prerequisite for quantum mechanics at my university and figured it was due to some math technique or physics concepts learned in classical mechanics. Perhaps langrangian and hamiltonions show up again? Thanks! 
August 17th, 2018, 07:03 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,430 Thanks: 1315 
google Poisson Brackets. They are a way of expressing classical mechanics that when transitioning to quantum mechanics are directly replaced with the commutator operator. That's about the limit of my understanding so I'll leave you to read. 
August 17th, 2018, 07:04 PM  #3 
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics  Thanks romsek! Much appreciated!

August 17th, 2018, 07:05 PM  #4  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,159 Thanks: 878 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
August 18th, 2018, 04:01 AM  #5 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
Although you have asked this in Physics, this is a Maths website and you mentioned university maths so here are a couple of references, that move you on from classical to quantum maths. The Mathematical Foundations of Quantum Mechanics G W Mackey (Professor of Mathematics, Harvard) This little monograph are lecture notes, unfortunately without index. However, treats QM using the standard mathematical paradigm of Axioms  lemma  theorem. The Mathematical Principles of Quantum Mathematics D F Lawden This book has both an index and a sound bibliography and goes right up to relativistic QM (Dirac's equation). Mathematics of Classical and Quantum Physics Byron and Fuller Excellent thick book includes good treatment of ladder operators. Quantum Mechanics Mandl (part of the Manchester Physics Series) More of Physics book, so includes lots of experimental verification data. Quantum Mechanics Davies and Betts Undergrad notes series chatty with lots of insight to the basic equations and processes. Romsek mentioned operators and commutators, which are very important and and are treated classically in both Goldstein and Corben & Stehle. Also look at Matrix methods (Heisenberg) and eigenvalues. Remember when you come to discuss 'spin' and 'isospin' that these are definitely different from classical mechanical spin, and have no direct counterpart in classical mechanics. Classical spin is included under the heading of angular momentum in QM and it is important to distinguish. Which classical textbook do you have? I can try to pick out important passages for you. Last edited by skipjack; August 18th, 2018 at 05:56 AM. 
August 20th, 2018, 07:04 AM  #6  
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics  Quote:
 
August 20th, 2018, 07:06 AM  #7  
Senior Member Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics  Quote:
My classical textbook is "classical dynamics of particles and systems" by Stephen T. Thornton. Would you happen to be familiar with this text?  
August 20th, 2018, 10:44 AM  #8 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271  

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