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August 17th, 2018, 05:18 PM   #1
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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!
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August 17th, 2018, 08:03 PM   #2
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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.
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August 17th, 2018, 08:04 PM   #3
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Originally Posted by romsek View Post
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.
Thanks romsek! Much appreciated!
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August 17th, 2018, 08:05 PM   #4
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Quote:
Originally Posted by SenatorArmstrong View Post
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!
Yes, both Hamiltonian and Lagrange methods are used. There is also a large amount of Linear Algebra involved with QM. The whole Hilbert space concept is based on it. That and some differential equation work. The solution for the electron states in the Hydrogen atom, for example, depends on polynomial series solutions that you probably will not have seen before.

-Dan
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August 18th, 2018, 05:01 AM   #5
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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.
Thanks from topsquark and SenatorArmstrong

Last edited by skipjack; August 18th, 2018 at 06:56 AM.
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August 20th, 2018, 08:04 AM   #6
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Originally Posted by topsquark View Post
Yes, both Hamiltonian and Lagrange methods are used. There is also a large amount of Linear Algebra involved with QM. The whole Hilbert space concept is based on it. That and some differential equation work. The solution for the electron states in the Hydrogen atom, for example, depends on polynomial series solutions that you probably will not have seen before.

-Dan
Thanks! I feel pretty comfortable with what you mentioned except for the polynomial series solutions to the electron states in the hydrogen atom. I appreciate the response!
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August 20th, 2018, 08:06 AM   #7
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Originally Posted by studiot View Post
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.
Thanks a ton for all these excellent references/resources.

My classical textbook is "classical dynamics of particles and systems" by Stephen T. Thornton. Would you happen to be familiar with this text?
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August 20th, 2018, 11:44 AM   #8
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Quote:
Originally Posted by SenatorArmstrong View Post
Thanks a ton for all these excellent references/resources.

My classical textbook is "classical dynamics of particles and systems" by Stephen T. Thornton. Would you happen to be familiar with this text?
Unfortunately that's a new one to me.
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