Derivation of momentum operator

1. Jan 27, 2015

pcflores

hello,

i am trying to learn the derivation of the momentum operator and i found 2 ways of deriving it. one is using fourier transform and the other is taking the time derivative of the expectation value of x.

i just want to know what is the physical interpretation of the time rate of change of <x>

thank you

2. Jan 27, 2015

Staff: Mentor

Its related to Stones Theorem and infinitesimal generators:
http://en.wikipedia.org/wiki/Stone's_theorem_on_one-parameter_unitary_groups

If you really want to understand the momentum operator get a hold of a copy of Ballentine - Quantum Mechanics - A Modern Development and have a look at chapter 3:
https://www.amazon.com/Quantum-Mechanics-Modern-Development-Edition/dp/9814578584

Its true basis is the symmetries of Galilean relativity - but that revelation is something you need to discover for yourself - its very profound and deep. Revelations like that are best understood by working through the detail.

Then get a hold of Landaus classic on Mechanics:
https://www.amazon.com/Mechanics-Third-Edition-Theoretical-Physics/dp/0750628960

It shows classical mechanics has exactly the same basis. The symmetries of Galilean relativity is the key to both just as the symmetries of the Lorentz transformations is the key to relativity.

Thanks
Bill

Last edited by a moderator: May 7, 2017
3. Jan 27, 2015

pcflores

thank you

Last edited by a moderator: May 7, 2017
4. Jan 27, 2015

Staff: Mentor

You are most welcome.

Also see if you can have a look at Landau as well - you may have missed that bit since I added it later.

Thanks
Bill