Virtually every treatment of quantum mechanics brings up the canonical commutation relations (CCR); they go over what the Poisson bracket is and how it relates to a phase space / Hamiltonian mechanics, and then say "then, you replace that with ih times the commutator, and replace the dynamical variables with Hermitian operators, and TADA! You now have TEH QUANTUM MACHANICHS"...sometimes there's a vague attempt to explain it away by pointing to the Fourier-duality of the canonical variables (p/q), or if they really want to put the argument to rest, they say "by the Stone-von Neumann theory".(adsbygoogle = window.adsbygoogle || []).push({});

But...that doesn't really help me. Reading the Wikipedia article on Stone-von Neumann seems to have a lot of information, but I can't really make out a real explanation / derivation from it.

So -- can anyone explain this to a n00b? Here are a few points that I'm curious about in particular:

- what does a Poisson bracket have to do with a (anti-)commutator?

- I get how if you multiply a commutator by h, at a classical level it would look like the operators did indeed commute -- but why that particular value? And where does the imaginary unit come in?

- What does this have to do with the change from a finite-dimensional phase space to an infinite-dimensional Hilbert space, and the change from variables in this phase space to Hermitian operators on the Hilbert space?

...though these are by no means anywhere near the only questions -- they're more to just maybe get the conversation started. Also, any pointers to books / references would be great.

Thanks,

Justin

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Canonical Commutation Relations: Why?

Loading...

Similar Threads for Canonical Commutation Relations |
---|

I Commuting Operators and CSCO |

A Commutator vector product |

Commutation relation |

I Solving the Schrödinger eqn. by commutation of operators |

I Canonical Quantization |

**Physics Forums | Science Articles, Homework Help, Discussion**