# What does canonical mean?

1. Aug 6, 2007

### captain

what does it mean in quantum mechanics when they say canonical variables or canonical momentum? what is the difference from regular momentum?

2. Aug 6, 2007

### Norman

To paraphrase: Any two variables whose Poisson Bracket (or Commutator in Quantum) that give a delta are canonical. A symmetry in one canonical variable implies the other is conserved. For example, translational invariance and conservation of momentum.

Cheers,
Norm

3. Aug 6, 2007

### olgranpappy

Not exactly. For example, in single particle quantum mechanics in the presence of an external electromagnetic field the canonical variables are the position $$\vec x$$ and the *canonical* momentum $$\vec p$$ which satisfy:
$$[x_j,p_k]=i\hbar\delta_{jk}\;.$$

But, if the above holds, it should also be obvious that the *mechanical* momentum $$\vec \pi =\vec p-e\vec A$$ where $$e$$ is the charge also satisfies:
$$[x_j,\pi_k]=i\hbar\delta_{jk}\;.$$

Thus, both sets of variables satisfy canonical communtation relations even though only the set (x,p) are called "canonical variables."

4. Aug 7, 2007

5. Aug 7, 2007

### captain

thanks for your help