Editing Canonical commutation relation (section)

== Generalizations ==
It can be shown that
<math display="block">[F(\vec{x}),p_i] = i\hbar\frac{\partial F(\vec{x})}{\partial x_i}; \qquad [x_i, F(\vec{p})] = i\hbar\frac{\partial F(\vec{p})}{\partial p_i}.</math>

Using <math>C_{n+1}^{k} = C_{n}^{k} + C_{n}^{k-1}</math>, it can be shown that by [[mathematical induction]]
<math display="block">\left[\hat{x}^n,\hat{p}^m\right] = \sum_{k=1}^{\min\left(m,n\right)}{ \frac{-\left(-i \hbar\right)^k n!m!}{k!\left(n-k\right)!\left(m-k\right)!} \hat{x}^{n-k} \hat{p}^{m-k}} = \sum_{k=1}^{\min\left(m,n\right)}{ \frac{\left(i \hbar\right)^k n!m!}{k!\left(n-k\right)!\left(m-k\right)!} \hat{p}^{m-k}\hat{x}^{n-k}} ,</math>
generally known as McCoy's formula.<ref>McCoy, N. H. (1929), "On commutation formulas in the algebra of quantum mechanics", ''Transactions of the American Mathematical Society'' ''31'' (4), 793-806 [https://pdfs.semanticscholar.org/1bc1/688c10bbb6d6630e647f675695a822f2a380.pdf online]</ref>

In addition, the simple formula
<math display="block">[x,p] = i\hbar \, \mathbb{I} ~,</math>
valid for the [[Canonical quantization|quantization]] of the simplest classical system, can be generalized to the case of an arbitrary [[Lagrangian (field theory)|Lagrangian]] <math>{\mathcal L}</math>.<ref name="town">{{cite book |first=J. S. |last=Townsend |title=A Modern Approach to Quantum Mechanics |url=https://archive.org/details/modernapproachto0000town |url-access=registration |publisher=University Science Books |location=Sausalito, CA |year=2000 |isbn=1-891389-13-0 }}</ref> We identify '''canonical coordinates''' (such as {{mvar|x}} in the example above, or a field {{math|Φ(''x'')}} in the case of [[quantum field theory]]) and '''canonical momenta''' {{math|&pi;<sub>''x''</sub>}} (in the example above it is {{mvar|p}}, or more generally, some functions involving the [[derivative]]s of the canonical coordinates with respect to time):
<math display="block">\pi_i \ \stackrel{\mathrm{def}}{=}\ \frac{\partial {\mathcal L}}{\partial(\partial x_i / \partial t)}.</math>

This definition of the canonical momentum ensures that one of the [[Euler–Lagrange equation]]s has the form
<math display="block">\frac{\partial}{\partial t} \pi_i = \frac{\partial {\mathcal L}}{\partial x_i}.</math>

The canonical commutation relations then amount to
<math display="block">[x_i,\pi_j] = i\hbar\delta_{ij} \, </math>
where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]].