# Canonical momentum operators in curvilinear coordinates

What is the quantum canonical momentum operator corresponding to arbitrary canonical position. For example, in Cartesian coordinates ($x^i$), the canonical momentum operator with respect to each $x^i$ is $-i\hbar \frac{\partial}{\partial x^i}$. For arbitrary canonical coordinates $q^i$, would the corresponding canonical momentum operators just be $-i\hat{\mathbf q}^i\cdot\hbar \nabla$?

• In general, there is nothing such as "the" unique canonical momentum for a coordinate $q^i$. A canonical momentum is defined in such a way that its commutator with $q^i$ is nonzero, but it is zero for other $q$'s. One may redefine the momenta and coordinates in many ways, the so-called canonical transformations en.wikipedia.org/wiki/Canonical_transformations , without changing the commutator. The idea that one always has a unique one-to-one map is OK to count the degrees of freedom but it's fundamentally flawed if one wants to study the full physical system. Mar 20, 2011 at 19:03
• Please define what you mean by $\hat{\mathbf q}^i$. Does your manifold have a metric? Moreover, $-i\hat{\mathbf q}^i\cdot\hbar \nabla$ has unusual "index up", where momentum $p_i$ traditionally has "index down". I recommend you to simply write $-i\hbar\nabla_{i}$ instead. Mar 21, 2011 at 22:39

If you use curvilinear coordinates you basically need to transform your derivatives likewise, ie, you need to use the Jacobian of the coordinate transformation you used to go from Cartesian coords to the curvilinear system you have at hands.

More generally, you can think in terms of Quantum Mechanics over a curved manifold, in which case you would simply use the covariant derivative.

• There isn't a unique covariant derivative on a manifold! So which one?
– MBN
Mar 21, 2011 at 4:32
• @MBN generally one works with a metric compatible connection, i.e. $\nabla g_{\mu\nu} = 0$. Given a metric, this equation determines the corresponding connection.
– user346
Mar 21, 2011 at 6:25
• @Deepak: I know that there is unique such connection (plus the condition to be trosion free). The question is not about manifolds with a choice of metric, so the answer should say something about it, otherwise it is incomplete and misleading.
– MBN
Mar 21, 2011 at 13:55
• @MBN: the last equation in the question has an inner product, which led me to believe that i could make some "general Physical assumptions", such as the existence of a metric and of a metric-compatible connection. Mar 21, 2011 at 15:44
• Fair enough.
– MBN
Mar 21, 2011 at 18:06