# Ambiguity in second quantized current operator

In Mahan eq. 1.193 I see an expression for the second quantized current operator of the form:

$$j(r)=\frac{e}{2mi}[\psi^\dagger(r)\nabla\psi(r)-\psi(r)\nabla\psi^\dagger(r)]$$

However, in other online sources I instead see expressions like

$$j(r)=\frac{e}{2mi}[\psi^\dagger(r)\nabla\psi(r)-(\nabla\psi^\dagger(r))\psi(r)]$$

Are these expressions equivalent? It seems to me that the $$\psi(r)$$ and $$\nabla\psi^\dagger(r)$$ do not commute, so I am surprised to see both definitions in different places online.

• The right definition is the one that makes $j(r)$ hermitian. That should not be too difficult to check. – Hans Moleman Sep 7 at 22:47

The correct definition is the first one, you can easily check that it gives a physically correct hermitian current density $$j(r) = j(r)^\dagger$$. As hinted by Hans Moleman in the comment to your question.
Online resources might be quoting the second expression as a special case, only valid in certain situations. $$\psi(r)$$ and $$\nabla\psi(r)$$ do commute, for instance, for $$\psi(r) = \mathrm{e}^{\mathrm{i}(kr-\omega t)}$$.
In general, any spin-$$0$$ particle/field, described by a scalar, will obey the commutation relation. And usually spin-$$0$$ are the first things books do when explaining QFT and second quantisation, so I would not be surprised if that's the case for the resources you are looking at.