I needed to work out the commutator , where

is the Hamiltonian operator and is the 3D position vector operator. It is difficult to find any textbook or online source that explicitly goes through the calculation of this three-dimensional case (in fact, I have not been able to find any) so I am recording my calculation step-by-step in this note.

The commutator is a vector operator with components

and

To evaluate these, note that the momentum operator (in position space) is

and so we have

Looking at the -component of we therefore have

Since multiplication is commutative we have . I will now show that we also have

To see this, let us first work out in detail the effect of on a wavefunction . We have

where the last equality follows from the fact that does not depend on . Thus, .

We can now easily show that because using the basic result for commutators that

(easy to prove by writing out the terms in full) we find that

Identical arguments show that . Thus, we can conclude that

It now only remains to work out and we can do this by first working out in detail the effect of on a wavefunction (this is of course the `canonical commutation relation’ of quantum mechanics). We have

where the last equality follows from the fact that is the same as multiplying by so its derivative with respect to equals . Thus, . Then we find that

and we can conclude that

Identical arguments show that

and

Thus, we finally reach our desired expression for the Hamiltonian-position commutator in three dimensions:

As an application of this result we will consider the problem of working out the expectation of position and velocity for a quantum particle. In three-dimensional space the quantum wave function is

and we obtain the probability density function as

The wavefunction satisfies the time-dependent Schrödinger equation

where is some potential energy function. We can write the Schrödinger equation in operator form using Dirac notation as

where

is the Hamiltonian operator (the Hamiltonian form of total energy) and

is the total energy operator. Note that the complex conjugate of the wavefunction satisfies the Schrödinger equation written in Dirac notation as

In quantum mechanics we find the expected position of the particle by integrating the position operator over all space, sandwiched between and . Thus, letting we have

where in the last term I have switched to using Dirac notation which will be useful shortly. The expected velocity can then be obtained by differentiating this integral with respect to . We get

The second triple integral on the right-hand side is zero because the position operator does not depend on time. The integrand in the first triple integral can be manipulated by using the operator form of the Schrödinger equation and Dirac notation to write

and

Thus, we have

where the last two equalities follow from the fact that is the commutator of and , which is equal to as we saw above. Therefore the expected velocity of a quantum particle looks a lot like the velocity of a classical particle (momentum divided by mass). The idea that quantum mechanical expectations exhibit Newtonian-like behaviour is the essence of the Ehrenfest Theorem of quantum mechanics.