Question

Got the following question in quantum mechanics and I'm not sure what to do:
Calculate the commutator (see picture)

Answer 1

This commutator is 0 because the Hamiltonian you've given is just basically the kinetic energy in 3D. The thing that might be confusing you is that you've written the magnitude of r to be sqrt(x^2+y^2+z^2) and you're possibly thinking that this *is the r* you need to put in. But that's just the magnitude of r, r should be an linear combination of all three position operators x, y and z. If you write out the r over the position operators, and the Hamiltonian given also, in let's say Cartesian space (which you imply should be used) then:
$$r = X\hat i + Y\hat j + Z\hat k\\
H = -\frac{\hbar}{2m}\nabla^2 = \frac{\hbar}{2m}p^2$$
Because all the position operator commute this is something that you can do in 3 steps, just calc the commutator for each coordinate individually and then bring them back together. Otherwise you can use the commutator property that:
$$[A+B+C, D] = [A, D]+[B, D]+[C, D]$$
Once you've separated them you're stuck with the (just for one this time, too much writing):
$$\frac{1}{2m} [x,p^2] = \frac{1}{2m} \left( [x,p]p + p[x,p] \right)$$
At which point you've to remember that:
$$[x, p_x] = \frac{i \hbar}{m} $$

Answer 2

sry, last line:
$$[x, p_x] = i\hbar$$

Answer 3

@ljetibo hmm okay thank you, but why can you replace nabla^2 with p^2 and change the sign from - to +?

Answer 4

Because the momentum operator is by definition, well - exactly that... I did make a stupid error in typing the signs and squares:
$$H = -\frac{\hbar^2}{2m}\nabla^2 = -\frac{p^2}{2m}$$
because
$$p_x = -i\hbar\frac{\partial}{\partial x}$$
so
$$p_x^2 = -\hbar^2\frac{\partial^2}{\partial x^2}$$
and you know the different axes commute so they're independed and you just roll it all up into a single equation, but can be done as 3 separate ones.
$$p^2 = -\hbar^2\left (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)$$
and that's the definition of Laplace operator.
$$\Delta f = \sum_{i=1}^n \frac {\partial^2 f}{\partial x^2_i}$$
where the xi are actually all coordinates you have, polar cartesian, spherical, generalized etc etc...

Sorry for bad typos.