Calculating the Commutator of H and r in 3D - What is the Correct Solution?

[rcross5]

Homework Statement

$$[\hat{H},\vec{r}]= ?$$

The Attempt at a Solution

The answer is given, and I KNOW that factor of 6 shouldn't be there. The answer should be

$$-\frac{\hbar^2}{m} \nabla$$

Anyway I've always been lurking these forums and I enjoy the discussions here, but this factor is really really bugging me and I was hoping you guys might be able to catch my probably simple mistake! Thanks :)

 

[ideasrule]

You made a few mistakes in your solution. For example, the Laplacian is not the gradient of a gradient; it's the divergence of a gradient. Also, the gradient of a vector has no well-defined meaning in multivariable calculus, so you have to be very careful when dealing with them.

The easiest way to do this problem is to realize that [H,r] can be split into three parts: [H,x], [H,y], and [H,z]. Calculate each part separately, and combine the result into a vector. Note that after you get [H,x], you can just replace all the x's with y's to get [H,y], since all your equations are symmetrical in the Cartesian coordinates.

 

[rcross5]

I guess I'm a bit rusty on my vector calculus! That's what I get for trying to be clever ;)
I'm too tired right now but if I have time I want to see if it's possible to directly solve it through like that. For now though, I've solved it by separating the components. Thanks!