Solving Operators and Ordering in 3D Electron Energies

[Lee]

In my question I have to find what the commutation of a electrons kinetic and potentials energys are, in 3 Dimensions. I have started by finding the kinetic operator T and the potential energy from coloumbs law. I have then applied commutation brackets and I'm at the stage where I'm solving the commutation bracket for the x-direction. (and then apply symmetry for my 2 other axis) My question is, as we have to retain order when dealing with operators, how do I 'deal' with my

$$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } xi \hbar \pd {} {x} {}$$

I presume I can't just differentate the x as I need to preserver order, does this just sit like this till I can 'deal' with it?

 

[nrqed]

In my question I have to find what the commutation of a electrons kinetic and potentials energys are, in 3 Dimensions. I have started by finding the kinetic operator T and the potential energy from coloumbs law. I have then applied commutation brackets and I'm at the stage where I'm solving the commutation bracket for the x-direction. (and then apply symmetry for my 2 other axis) My question is, as we have to retain order when dealing with operators, how do I 'deal' with my

$$\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } xi \hbar \pd {} {x} {}$$

I presume I can't just differentate the x as I need to preserver order, does this just sit like this till I can 'deal' with it?

In calculating commutators of differntial operators, it is convenient to apply th commutator on a "test function", which is is just som arbitrary function of x, y and z that must be removed at the very end of the calculation.

So if you have two operators A and B (which are differential operators) and you want to compute their commutator, just consider
$$[A,B] f(x,y,z) = AB f(x,y,z) - BA f(x,y,z)$$
Apply all the derivatives and at the very end, remove the test function.

 

[Hurkyl]

You can apply a commutation relation if you wanted to reverse the order.

But remember that "x" is an operator, not a function. So

$$\frac{\partial}{\partial x} x \neq 1$$

Instead, it's supposed to be the operator

$$\psi(x, y, z, t) \rightarrow \frac{\partial (x \psi(x, y, z, t))}{\partial x}$$