# Angular momentum commutation relation, extra terms?

## Homework Statement

What is the commutation relation between the x and y componants of angular momentum L = r X P

None.

## The Attempt at a Solution

I do r X p and get the angular momentum componants:

$$L_{x} = (-i \hbar) (y \frac{d}{dz} - z \frac{d}{dy})$$
$$L_{y} = (-i \hbar) (z \frac{d}{dx} - x \frac{d}{dz})$$
$$L_{z} = (-i \hbar) (x \frac{d}{dy} - y \frac{d}{dx})$$

then when I attempt to put into the commutation relation [Lx,Ly] it comes out very complicated.

My question:

I found a derivation BUT it has extra terms in it (circled in red)

Why are there 5 terms when there should only be 4 when multiplying out 2 brackets? It's important because those 2 extra terms enable the correct solution. Or have I missed something?

Thankyou for any help in advance.

Source of derivation: http://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect8.pdf

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## Homework Statement

What is the commutation relation between the x and y componants of angular momentum L = r X P

None.

## The Attempt at a Solution

I do r X p and get the angular momentum componants:

$$L_{x} = (-i \hbar) (y \frac{d}{dz} - z \frac{d}{dy})$$
$$L_{y} = (-i \hbar) (z \frac{d}{dx} - x \frac{d}{dz})$$
$$L_{z} = (-i \hbar) (x \frac{d}{dy} - y \frac{d}{dx})$$

then when I attempt to put into the commutation relation [Lx,Ly] it comes out very complicated.

My question:

I found a derivation BUT it has extra terms in it (circled in red)

Why are there 5 terms when there should only be 4 when multiplying out 2 brackets? It's important because those 2 extra terms enable the correct solution. Or have I missed something?

Thankyou for any help in advance.

Source of derivation: http://www2.ph.ed.ac.uk/~ldeldebb/docs/QM/lect8.pdf

There are four terms when you multiply the brackets. However, one of them can be rewritten with two terms, namely
$$y\partial_z z \partial_x = y \partial_x + yz \partial_z \partial_x.$$
This is just the product rule for differentiation.

There are four terms when you multiply the brackets. However, one of them can be rewritten with two terms, namely
$$y\partial_z z \partial_x = y \partial_x + yz \partial_z \partial_x.$$
This is just the product rule for differentiation.

thats great, many thanks for your help!!