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I have tried for hours to calculate the commutator of angular momentum in the differential form, but I cannot get the correct answer. This is my first experience with actually checking if two operators commutes, so there may be some beginner's misunderstandings that causes the problem.

I know from my textbook the correct answer:

[itex]

[\hat{L}_x,\hat{L}_y] = \frac{\hbar}{i}\hat{L}_z = \frac{\hbar}{i} \left( x \frac{\partial \psi}{\partial y} - y \frac{\partial \psi}{\partial x} \right)

[/itex]

The angular momenta are defined as

[itex]

\hat{L}_x = y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y} \\

\hat{L}_y = z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z} \\

\hat{L}_z = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}

[/itex]

Sets up the commutator expression, and writes it out:

[itex]

[\hat{L}_x,\hat{L}_y] = \left[ y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y} , z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z} \right]

[/itex]

From what I assume is a well-known algebraic rule for handling commutators, I get the following:

[itex]

\left[y \frac{\hbar}{i} \frac{\partial}{\partial z}, z \frac{\hbar}{i} \frac{\partial}{\partial x} \right] - \left[y \frac{\hbar}{i} \frac{\partial}{\partial z}, x \frac{\hbar}{i} \frac{\partial}{\partial z} \right]

- \left[z \frac{\hbar}{i} \frac{\partial}{\partial y}, z \frac{\hbar}{i} \frac{\partial}{\partial x} \right]

+ \left[z \frac{\hbar}{i} \frac{\partial}{\partial y}, x \frac{\hbar}{i} \frac{\partial}{\partial z} \right]

[/itex]

It was straight forward to find that the two middle terms commute, i.e. they equal zero. Hence, we are left with:

[itex]

\left[y \frac{\hbar}{i} \frac{\partial}{\partial z}, z \frac{\hbar}{i} \frac{\partial}{\partial x} \right]

+ \left[z \frac{\hbar}{i} \frac{\partial}{\partial y}, x \frac{\hbar}{i} \frac{\partial}{\partial z} \right] \phantom{................} [1]

[/itex]

Upon evaluating these terms, I let them act on a function [itex] \psi [/itex]. Looking at the first term:

First I factored the complex fraction out. Then I wrote out the commutator and let each differential fraction act on [itex] \psi [/itex]:

[itex]

\frac{\hbar}{i} \left(y \frac{\partial \psi z}{\partial z} \frac{\partial \psi}{\partial x} - yz \frac{\partial \psi }{\partial x} \frac{\partial \psi}{\partial z} \right)

[/itex]

The y in the second term above can be factored out from the differentials, because the differential is not done with respect to y. In the first term, however, the z must be differentiated along with psi by using the product rule. Doing this, and cancelling equal terms of opposite sign, yields:

[itex]

\frac{\hbar}{i} y \psi \frac{\partial \psi}{\partial x}

[/itex]

Repeating this calculation for the second term in [1] yields

[itex]

-\frac{\hbar}{i} x \psi \frac{\partial \psi}{\partial y}

[/itex]

Substituting into [1] yields

[itex]

[\hat{L}_x,\hat{L}_y] \psi = \frac{\hbar}{i} \left( y \frac{\partial \psi}{\partial x} - x \frac{\partial \psi}{\partial y} \right) \psi

[/itex]

Now, comparing with the correct answer:

i) My signs are opposite of what they should be

ii) More importantly, I have a [itex] \psi [/itex] in the differentials which should not be there. My question is, "why, and how do I rid my answer of it?".

I hope I have sufficiently showed what I have done. I have checked my notes several times, and I cannot get rid of those extra psi's in my answer.

I would appreciate any help!

Regards,

Anders

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# Calculating Commutator of Differential Angular Momentum

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