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sorry if this looks ugly but I couldn't find out how to write out bras and kets on the Latex thing.

I have these inner products

<f|g> = i<x|(AB - A<B> - <A>B + <A><B>)|x>

and

<g|f> = -i<x|(BA - B<A> - <B>A + <A><B>)|x>

where |x> is some arbitrary ket and A and B do not commute.

I'm trying to follow a derivation of the uncertainty principle that has a step which involves getting

<f|g> + <g|f> = i<x|[A,B]|x>,

but I'm not sure how to add the two things together.

I tried to see if it was allowed that, say, <x|M|x> + <x|N|x> = <x|(M+N)|x>, but that didn't work, and I tested this with a real numerical example so that definitely doesn't work. I need to somehow end up with

<f|g> + <g|f> = i<x|(AB - BA)|x>

but I haven't seen how objects like these add together before.

How is this done?

I have these inner products

<f|g> = i<x|(AB - A<B> - <A>B + <A><B>)|x>

and

<g|f> = -i<x|(BA - B<A> - <B>A + <A><B>)|x>

where |x> is some arbitrary ket and A and B do not commute.

I'm trying to follow a derivation of the uncertainty principle that has a step which involves getting

<f|g> + <g|f> = i<x|[A,B]|x>,

but I'm not sure how to add the two things together.

I tried to see if it was allowed that, say, <x|M|x> + <x|N|x> = <x|(M+N)|x>, but that didn't work, and I tested this with a real numerical example so that definitely doesn't work. I need to somehow end up with

<f|g> + <g|f> = i<x|(AB - BA)|x>

but I haven't seen how objects like these add together before.

How is this done?

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