I'm preparing for an exam at the moment and in one of the past exams the is a question asking to prove that the hamiltonian operator can be expressed in terms of the ladder operators.(adsbygoogle = window.adsbygoogle || []).push({});

The solution is this

(The minus sign didn't come out in the last line, and obviously there is one more step that I left of the end, but you get the picture.)

Following this from the second to the third line seems to imply

But I would expect

because the derivative of x wrt x is 1. My maths isn't terribly good, and there is obviously something simple that I am missing, but I've been staring at this for days and can't seem to get it. Could someone please explain what I'm missing?

I could answer this question if it comes up on the exam, because I can remember how it goes, but I want to actually be able to understand it.

Thanks in advance.

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# Hamiltonian in terms of ladders, question.

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