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The hydrogen atom Hamiltonian is

$$H=\frac{p^2}{2m} -\frac{e^2}{r}\tag{1}$$

with e the elementary charge,m the mass of the electron,r the radius from the nucleus and p,the momentum. Apparently we can factorize H $$H=\gamma +\frac{1}{2m}\sum_{k=1}^{3}\left(\hat p_k+i\beta\frac{\hat x_k}{r}\right)\left(\hat p_k-i\beta\frac{\hat x_k}{r}\right)\tag{2}$$

for suitable constants β and γ that you can calculate. I assume the operator identity:

$$\hat{A}^2+\hat{B}^2=(\hat{A}-i\hat{B})(\hat{A}+i\hat{B})-i[\hat{A},\hat{B}]$$was used.

Can someone explain to me how we can start with formula (1) and make the position operator appear in (2)?

Here is the source https://ocw.mit.edu/courses/physics...all-2013/lecture-notes/MIT8_05F13_Chap_09.pdf pages 33-34

Thank you.

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# I How to factorize the hydrogen atom Hamiltonian?

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