# Ladder Operators for this Hamiltonian $\widehat{H}$

how to find the ladder operators for this hamiltonian: $$\widehat{H}=a\widehat{A}^2 + b\widehat{B}^2$$ where $$a$$ and $$b$$ are two real and positive constants.

And how to write the hamiltonian in function of the two ladder operators?

Actually the answer is:$$a_-=\frac{1}{\sqrt{2a}}\widehat{A}+i\frac{1}{\sqrt{2b}}\widehat{B}$$ and $$a_+=\frac{1}{\sqrt{2a}}\widehat{A}-i\frac{1}{\sqrt{2b}}\widehat{B}$$

And the condition on the commutator $$\widehat{A}$$ and $$\widehat{B}$$ for having: $$[a_-,a_+]=\widehat{1}$$, I found: $$[\widehat{A},\widehat{B}]=i\sqrt{ab}$$

but I couldn't reach them.

• You need to define the commutation relations of $\hat A,\hat B$ for ladder operators to make sense – user619894 Dec 31 '19 at 11:51
• ok done @user721481 – walid Dec 31 '19 at 11:59
• To get the commutation you had to compute $\hat A,\hat B$ in terms of $a_{+,-}$ so you are basically done. – user619894 Dec 31 '19 at 12:16
• Cool thank you it was a good hint. I was trying to find how to factorize the expression of H to deduce the ladder operators? – walid Dec 31 '19 at 13:03
• Does the question mark mean you are still having trouble? If so I will elaborate. – user619894 Dec 31 '19 at 13:18

Moving this to an answer as in is too long for a comment: Given a generic Hamiltonian $$a\widehat{A}^2 + b\widehat{B}^2$$ is not enough information to construct ladder operators. There is no reason to expect that the commutation relation of two conjugate linear combinations of $$A,B$$ is 1. So you need to be supplied with either the commutation relations of $$[A,B]$$ or the explicit construction of $$A,B$$ in terms if $$a_{+},a_{-}$$.

In the question you say that

$$a_-=\frac{1}{\sqrt{2a}}\widehat{A}+i\frac{1}{\sqrt{2b}}\widehat{B}$$

$$a_+=\frac{1}{\sqrt{2a}}\widehat{A}-i\frac{1}{\sqrt{2b}}\widehat{B}$$

So you can plug back in and get the Hamiltonian in terms of $$a_{+}, a_{-}$$ but the Hamiltonian will now contain mixed terms: $$\alpha a_{+}^2+\beta a_{-}^2+\gamma a_{+} a_{-}+\delta a_{-} a_{+}$$ for some parameters $$\alpha,\beta,\gamma,\delta$$.

on the other hand, if you are given $$[\widehat{A},\widehat{B}]=i\sqrt{ab}$$ , then the construction of a pair of conjugate operators that have a commutation of $$1$$ is also direct.

• I see, actually, they don't ask me to find a+ and a- they are given to work with them. But I try to refind them from [A, B]=i√(ab), to do so I get stuck on how to write H as a product, like $x^2+y^2=(x+iy)(x−iy)$ – walid Dec 31 '19 at 15:03