Hi everyone. I have problem with Advanced Quantum Mechanics exam preparation. the question is to Diagonalize the operator (a(p^2)+b(x^2))^n where x and p are position and momentum operators satisfying [x,p]=ihbar , a and b are non-zero real numbers and n a positive non-zero integer.

Question

Hi everyone. I have problem with Advanced Quantum Mechanics exam preparation. the question is to Diagonalize the operator (a(p^2)+b(x^2))^n  where x and p are position and momentum operators satisfying [x,p]=ihbar , a and b are non-zero real numbers and n a positive non-zero integer.

algorithm · Answer

Diagonalize as in?

anonymous · Answer

will u tell me wht is mean by diagonalize..??

anonymous · Answer

I  mean to act it on a basis set in which this operator is diagonal. 
Actually I could solve it meanwhile. 
this operator, without the power n, is the Hamiltonian of the general harmonic oscillator and the basis set for it to be diagonalized would be the energy eigenstates. then I solved the eigenvalue problem using the expressions to transform X and P to ladder operators and found the eigenvalues. 
Thank you anyways!

kainui · Answer

I'm curious what that looks like. I've diagonalized matrices before but I can't quite figure out how you would diagonalize these continuous operators.

I've solved the harmonic oscillator before by factoring it as $H=(i\sqrt{a}p+\sqrt{b}x)(-i\sqrt{a}p+\sqrt{b}x) -i \sqrt{ab}[p,x]$ or whatever if I messed up a little algebra. From here I used ladder operators to get all the eigenvalues and eigenstates. So is this really all that it means? Since I'm imagining normalizing the eigenstates,

$$H=E^{\dagger} D E$$

but I'm not really sure if this is the same as what you've done is it? After this of course you get this like you wanted, but I feel like I'm doing a different way than you so I'm curious. 

$$H^n=E^\dagger D^n E$$