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Homework Statement
Two quantum mechanical operators obey the following commutation relation.
[\hat{A},\hat{B}]=i
Given this commutation relation which of the following are true or false? Justify your answers.a) The two observables are simultaneously diagonalizable.
b) The two satisfy a Heisenberg uncertainty relation that has the form
\left<(\Delta\hat{A})^2\right>\left<(\Delta\hat{B})^2\right>\ge\frac{1}{4}
c) They are spin operators.
d) The Hamiltonian \hat{H}=\hat{A}^2+\hat{B}^2 could describe a harmonic oscillator system.
Homework Equations
The Attempt at a Solution
a) False - simultaneously diagonalizable ==> simultaneously observable. Since they don't commute, they aren't simultaneously observable and can't be simultaneously diagonalizable.
b) No idea how to get there from here
c) They could be, but don't have to be. They satisfy a commutation relation for angular momentum (as do spin operators), but other angular momentum operators also satisfy the relation.
d) True. Dropping a factor of \hbar, A and B could be position and momentum operators (respectively), which gives a Hamiltonian of H=X^2+P^2.
I could use a hint or two on b, as well as someone verifying my logic for a, c, and d. Thanks!