[PhD Qualifier] Commutation relation

AI Thread Summary
The discussion revolves around the commutation relation of two quantum mechanical operators, specifically [A, B] = i. It is concluded that the observables are not simultaneously diagonalizable due to their non-commuting nature, making statement (a) false. Statement (b) is affirmed as true, with the justification involving the application of the Schwartz inequality to derive the Heisenberg uncertainty relation. Statement (c) is uncertain as the operators could represent spin but are not limited to it, while statement (d) is confirmed as true, suggesting the Hamiltonian could describe a harmonic oscillator. The conversation highlights the need for clarity in understanding the implications of commutation relations in quantum mechanics.
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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!
 
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For b., and any other heisenburg type relation, you need to invoke the schwartz inequality.
 
I suppose I can answer (b) using my observation from (d).

Let A=\frac{X}{\hbar}, B=P. Heisenberg's relation is \Delta x\Delta p \ge \frac{\hbar}{2}, so \Delta a\Delta b\ge\frac{1}{2}. Square it to get "true".

Can someone verify these answers for me?
 
I will tell you that b is true, but that's a tremendously ad hoc way to determine it.

If you have Griffith's "Quantum Mechanics" he performs the full derivation the uncertainty principle for two arbitrary operators.
 
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