Harmonic oscillators and commutators.

[Beer-monster]

Homework Statement

If we have a harmonic oscillator with creation and annhilation operators $a_{-} a_{+}$, respectively. The commutation relation is well known:

$$[a_{+},a_{-}] = I$$

However, if we have two independent oscillators with operators $a'_{-} a'_{+}$

As the operators are the same function would they commute if mixed.

i..e is $$[a_{+},a'_{-}] = I$$

Still valid?

 

[vela]

When you say two operators A and B commute, it means AB=BA so that [A,B]=AB-BA=0.

As to your question about the commutation relation, the answer is no. The creation and annihilation operators for the one oscillator commute with the operators for the other oscillator.

 

[Beer-monster]

So $[a_{+},a'_{-}] = 0$

Is there any reason for this beside the operators for one oscillator have no effect on the states of another?

Would assume that any other mix of these operators would commute?

e.g [a'_{-}a_{+},a'_{+},a_{+}] = 0

 

[vela]

The position and momentum operators, x₁ and p₁, for one oscillator commute with the position and momentum operators, x₂ and p₂ for the second oscillator:
\begin{align*}
[x_i, x_j] &= 0 \\
[p_i, p_j] &= 0 \\
[x_i, p_j] &= i\hbar\delta_{ij}
\end{align*}It follows from these that a primed and unprimed operator commute.