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Commuting operators

You might have heard of commuting and non commuting operators. What do they mean?

If \begin{eqnarray*}
{\hat {A}} ({\hat {B}} \psi) & \neq & {\hat {B}} ({\hat {A}} \psi)
\end{eqnarray*}, then we say that the operators  and B do not commute. The corresponding physical observables A and B are then said to be complementary observables.

The quantity \begin{eqnarray*}
{\hat {A}} {\hat {B}} & - & {\hat {B}} {\hat {A}}
\end{eqnarray*} is expressed as [Â,B] and called the commutator of the operators  and B.

So when two operators do not commute

\begin{eqnarray*}[{\hat {A}},{\hat {B}}]& \neq & 0

and when two operators do commute

\begin{eqnarray*}[{\hat {A}},{\hat {B}}]& = & 0

In particular, quantum mechanical operators are postulated to satisfy the following commutation relations:

\begin{eqnarray*}[{\hat {q}},{\hat {p_{q^{/}}}}]& = & i \hbar \delta_{qq^{/}}

\begin{eqnarray*}[{\hat {q}},{\hat {q^{/}}}]& = & 0

\begin{eqnarray*}[{\hat {p_{q}}}, {\hat {p_{q^{/}}}}]& = & 0

The q's are the cartesian coordinates and the pq's are the corresponding conjugate momenta. Needless to mention, they are a set of complementary observables and as we shall see soon, obey Heisenberg's uncertainty principle.


Abhijit Poddar