Aug 30, 2013

On the conserved observable in a truncated subspace

In quantum mechanics, a time-independent observable $A$ is a conserved quantity (constant of motion) if, according to Heisenberg equation, it commutes with the Hamiltonian, \[
\frac{\mathrm{d}}{\mathrm{d}t} A = i [H, A] \quad \Rightarrow \quad [H, A] = 0.
\] An interesting question is that if $A$ is still a conserved quantity in a truncated subspace.

Let's $P$ be the projection operator for the truncated subspace, $P^2 = P$. In the truncated subspace, the operators $H$ and $A$ are replaced with $P H P$ and $P A P$ respectively. Then, \[
[ P H P, P A P ] = P H P A P - P A P H P = P [H, A] P + P [ [P, H], [P, A] ] P = P [ [P, H], [P, A] ] P
\] Therefore, in the truncated space, the truncated observable is not necessarily a conserved quantity. There are two important exceptions:

  1. The truncation is consistent with the eigensubspaces of the Hamiltonian, i.e. $[P, H] = 0$, $A$ is still conserved. This means solving the problem first. 
  2. The truncation is consistent with the observable $A$, namely $[A, P] = 0$. 

Example: the shell model. The basis space is generated with 3D harmonic oscillator Hamiltonian $G$. A cutoff is performed on its eigensubspaces (all the degenerate eigensubspaces have to be included). Then the angular momentum $[G, J] = 0$, so $[P, J] = 0$. Therefore, in the truncated subspace, the angular momentum is still a good quantum number (a conserved quantity). 

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