Ladder operators, in quantum mechanics, could be used to derive eigenkets of energy with different eigenvalues (i.e. the system's energy) by raising and lowering it.
However, is there a proof that it proves a COMPLETE set of eigenkets?
Asked
Active
Viewed 493 times
1 Answers
1
Relying only on commutation relations of $a$ and $a^\dagger$ it is not possible to prove completeness of the produced basis by the action of $a^\dagger$ on $\psi_0$, since counterexamples can be constructed easily. If you instead refer to the whole theory, assuming that the Hilbert space is $L^2(\mathbb R, dx)$ and $a$ and $a^\dagger$ are constructed as you know out of $X$ and $P$, you can prove completeness observing that (1) $\psi_0$ is the first Hermite function and (2) the recurrence formula of $\psi_n$, $\sqrt{n+1}\psi_{n+1} = a^\dagger \psi_n$ is the same as for the known Hilbert basis of Hermite functions. This proves completeness.
Valter Moretti
- 80,330