# Eigenvalue of lowering operator

How to prove that eigenvalue of lowering operator is zero?

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ShayanJ
Gold Member
Eigenvalue of lowering operator is zero only on the ground state and this is so by the definition of ground state.
For other eigenvectors of the lowering operator(coherent states), eigenvalues are non-zero.

Ground state or the absolute minimum state of the system is "defined" as the state which gives zero when operated by the
lowering operator.

It is easy to consider the H.OSc. and operate with the lowering operator till it gets to the min = h-bar omega/2 and if you apply furthe the lowering operator then you get zero. Of course the system maybe different in different situation and hence the minimum state may vary.

watch out that minimum state or the vacuum state is not the zero state, a zero state does not exists in nature.
The proof is kind of trival. google on Fock space and there you get your proof.