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If I implement the annihilation operator physically and my system is in the $ |0\rangle$ state, what does it mean to apply this operator on this state? What does it mean for the state to be annihilated?

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  • $\begingroup$ That $|0\rangle$ is in the kernel of the annihilation operator? Physically? It gets you out of your Hilbert space. $\endgroup$ Commented Apr 9, 2022 at 21:52

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Since the Schrodinger equation is unitary (meaning it preserves the norm of the wavefunction), and since $|0\rangle$ has unit norm while $0$ has zero norm, the wavefunction will never evolve from $|0\rangle$ to $0$.

The equation $a|0\rangle=0$ is an algebraic statement about the spectrum of the quantum harmonic oscillator (QHO), it doesn't represent a physical process. A physical interpretation of this equation is that there exists a lowest energy state in the spectrum of the QHO.

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