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The displacement operator generates the coherent state out of the vacuum.

$$\hat D(\alpha)|0\rangle = |\alpha\rangle$$

but I am wondering what the meaning of a displacement operator acting upon a number state with $n \neq 0$ is.

i.e $$\hat D(\alpha)|1\rangle$$

Is this meaningful? I have not been able to find any mention of this online

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It is indeed meaningful; they're called, surprisingly enough, displaced number states, and they've appeared on this site e.g. here and here. It's hard to say anything else about them without knowing what exactly you want to know about them, though.

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  • $\begingroup$ My apologies, it seems obvious now that you say it, but for whatever reason my search terms weren't picking this up. I think I can find the general information I need online now. Thank you $\endgroup$ – jm22b Dec 7 '17 at 10:54
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    $\begingroup$ @jm22b No worries. It happened to me, too, except that I had done all of the calculations in §3.4 and 3.5 of my bachelor's thesis by the time I hit on the right search term. That did teach me the value of a thorough literature search before you calculate, though ;-). As the saying goes: "A few weeks of developing and testing can save a whole afternoon in the library". $\endgroup$ – Emilio Pisanty Dec 7 '17 at 10:59

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