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Suppose $\mathcal{B}=\{|0\rangle, |1\rangle, |2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} |\Psi\rangle = c_{0}|0\rangle + c_{1}|1\rangle + c_{2}|2\rangle + ... \end{equation} Besides the normalization condition \begin{equation} \langle\Psi|\Psi\rangle = |c_{0}|^2+ |c_{1}|^2+ |c_{2}|^2+...=1, \end{equation} are there any other mathematical constrains on the complex amplitudes $c_0,c_1,c_2,...$?

Also, is there a protocol, perhaps from quantum optics, for creating an arbitrary $|\Psi\rangle$ as defined above?

For example, the complex amplitudes which define the state can be taken as \begin{equation} \{c_n\}=\{ \frac{1}{\sqrt{\zeta\left(3\right)}}\frac{1}{\left(1+n\right)^{3/2}} \}, \end{equation} where $\zeta\left(3\right)\approx 1.202$ is given by the Riemann zeta function. In this case, the normalization condition holds. Also, we get a nice finite value for the mean energy \begin{equation} \langle\Psi|\hat{n}|\Psi\rangle \approx0.019, \end{equation} where $\hat{n}=\hat{a}^{\dagger}\hat{a}$ is the number operator (i.e. the Hermitian operator corresponding to the eigen-basis $\mathcal{B}$). However,the variance in the energy \begin{equation} \text{Var}[\hat{n}]=\langle\Psi|\hat{n}^2|\Psi\rangle-\langle\Psi|\hat{n}|\Psi\rangle^2 \end{equation} is infinite. Can such a state exist in nature?

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    $\begingroup$ There aren't any other constraints on the complex coefficients. Could you clarify what it is you're trying to do, or what problem you're trying to solve? $\endgroup$ – Kyle Arean-Raines Jun 18 '15 at 14:05
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    $\begingroup$ Yes, I get $\langle \Psi | \hat{n} | \Psi \rangle$ to be finite and $\langle \Psi | \hat{n}^2 | \Psi \rangle$ diverges. For example, take $c_{n} = \sqrt{\frac{1}{\zeta(3)}}\sqrt{\frac{1}{\left(1+n\right)^3}}$, where $\zeta(3)$ is the Riemann zeta function. Then $\langle \Psi | \hat{n}^2 | \Psi \rangle$ will indeed diverge. $\endgroup$ – Mustapha Mond Jun 18 '15 at 15:00
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    $\begingroup$ Ah yes, that was dumb. Thanks for the correction :) $\endgroup$ – Kyle Arean-Raines Jun 18 '15 at 15:08
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    $\begingroup$ My only advice is to compare it to the standard college physics paradigm for this sort of thing, as @CRDrost recommends: To wit, the sole bound state of the δ-function potential has a kink at the origin, which amounts to the long tails of its Fourier transform the Cauchy distribution, which is likewise momentless. So, the kink at the origin of time of your wave-function will be useful or not. But QM is blasé and inured to this sort of thing.... $\endgroup$ – Cosmas Zachos Jul 1 '17 at 15:51
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    $\begingroup$ In any case, Apéry's constant is bread-and-butter in theoretical physics, and pops up all over the scene. But, as I indicated, it behooves you to compute the energy F.T. of this, so stick in the phase factors exp(-i E(n) t)s and study the evident Ψ(t) s. $\endgroup$ – Cosmas Zachos Jul 1 '17 at 16:13

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