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Suppose $\mathcal{B}=\{\lvert 0\rangle, \lvert 1\rangle, \lvert 2\rangle, ... \}$ is the energy eigen-basis of a quantum simple harmonic oscillator. I want to create the state \begin{equation} |\Psi\rangle = c_{0}\lvert 0\rangle + c_{1}\lvert 1\rangle + c_{2}\lvert 2\rangle + ... \end{equation} Besides the normalization condition \begin{equation} \langle\Psi\mid\Psi\rangle = \lvert c_{0}\rvert^2+ \lvert c_{1}\rvert^2+ \lvert c_{2}\rvert^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?

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  • $\begingroup$ 1) Nope. 2) Not sure what you are taking about creating an arbitrary $|\Psi\rangle$. Sometimes people (e.g those works on lasers) find it more convenient to use quantum coherent states instead of $|0\rangle, |1\rangle, |2\rangle,\ldots$ as a basis. $\endgroup$ – achille hui Jun 18 '15 at 14:13
  • $\begingroup$ @achillehui Thanks for the comment. To clarify: I have a set of complex amplitudes $\{c_0,c_1,c_2,...\}$ that satisfy the normalization condition. However, the resulting $|\Psi\rangle$ from such amplitudes is quite peculiar: It has a finite average energy, $\langle \Psi|\hat{n}|\Psi\rangle$, but the variance, $\langle \Psi|\hat{n}^2|\Psi\rangle - \langle \Psi|\hat{n}|\Psi\rangle^2$, in the energy is infinite. Is this physically possible? Here I have assumed that $a^{\dagger}a=\hat{n}$ is the number operators: i.e. $\hat{n}|n\rangle = n|n\rangle$ for all $n=0,1,2...$ $\endgroup$ – Mustapha Mond Jun 18 '15 at 14:38
  • $\begingroup$ If it is a pure quantum state driven by some external source, I won't be surprised you can get something with infinite variance. The real thing you should pay attention is in the approximation to write down the Hamilton of your system, whether there are terms one usually ignore become significant when the variance becomes infinite. This can change the dynamic of the system. $\endgroup$ – achille hui Jun 18 '15 at 14:55
  • $\begingroup$ The "free Hamiltonian" governing the dynamics of this $|\Psi\rangle$ will simply be $H \sim \hat{n}$. More specifically, the complex amplitudes are given by $c_n = \sqrt{\frac{1}{\zeta(3)}} \sqrt{\frac{1}{\left(1+n\right)^3}}$. I am wondering if the resulting $|\Psi\rangle$ is physically possible? $\endgroup$ – Mustapha Mond Jun 18 '15 at 15:17
  • $\begingroup$ The state is physically possible. However, if you make it in contact with the environment, say a heat bath, the large $c_n$ for large $n$ will make it highly unstable. If you need it in a real world experiment, then you need to worry about its life time. If you need it on paper, I don't see any problem of that. $\endgroup$ – achille hui Jun 18 '15 at 15:24