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One has to specify the single particle basis for the Fock state one chooses. For instance, a fock state $$|\textbf{n}\rangle = |n_1, n_2, n_3...>$$ can be written in single particle plane-wave basis with $$ \langle\textbf{r}|a_i^\dagger|0\rangle = e^{-i\textbf{q}_i\textbf{r}} $$ or
in harmonic oscillator states as: $$\langle\textbf{r}|a_i^\dagger|0\rangle = \phi_i(\textbf{r})$$ where $\phi_i(\textbf{r})$ is harmonic oscillator state in $i$th mode

But it is not clear to me from the Quantum Many Body Physics course I studied which, like other Physics courses does not rigorously teach the formal mathematics behind fock state description, how one might write a fock state where single particle states are let's say gaussian wavepackets.

Here, the single particle basis would not be orthogonal and hence the fock state would itself be not orthogonal.

My question is:

Is it possible to write fock states in any (even non-orthogonal) single particle basis states ?

If yes, how?

Lost
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  • A vector in Fock space is a sequence of $n$-particle states: $F\ni \psi=(\psi_n)_{n\in \mathbb N_0}$. You don't have to refer to any single-particle states/bases. What, I think, you are referring to is the fact that if you have an orthonormal single-particle basis, then the corresponding occupation number basis is a basis of the $n$-particle Hilbert space. The union of these bases then is a basis for the corresponding Fock space. Every choice of single-particle ONB gives hence rise to an ONB in Fock space. – Tobias Fünke Nov 18 '23 at 09:55
  • Linked. – Cosmas Zachos Nov 18 '23 at 13:10
  • There is no indication in your question about your appreciation that r is not an eigenvalue of a well-defined operator, but a thoroughly different label, as it is normally in QFT and Fock space. Several of your expressions are problematic/undefined. Are you bouncing off a QFT text? – Cosmas Zachos Nov 19 '23 at 14:26
  • @CosmasZachos Well, this depends if they consider relativistic or non-relativistic QFT... For the latter there is no problem at all. They write "Quantum Many Body Physics course", so I'd assume non-relativistic QFT here, but of course OP should clarify. – Tobias Fünke Nov 19 '23 at 21:30
  • @CosmasZachos Hi the course was not on QFT but on quantum many body Physics of ultracold gases. There the need to define fock states was minimal (like I defined in my question). Here $|\textbf{r}\rangle$ is the eigenket of position operator $ \hat r $. My intent was to say that fock state requires a single particle basis which is formally specified by creating one particle and taking its projection on the position basis. Maybe I cannot get away now studying the formal math from 1st quantization to 2nd quantization to understand all this. – Lost Nov 20 '23 at 06:45

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