Combining two finite number fock spaces into one

Say I have two separate systems of identical Bosons, one with N Bosons the other with M. System one is described by a state $|\psi_1\rangle$ the other with $|\psi_2 \rangle$ which are expressed in a Fock space like

$|\psi_1\rangle = \sum_{n_1,...,n_{max}} \alpha(n_1,..,n_{max}) |n_1,n_2,..,n_{max}\rangle$

$|\psi_2\rangle = \sum_{n_1,...,n_{max}} \beta(n_1,..,n_{max}) |n_1,n_2,..,n_{max}\rangle$

where $|n_1,n_2,..,n_{max}>=\prod_{k=1}^{max} \frac{(a^{\dagger}_k)^{n_k}}{\sqrt{n_k!}} |vac\rangle$

with "max" denoting the maximum occupied mode, $\alpha$ and $\beta$ some constants depending on each of the values (zero if $\sum_{k} n_k$ is not equal to $N$ for $\alpha$ or $M$ for $\beta$) and the wavefunction satisfying all the usual normalisation conditions.

At someone point I wish to bring these two subsystems together, this state can be expressed as an $N+M$ body Fock space.

$|\psi_{total}\rangle = |\psi_1\rangle \otimes|\psi_2\rangle$

For distinguishable particles this is fairly trivial, however the symmetry makes it somewhat unclear (to me) how to do this and give states with the appropriate amplitudes.

Can anyone tell me, or point me to an appropriate book or paper?

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– Michael Brown Feb 19 at 15:47

1 Answer

Possible answer to the question (if anyone could confirm this that would be great)

$$|\psi_1⟩⊗|\psi_2⟩ \propto \sum_{m_1,..,m_{max},n_1,...,n_{max}} \beta(m_1,..)\alpha(n_1,..) \prod_{k=1}^{max} \frac{(a^{\dagger}_k)^{n_k+m_k}}{\sqrt{n_k! m_k!}} |vac \rangle$$

Which seems to give the correct weighting with symmetry but does require a correction to the normalisation.

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