36.4 Multi-particle wavefunctions

36.4 Multi-particle wavefunctions

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To recover the conventional formalism in which an N-particle state is described by a wavefunction

$${\psi }_N(p_1,p_2,\dots ,p_N)$$

symmetric in the N arguments, one needs to recall (see chapter 9) that the tensor product of the vector space of functions on a set $X_1$ and the vector space of functions on a set $X_2$ is the vector space of functions on the product set $X_1\times X_2$ . The symmetric tensor product will be the symmetric functions. Applying this to whatever space ${\mathcal{H}}_1$ of functions on $R$ we choose to use, the symmetric tensor product $S^{\bar{N}}({\mathcal{H}}_1)$ will be a space of symmetric functions on ${\mathbf{R}}^N$ . For details of this construction, see for instance chapter 5 of [17].

From the point of view of distributional operators $a(p),a^{†}(p)$ , given an arbitrary state $|\psi ⟩$ in the multi-particle state space, the momentum space wavefunction component with particle number $N$ can be expressed as

$${\psi }_N\left(p_1,p_2,\dots ,p_N\right)=⟨0|a\left(p_1\right)a\left(p_2\right)\dots a\left(p_N\right)|\psi ⟩$$

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