Systems of Fermions

In this section the Bayesian approach for inverse problems will be applied to many-body systems. To be specific, we will study the simultaneous measurement of the positions of $N$ particles. We assume the measurement result to be given as a vector $x_i$ consisting of $N$ single particle coordinates $x_{i,j}$. The treatment can easily be generalized to partial measurements of $x_i$ by including an integration over components which have not been observed. The likelihood for $v$, if measuring a vector $x_i$ of coordinates, becomes for a many-body system

$$p(x_i\vert\hat x,v) = {\rm Tr} \Big(\mbox{$\vert\, x_{i,1}... ...x_{i,N}\!><\! x_{i,1},\cdots x_{i,N} \, \vert$}\, \rho \Big) ,$$ (61)

which is now a thermal expectation with respect to many-body energies $E_\alpha$

$$p(x_i\vert\hat x,v) = \sum_\alpha p_\alpha \vert\psi^{(N)}_... ...\vert^2 = <\vert\psi^{(N)}(x_{i,1},\cdots x_{i,N})\vert^2 > .$$ (62)

In particular, we will be interested in fermions for which the wave functions $\psi_\alpha$ and $\mbox{$\vert\,x_{i,1},\cdots x_{i,N}\!>$}$ have to be antisymmetric. Considering a canonical ensemble, the density operator $\rho$ has still the form of Eq. (15), but with $H$ replaced now by a many-body Hamiltonian. For fermions, it is convenient to express the many-body Hamiltonian in second quantization, i.e., in terms of creation and annihilation operators [76,77]. A Hamiltonian with one-body part $T$, e.g., $T$ = $-(1/2m)\Delta$, and two-body potential $V$ can so be written

$$H = T + V = \sum_{ij}T_{ij} \,a^\dagger_i a_j + \frac{1}{4}\sum_{ijkl} V_{ijkl} \, a^\dagger_i a^\dagger_j a_l a_k ,$$ (63)

with antisymmetrized matrix elements $V_{ijkl}$. Hereby, $a_\alpha a_\gamma^\dagger +a_\gamma^\dagger a_\alpha$ = $\mbox{$<\!\varphi_\alpha\,\vert\,\varphi_\gamma\!>$}$ is equal to the overlap of the one-body orbitals $\mbox{$\vert\,\varphi_\alpha\!>$}$ = $a_\alpha^\dagger\mbox{$\vert\,0\!>$}$ and $\mbox{$\vert\,\varphi_\gamma\!>$}$ = $a_\gamma^\dagger\mbox{$\vert\,0\!>$}$ which are created or destroyed by the operators $a_\gamma^\dagger$ or $a_\alpha$, respectively. Furthermore, $a_\alpha^\dagger a_\gamma^\dagger +a_\gamma^\dagger a_\alpha^\dagger$ =0, $a_\alpha a_\gamma +a_\gamma a_\alpha$ =0. A two-body eigenfunction of the Hamiltonian (63) can for example be expanded as follows

$$\mbox{$\vert\,\psi^{(2)}_\alpha\!>$} = \sum_{\alpha,\gamma}... ...ha,\gamma} \mbox{$\vert\,\varphi_\alpha,\varphi_\gamma\!>$} ,$$ (64)

where $\mbox{$\vert\,\varphi_\alpha,\varphi_\gamma\!>$}$ = $a_\alpha^\dagger a_\gamma^\dagger\mbox{$\vert\,0\!>$}$ denotes a Slater determinant being an antisymmetrized wavefunction.

The symmetrized version of a potential, local in relative coordinates, is

$$V_{x_1 x_2 x^\prime_1 x^\prime_2} =v(\vert x_1-x_2\vert) \B... ...prime) - \delta(x_1-x_2^\prime) \delta(x_2-x_1^\prime) \Big) .$$ (65)

Here we can always choose $v(0)$ = 0. Now, assume we are interested in the reconstruction of $v(x)$ for $x>0$. Solving the stationarity equation of the maximum posterior approximation analogous to Eq. (44) of Sect. 3, the prior terms remains unchanged and only the likelihood terms have to be adapted. Using

$$\delta_{v(x)} v(\vert x_1-x_2\vert) = \delta (x-\vert x_1-x_2\vert) ,$$ (66)

we find

$$\delta_{v(x)} H = \frac{1}{2}\sum_{x_1} a^\dagger_{x_1} a^\... ...{x_1} a^\dagger_{x_1} a^\dagger_{x_1+x} a_{x_1+x} a_{x_1} ,$$ (67)

where $x>0$, and can write, similar to the one-body case,

$$\delta_{v(x)} E_\alpha = \frac{<\!\psi_\alpha\,\vert\,\delta_{v(x)} H\,\vert\,\psi_\alpha\!>} {\mbox{$<\!\psi_\alpha\,\vert\,\psi_\alpha\!>$}} ,$$ (68)

$$\mbox{$\vert\,\delta_{v(x)} \psi_\alpha\!>$} = \sum_{\gamma \atop E_\gamma\ne E_\alpha} \frac{1}{E_\alpha-E_\gam... ...ma\!><\! \psi_\gamma \, \vert$} \delta_{v(x)} H\mbox{$\vert\,\psi_\alpha\!>$} .$$ (69)

From this the functional derivatives of the likelihoods, $\delta_{v(x)} p(x_i\vert\hat x,v)$, can be obtained. However, a direct numerical or analytical solution of the full inverse many-body equations is usually not feasible. To deal with this problem, a mean field approach will be developed in the next section.