Likelihood in the canonical ensemble

From now on we will consider a quantum mechanical canonical ensemble at temperature $1/\beta$. Such a system is described by a density operator

$$\rho = \frac{e^{-\beta H}}{{\rm Tr} \, e^{-\beta H} } ,$$ (15)

$H$ denoting the Hamiltonian of the system. Specifically, we will focus on repeated measurements of a single particle in a heat bath of temperature $1/\beta$ with sufficient time between measurements to allow the heat bath to reestablish the canonical density operator. For (non-degenerated) particle coordinates $x_i$ the likelihood for $\rho$ becomes the thermal expectation

$$p(x_i\vert\hat x,\rho) ={\rm Tr} \left( P_{\hat x} (x_i) \r... ...lpha \vert\phi_\alpha(x_i)\vert^2 =<\vert\phi (x_i)\vert^2> ,$$ (16)

$<\cdots>$ denoting the thermal expectation with probabilities

$$p_\alpha = \frac{e^{-\beta E_\alpha}}{Z}, \quad Z= \sum_\alpha e^{-\beta E_\alpha} ,$$ (17)

and energies and orthonormalized eigenstates

$$H\mbox{$\vert\,\phi_\alpha\!>$}=E_\alpha\mbox{$\vert\,\phi_\alpha\!>$} .$$ (18)

In particular, we will consider a hermitian Hamiltonian of the standard form $H$ = $T + V$, with kinetic energy $T$, being $1/(2m)$ times the negative Laplacian $-\Laplace$ for a particle with mass $m$ (setting $\hbar$ = $1$), and local potential $V(x,x^\prime)$ = $v(x) \delta (x-x^\prime )$. Thus, in one dimension

$$H(x,x^\prime)= \left(-\frac{1}{2m}\frac{\partial^2}{\partial x^2} +v(x) \right)\delta (x-x^\prime ) ,$$ (19)

where the $\delta$-functional is usually skipped.

For $n$ independent position measurements $x_i$ the likelihood for $\rho(v)$, and thus for $v$, becomes (writing now $p(x_i\vert\hat x,\rho)$ = $p(x_i\vert\hat x,v)$)

$$p(x_T\vert\hat x,v) =\prod_{i=1}^n p(x_i\vert\hat x,v) =\prod_{i=1}^n <\vert\phi(x_i)\vert^2> .$$ (20)

We remark that it is straightforward to allow $\beta$ to vary between measurements.