Classical representation of a quantum system at equilibrium: Applications

In the preceding paper, the structure and thermodynamics of a given quantum system was represented by a corresponding classical system having an effective temperature, local chemical potential, and pair potential. Here, that formal correspondence is implemented approximately for applications to two quantum systems. The first is the electron gas (jellium) over a range of temperatures and densities. The second is an investigation of quantum effects on shell structure for charges confined by a harmonic potential.

Figure 1

Demonstration of crossover for $r^{*}▵(t,r_s,r^{*})$ to Coulomb with effective coupling constant ${\Gamma }_e(t,r_s)$ given by Eq. (13), for $r_s=5$ and $t=0.5,1,10$. Also shown are the corresponding results for $r^{*}{▵}^{\mathrm{PDW}}(t,r_s,r^{*})$.

Figure 2

Quantum RPA pressure $p_{\mathrm{RPA}}$ at $t=0$ as a function of $r_s$.

Figure 3

Classical reduced temperature $T_c/T_F$ as a function of t for $r_s=0,1,3$, and 4.

Figure 4

Dimensionless classical chemical potential ${\mu }_c/E_F$ as a function of $t$ for $r_s=1,3,5$.

Figure 5

Radial distribution function $g\left(r^{*}\right)$ for $r_s=6$ at $t=0.5,1,4,$  and 8. Also shown are the results of PIMC.

Figure 6

Radial distribution function $g\left(r^{*}\right)$ for $r_s=6$ at $t=0.5,1,4$, and 8. Also shown are the results of PDW.

Figure 7

Radial distribution function $g\left(r^{*}\right)$ for $t=0$ at $r_s=1,5,\mathrm{and}10$. Also shown are results from PIMC and diffusion Monte Carlo. The PIMC and diffusion Monte Carlo plots are indistinguishable.

Figure 8

Dimensionless classical pressure $p_c/\left(n{\epsilon }_F\right)$ as a function of $t$ for $r_s=1,3,\mathrm{and}5$. Also shown are the corresponding modified RPA results.

Figure 9

Effective classical trap potential $▵{\beta }_c{\mu }_c\left({\mathbf{r}}^{*}\right)\equiv [{\left({\beta }_c{\mu }_c\right)}^{\left(0\right)}\left(\mathbf{0}\right)-{\left({\beta }_c{\mu }_c\right)}^{\left(0\right)}\left({\mathbf{r}}^{*}\right)]/\Gamma$ as a function of $r^{*}$ for $t=0.5,10$. Also shown is the harmonic potential.

Figure 10

Comparison of $-c\left(r^{*}\right)$ and ${\Delta }_K\left(r^{*}\right)$ at $t=0.1,0.27$ both corresponding to $\Gamma =3$. Also shown is the Coulomb limit $\beta q^2/r$.

Figure 11

Diffraction mean-field approximate density profile for $\Gamma =3$ and $t=0.1,0.5,\mathrm{and}1$.

Figure 12

Exchange mean-field approximate density profile for $r_s=5$ and $t=0.5,1,2,5,\mathrm{and}10$.