Example: Particle in an External Field

As a first example, we apply this method to the problem of one particle in an external potential

$${\mathcal{H}}= - \lambda {\bf\nabla}^{2} + V(r)\;\;.$$ (144)

The one-particle density matrix will be approximated as a Gaussian with mean ${\bf m}$, width $w$ and amplitude factor $D$,

$$\rho_1({\bf r},{\bf r}',\beta) = (\pi w)^{-3/2} \: \mbox{exp} \left\{ -\frac{1}{w} ({\bf r}-{\bf m})^2 + D \right\}$$ (145)

as variational parameters. The initial conditions at $\beta\longrightarrow 0$ are $w= 4\lambda\beta$, ${\bf m}={\bf r}'$ and $D=0$ in order to regain the correct free particle limit, Eq. 3.23. For this ansatz $H$, defined in Eq. 3.20 as

$$H\equiv\int \rho{\mathcal{H}}\rho\;{\bf d}{\bf r}= \left( \frac{3\lambda}{w}+V^{[0]} \right) \frac{e^{2D}}{(2\pi w)^{3/2}}$$ (146)

where

$$V^{[n]}\equiv \left({2\over \pi w}\right)^{3/2}\int ({\bf r}-{\bf m})^{n} V(r) e^{-2({\bf r}-{\bf m})^{2}/w} {\bf d}{\bf r}$$ (147)

and

$$N\equiv \int \rho \rho' {\bf d}{\bf r}= [\pi (w+w')]^{-3/2} ... ...left\{-({\bf m}-{\bf m}')^{2}/(w+w')\right\}\exp( D+D') \quad.$$ (148)

From Eq. 3.19, the equations for the variational parameters are,

$$\dot{w} = 4 \lambda + 2 w V^{[0]} - \frac{8}{3} V^{[2]}$$ (149)

$$\dot{{\bf m}} = - 2 {\bf V}^{[1]}$$ (150)

$$\dot{D} = \frac{1}{2} V^{[0]} - \frac{2}{w} V^{[2]}\quad\;.$$ (151)

In absence of a potential, the exact free particle density matrix is recovered. The harmonic oscillator case is also correct since the Gaussian approximation is exact there. For a hydrogen atom, $\lambda=1/2$, $V(r)=-1/r$ and

$$V^{[0]} = -\frac{1}{m}\mbox{erf}\left(m\sqrt{2/w}\right)$$ (152)

$${\bf V}^{[1]} = {{\bf m}\over m^3}{w\over 4}\left[ \mbox{erf}\left(m\sqrt{2/w}\right)- \sqrt{8\over \pi w}e^{-2m^2/w}\right ]$$ (153)

$$V^{[2]} = \sqrt{w\over 2\pi}e^{-2m^2/w}+ {3 w\over 4}V^{[0]}\quad.$$ (154)

At low temperature, the density matrix as a function of ${\bf r}$ goes to the ground state wave function as discussed in more detail in the next section. One expects this to be a fixed point of the dynamics of the parameters ${\bf m}$ and $w$ determined by $\dot{{\bf m}}=0$ and $\dot{w}=0$ while $\dot{D}=-E_0$. The $\beta\to \infty$ fixed point: ${\bf m}=0$, $w=9\pi /8$, $\dot{D}=4/3\pi$ corresponds to the well known Rayleigh-Ritz variational result for a Gaussian trial wave function

$$\Psi_0({\bf r})=\left(4\over 3\pi\right)^{3/2}\exp(-8 r^{2}/9\pi)\;.$$ (155)

In ground state variational studies, addition of two more Gaussians brings the ground state energy to within $0.6$% of the exact value and similar improvement would be obtained here.

Results at finite $\beta$ require a numerical solution, which is illustrated in the figure below comparing the Gaussian variational density matrix with the exact (Pollock, 1988) and the free particle density matrix at several temperatures for the initial condition ${\bf r}'=1$. At high temperatures ($\beta=0.05$ and $\beta=0.25$) the Gaussian approximation correctly reproduces the limiting free particle density matrix. At lower temperatures, the cusp in the exact density matrix due to the Coulombic singularity at the proton becomes evident and the peak shifts to the origin somewhat faster than the Gaussian variational approximation. As $\beta$ increases the exact result grows faster than the variational since the correct energy, $-0.5$, is lower than $-4/3\pi$ but the Gaussian variational approximation remains rather accurate for $r>1$. The free particle density matrix remains centered at ${\bf r}=1$ and beyond $\beta=0.5$ ($T=54.4$ eV) bears little resemblance to the correct result.

Figure 3.1: Comparison of the Gaussian variational approximation (circles) with the exact density matrix $\rho({\bf r},{\bf r}';\beta)$ (solid line) for a hydrogen atom. The free particle density matrix (dashed line) is also shown. The plotted $r$ is along the line from the proton at the origin (marked by the vertical bar) through the initial electron position ${\bf r}'=1$.