Comment on "The hard sphere quantum propagator: exact results via partial wave analysis''

TL;DR

This paper discusses the particle-sphere propagator, highlighting that the Van Vleck-Gutzwiller approximation admits an exact analytic expression, contrasting prior numerical evaluations and partial wave expansions.

Contribution

It provides an exact analytic expression for the Van Vleck-Gutzwiller propagator in the particle-sphere collision problem.

Findings

The VG propagator can be expressed analytically using elementary functions.

This challenges the previous notion that the VG propagator requires numerical evaluation.

The result simplifies the analysis of quantum scattering with hard spheres.

Abstract

There is no known exact expression for the propagator of a non-relativistic particle colliding with a hard sphere. De Prunel\'e (2008 {\it J.~Phys.~A:~Math.~Theor.} {\bf 41} 255305) derived a partial wave expansion of the propagator and compared it against some known approximations, including the semiclassical Van Vleck-Gutzwiller (VG) propagator; the VG propagator was evaluated entirely numerically. Here we point out that the VG propagator for the particle-sphere problem admits an analytic expression in terms of elementary functions.

Equations

$$⟨{\mathbf{r}}_2|U_{\mathrm{VG}}|{\mathbf{r}}_1⟩=⟨{\mathbf{r}}_2|U_0|{\mathbf{r}}_1⟩-\frac{\sqrt{|D|}}{(2\pi i\hslash {)}^{3/2}}\exp \left(\frac{iS}{\hslash }\right),$$

$$S=\frac{M}{2t}{L}^2,L={\ell }_{+}+{\ell }_{-},$$

$${\ell }_{\pm }=\sqrt{a^2+r_1-2a{r}_1\cos \left(\theta /2\pm \alpha \right)}.$$

$$\frac{r_1\cos (\theta /2+\alpha )-a}{{\ell }_{+}}=\frac{r_2\cos (\theta /2-\alpha )-a}{{\ell }_{-}}.$$

$$\frac{r_1\sin (\theta /2+\alpha )}{{\ell }_{+}}=\frac{r_2\sin (\theta /2-\alpha )}{{\ell }_{-}}.$$

$$\sin \alpha \cos \alpha -u\sin \alpha +v\cos \alpha =0,$$

$$u=\frac{a}{2}\left(\frac{1}{r_1}+\frac{1}{r_2}\right)\cos \frac{\theta }{2},v=\frac{a}{2}\left(\frac{1}{r_1}-\frac{1}{r_2}\right)\sin \frac{\theta }{2}.$$

$$x=\tan \frac{\alpha }{2}$$

$$v{x}^4+2(1+u)x^3-2(1-u)x-v=0.$$

$$\alpha =-\frac{v}{1-u}-\frac{(1+2u)v^3}{6(1-u{)}^4}+\mathcal{O}(v^5).$$

Full text

Exact closed-form expressions for quantum propagators in time-dependent scattering systems are very rare. Even such a ‘simple’ scenario as a non-relativistic point particle colliding with a hard sphere has no known exact propagator. In [1], de Prunelé obtained an infinite-series partial wave expansion for the particle-sphere propagator and compared the accuracy of his expansion against the predictions of two available approximations: the approximation due to Cio and Berne [2] and the semiclassical (short-wavelength) Van Vleck-Gutzwiller (VG) propagator [3].

For the particle-sphere problem, the VG propagator takes the form (cf. (24) in [1])

[TABLE]

where $\langle\bm{r}_{2}|U_{0}|\bm{r}_{1}\rangle$ is the free-particle propagator, $S$ is the classical action corresponding to the collision path leading from point $\bm{r}_{1}$ to $\bm{r}_{2}$ in time $t$ (see figure 1), and $D=\det\left(-\frac{\partial^{2}S}{\partial\bm{r}_{2}\partial\bm{r}_{1}}\right)$ is the Van Vleck determinant. It is assumed that the origin $O$ of the coordinate frame coincides with the sphere centre (figure 1) and that points $\bm{r}_{1}$ and $\bm{r}_{2}$ do not lie in the geometric shadow of one another. The action is given by

[TABLE]

where $M$ is the particle mass, and $\ell_{+}$ (respectively $\ell_{-}$) is the distance between $\bm{r}_{1}$ (respectively $\bm{r}_{2}$) and the collision point $\bm{r}_{\mathrm{coll}}$, see figure 1. Path length $L$ is a function of the sphere radius $a$, lengths $r_{1}\equiv|\bm{r}_{1}|$ and $r_{2}\equiv|\bm{r}_{2}|$, and angle $\theta$ between $\bm{r}_{1}$ and $\bm{r}_{2}$. The absence of an explicit expression for $L(a,r_{1},r_{2},\theta)$ forced de Prunelé to compute action $S$ via numerical minimization and, subsequently, to evaluate matrix $\frac{\partial^{2}S}{\partial\bm{r}_{2}\partial\bm{r}_{1}}$ using finite differences. Below we show that function $L(a,r_{1},r_{2},\theta)$ is expressible in elementary functions.

Denoting the angle between $\bm{r}_{1}$ and $\bm{r}_{\mathrm{coll}}$ by $\theta/2+\alpha$ (see figure 1), we have

[TABLE]

An exactly solvable equation for the unknown angle $\alpha$ can be obtained in the following three steps. Firstly, we notice that the cosine of the angle between $\bm{r}_{\mathrm{coll}}$ and $(\bm{r}_{1}-\bm{r}_{\mathrm{coll}})$ is equal to the cosine of the angle between $\bm{r}_{\mathrm{coll}}$ and $(\bm{r}_{2}-\bm{r}_{\mathrm{coll}})$. This leads to $\bm{r}_{\mathrm{coll}}\cdot(\bm{r}_{1}-\bm{r}_{\mathrm{coll}})/(a\ell_{+})=\bm{r}_{\mathrm{coll}}\cdot(\bm{r}_{2}-\bm{r}_{\mathrm{coll}})/(a\ell_{-})$, or

[TABLE]

Secondly, we use the fact that $\alpha$ must extremize the length of the reflection path, $L=\ell_{+}+\ell_{-}$, i.e. $\partial L/\partial\alpha=0$. This leads to

[TABLE]

Thirdly, dividing (4) by (5), and so eliminating $\ell_{+}$ and $\ell_{-}$, and performing straightforward trigonometric manipulations, we obtain

[TABLE]

where

[TABLE]

Notice that $0<u<1$ and $-\frac{1}{2}<v<\frac{1}{2}$. The substitution

[TABLE]

transforms equation (6) into

[TABLE]

This quartic equation, and consequently equation (6), is exactly solvable (see, e.g., chapter 3 in [4]). While an explicit expression for $\alpha$ as a function of $u$ and $v$ is too lengthy to be presented in this Comment, it can be readily obtained using such symbolic packages as Mathematica or Maple. Here we only report a simple (but extremely accurate) approximate solution obtained by regarding $v\in\left(-\frac{1}{2},\frac{1}{2}\right)$ as a small parameter:

[TABLE]

The exact solution to equation (6), or the approximation given above, makes it possible to write an analytic expression for the VG propagator (1), thus avoiding the necessity of potentially expensive numerical computations.