De Broglie–Bohm Cycles. Free Relativistic One-Half Particles

Abstract

In the de Broglie–Bohm quantum theory, particles describe trajectories determined by the flux associated with their wave function. These trajectories are studied here for relativistic spin-one-half particles. Based in explicit numerical calculations for the case of a massless particle in dimension three space-time, it is shown that if the wave function is an eigenfunction of the total angular momentum, the trajectories—here called “de Broglie–Bohm cycles”—begin as circles of slowly increasing radius until a transition time at which they tend to follow straight lines. Arrival times at some detector, as well as their probability distribution are calculated, too. The chosen energy and momentum parameters are of the orders of magnitude met in graphene’s physics.

Appendices

Appendices

Appendix A: Notations and Conventions

Units used in this paper are adapted to the physics of graphene. Length, time and energy are given in nm, ns and meV, respectively. The critical velocity and the Planck constant take the values

$$\begin{aligned} c=10^6\,\hbox {nm ns}^{-1}, \quad \hbar =6.5821 \times 10^{-4}\hbox {meV ns}. \end{aligned}$$

(A.1)

Space-time coordinate are denoted by $x^\mu$, $\mu =0,1,2$, space coordinates by ${\textbf{x}}=(x,y)$, or $(r,{\phi })$. Space-time metric is $\eta _{\mu \nu }$ = diag$(1,-1,-1)$

Dirac matrices are chosen in terms of the Pauli matrices as

$$\begin{aligned} \gamma ^0=\sigma ^z,\quad \gamma ^1=\gamma ^0\sigma ^x ,\quad \gamma ^2=\gamma ^0\sigma ^y. \end{aligned}$$

(A.2)

The Dirac matrices $\alpha ^i=\gamma ^0\gamma ^i$ used in the non-relativistic formulation explicitly are

$$\begin{aligned} \alpha ^1=\sigma ^x,\quad \alpha ^2=\sigma ^y,\quad \beta =\sigma ^z. \end{aligned}$$

(A.3)

Appendix B: Some Useful Properties of the Bessel Functions

The general solution of the Bessel equation [41]

$$\begin{aligned} z^2 f''(z)+z f'(z) + (z^2-n^2)f(z) = 0, \end{aligned}$$

(B.1)

has the form

$$\begin{aligned} f(z)=C_1 J_n(z)+C_2 Y_n(z), \end{aligned}$$

(B.2)

where $J_n$ and $Y_n$ are the Bessel functions of the first [41], respectively second [41] kind, and $C_1$, $C_2$ are two arbitrary complex constants. We shall restrict ourselves to an integer index n.

The asymptotic behaviours of the Bessel functions at the origin are given by

$$\begin{aligned} \begin{array}{ll} J_n(x)\sim \dfrac{1}{n!}\left( \dfrac{x}{2}\right) ^n \quad &{}(0<x\ll 1,\ n\ge 0),\\ Y_n(x)\sim -\dfrac{(n-1)!}{\pi }\left( \dfrac{2}{x}\right) ^n \quad &{}(0<x\ll 1,\ n\ge 1),\\ Y_0(x)\sim \dfrac{2}{\pi } \log \left( \dfrac{x}{2}\right) \quad &{} (0<x\ll 1), \end{array} \end{aligned}$$

(B.3)

and at infinity by

$$\begin{aligned} \begin{array}{ll} J_n(x)\sim \sqrt{\dfrac{2}{\pi x}}\cos \left( x-\dfrac{(n+\frac{1}{2})\pi }{2}\right) \qquad &{}(x\gg 1,\ n\ge 0) ,\\ Y_n(x)\sim \sqrt{\dfrac{2}{\pi x}}\sin \left( x-\dfrac{(n+\frac{1}{2})\pi }{2}\right) \qquad &{}(x\gg 1,\ n\ge 0). \end{array} \end{aligned}$$

(B.4)

Functions with a negative index are related to those with a positive one by the identities

$$\begin{aligned} J_{-n}(z)=(-1)^n J_n(z),\qquad Y_{-n}(z)=(-1)^n Y_n(z). \end{aligned}$$

(B.5)

Under parity $z\rightarrow -z$, the function $J_n$ transforms as

$$\begin{aligned} J_n(-z)=(-1)^n J_n(z). \end{aligned}$$

(B.6)

An interesting orthogonality property is given by [41]

$$\begin{aligned} \displaystyle {\int }_{\!\!\!\!0}^R dr\,r\, J_n\left( \frac{z_{n,\alpha }\,r}{R}\right) J_n\left( \frac{z_{n,\beta }\,r}{R}\right) =\frac{R^2}{2}\left( J_{n+1}(z_{n,\alpha }\right) ^2 \delta _{\alpha \beta }, \end{aligned}$$

(B.7)

for $n\ge 0$, where $z_{n,\alpha }$ is the $\alpha \textrm{th}$ positive zero of the Bessel function $J_n(z)$ [42]. Moreover, any function f(r) defined in the interval $0\le r\le R$ with bounded variation and vanishing at the end point $r=R$ can be represented as a “Fourier Bessel series” [43] as

$$\begin{aligned} f(r)=\sum _{\alpha =1}^\infty c_\alpha J_n\left( \frac{z_{n,\alpha }\,r}{R}\right) , \end{aligned}$$

(B.8)

for any $n\ge 0$. The coefficients $c_\alpha$ can be calculated using the orthogonality formula (B.7).