Atomic Chirality

The chirality of an atom arising from the neutral current weak interaction between the electron and a nucleon can be shown by plotting the electron probability current density for a given atomic state, shown here for the 2p_1/2 state in hydrogen. Under a parity transformation, or equivalently under mirror reversal, the helicity of the streamlines is reversed: the atom is fundamentally handed.
(After R. A. Hegstrom et al, Am. J. Phys. 56 p1086, 1988).

The image is calculated as follows. The probability current density is given by
$J(x)=Re{\psi*(x) P \psi(x)}$
The weak interaction mixes s_1/2 and p_1/2 states, so that for hydrogen 2p_1/2
$\psi=2p1/2 + i \epsilon 2s1/2$
where $\epsilon$ is a real quantity given by

Here H^W is the weak interaction Hamiltonian, and $\epsilon$ is of order 10^-11 for hydrogen. Solving for J by substituting standard hydrogenic wavefunctions, the streamline is obtained by starting at an arbitrary point and following J in small increments.

In order for the helicity to be clearly visible, it is necessary to arbitrarily increase $\epsilon$ to a value of 0.1, equivalent to increasing the effect of the weak perturbation. Also, the curve shown is strictly only quasi-periodic; it does not return exactly to the starting point, so it has been slightly fudged. It traces out a toroidal surface. Without the weak perturbation ( $\epsilon$ =0) the streamlines are circular sections on the same torus, with no z component and no helicity, and are then symmetric under a parity transformation.

RBW (2 Dec 1997)