Atomic Chirality

Chiral atom
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 2p1/2 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 s1/2 and p1/2 states, so that for hydrogen 2p1/2
\psi=2p1/2 + i \epsilon 2s1/2
where \epsilon is a real quantity given by
(weak perturbation)
Here HW 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)