W h e n one meets the Schroedinger orbi- tals for the first time the following question is often asked, "How does a p electron get from one lobe to the other if it cannot pass through the nuclear plane?" This question can be dismissed in many ways. One can say that the node is just an infinitely thin mathe- matical plane having no physical significance (1). Another way is to show that it is an improper question (2) using the Heisenberg uncertainty principle: it is known that AEAt - h so that in a stationary state, where the energy is known exactly, time is completely indeterminate, and hence one cannot talk about an electron going from one place to the next since this in- volves the concept of time. The most satisfactory way to answer the question is to say that in the relativistic theory of the hydrogen atom there are no restrictive nodes a t all. This fact does not allow one to discuss the electron's motion classically, but it does show that one should not take pictorial representations of orbitals (in- cluding the ones in this article) too seriously since the shapes can he quite different in other, more complete, theory. This aspect of relativistic orbitals was dis- cussed in a recent article in THIS JOURNAL by Powell (8). The purpose of this brief note is to amplify Powell's article by presenting contour diagrams for various relativistic hydrogen orbitals. It should be notcd that the Dirac theory is not the definitive theory of the hydrogen atom. Levels with the same ,j but different 1 are not quite degenerate, hut the level with lower 1 lies slightly higher in energy (4). This is called the Lamb shift; its explanation lies in Presented in part a t the Canadian Undergraduate Physics Conference, November, 1967. Attila Szabo Harvard University Cambridge, Mossachusetts 02138 quantum electrodynamics, but it can be appreciated by noting that the relativistic electron undergoes a type of motion called Zitterleweyung (trembling-motion) (8). Since the electron is oscillating about its non-relativistic position (with all amplitude of about em) the electron is not seeing as small a potential as it should. The effect on the energy is most important when the electron is near the nucleus, and since an s electron spends more time near the nucleus than a p electron, this trembling-motion would raise its energy slightly above that of the p electron (5). Although this argu- ment is not quite correct, it can be made quantitative (6) and accounts for a large portion of the energy dif- ference. Contour Diagrams for Relativistic Orbitals Method of Computation We write the probability density 1 $1 as P,Pe where P, and Ps are functions of the polar coordinates r and 8, respectively. In order to conceptualize relativistic orbitals more easily, the sequence of energy levels of the hydrogen atom with a rough polar plot of Po, is given in Figure 1. The contour diagrams for orbitals which differ markedly from those of Schroedinger were constructed as follows (Fig. 2): a certain value of I $ I 2 was chosen, and a given angle Pe was evaluated using formulas from White's paper (7), and hence P, was calculated. From a plot of P, versus r, the values of r corresponding to the particular value of 8 were found and plotted as indicated. The entire procedure was then repeated using different values of 8 until enough points were obtained. The resulting graphs consist of contours of constant electron density. Although we used the relativistic form for P,, essentially the same 678 / Journal of Chemical Education