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The Relativistic Electron Wave Equation P europhosycs BULLETIN OF THE EUROPEAN PHYSICAL SOCIETY news J.A. Volume 8 Number 10 October 1977 The Relativistic Electron Wave Equation P. A. M. Dirac An abridged version of the lecture celebrating both the 75th. birthday of Professor Dirac and the 50th. anniversary of the publication of this fundamental equation of Nature, given on July 4 at the Conference on Particles Physics in Budapest. The full text will be published in the Conference Proceedings. Before the period that I want to The outline of Heisenberg’s method I was thinking over Heisenberg’s talk about, that is to say in the early was to set up a theory dealing with ideas, concentrating on the non-com­ 1920s, we had a period of frustration. only observable quantities. These mutation, and it occured to me rather We had the theory of the Bohr orbits observable quantities fitted into ma­ by accident that there was really a which worked very well for some sim­ trices, so he was led to consider ma­ great similarity between the commu­ ple problems, essentially for those pro­ trices, and he had the idea of consi­ tator of two quantities that do not com­ blems where only one electron was dering the matrix as a whole instead mute and the Poisson bracket which playing an important role. People of just dealing with particular matrix we have in classical mechanics. As a were trying to extend the theory to elements. Dealing with matrices one result of this similarity, the equations deal with several electrons, for exam­ is then directed to non-commutative in the new mechanics with non-com­ ple to the spectrum of helium, where algebra. mutation appeared as analogous to two electrons are concerned, but they Now it was really very difficult for the equations in the old mechanics of did not know how to do it. There were physicists to accept non-commutative Newton, when these old equations basic ambiguities in applying the rules algebra In those days. Heisenberg were expressed in the Hamiltonian of quantization and people did not himself had very grave doubts when form. On the strength of this analo­ know what to do. They could only pro­ he first noticed that his algebra was gy one immediately had a general ceed by making various artificial actually non-commutative, and he won­ connection between the old mecha­ assumptions and these assumptions dered very much whether he would nics and the new mechanics of were not very successful. not have to abandon the whole idea Heisenberg. Now this frustration is something because of the non-commutation. But That was the start of my work. It that one can understand again very still he found that It was unavoidable gave me a rather different handle well at the present time, because we and he had to accept it. from Heisenberg because I had the have a similar situation with regard I learned about this theory of non-commutation as the essential new to the relativistic quantum theory for Heisenberg in early September, 1925. feature. dealing with high energy particles. It was very difficult for me to appre­ Again we have this feeling that we ciate it at first, then I suddenly reali­ don’t know the basic rules. We know zed that the non-commutation was some rules which work only with a actually the most important idea that Contents limited degree of success and essen­ was introduced by Heisenberg. It was tially we are in a similar situation the one drastic new idea which would Relativistic Electron Wave where we don’t know what the correct provide the whole basis of any new Equation............................... 1 basic assumptions are that we can theory which one was going to cons­ Precision Determination of hold fast to. truct. Working with his matrices, Atomic Masses........................5 In 1925, the whole situation was Heisenberg was led to a new equation 9th European Conference on suddenly changed by Heisenberg who of motion for them, namely. Atomic Spectroscopy . 8 had a really brilliant idea. He was led Quantum Transport in to introduce the idea of non-commuta­ ih du/dt = uH — Hu, (1) Crystalline Semiconductors . 9 ting algebra into physics. This idea where u is some dynamical variable Public Attitudes to Science . 11 was most startling and most unexpec­ and H is a diagonal matrix which re­ Society News..........................12 ted. presents the energy. turophysics News is published monthly by the European Physical Society. © 1977. Reproduction rights reserved. 1 The idea of bringing in non-commu­ physical importance. There I was was then just a question of a mathe­ tation proved to be the key to deve­ wrong. Schrödinger did take these matical transformation to pass from loping a new mechanics, which ena­ waves seriously. He thought that they the Heisenberg theory to the Schrö­ bled one to escape from the frustra­ really would be associated with the dinger theory. They were two mathe­ tion that had been holding us up motion of an electron in an atom, but matically equivalent theories for the during the previous years. The result one would have to modify the wave same underlying physics. That under­ was a period of great activity among equation somewhat to take into ac­ lying physics is what we now call theoretical physicists at that time, count the electromagnetic field in quantum mechanics. great excitement together with great which the electron was moving. We then had a satisfactory situa­ activity. There was so much work to He tried to guess a good way to tion of one good theory. The result of do developing the new ideas and modify this equation (2) of de Broglie Schrodinger’s work was to introduce a seeing how the equations of the old keeping to the requirements of relati­ new concept, the wave function Ψ, mechanics could be translated into vity. Well, he was able to guess this which was a great help for the physical the new theory. One could get new equation : interpretation of the theory. It was results very easily and one had great found that if you take Ψ and suitably confidence that one was really getting normalize it, then |Ψ2| gives the proba­ somewhere. One had the possibility bility of finding the particle in any of developing the new theory in a place. general way and also of applying in to (3) One had to get used to the idea examples and working out equations. that the new mechanics only gave one These equations involved non-com- This equation reduces to the pre­ probabilities and did not give one the mutating quantities and there was the vious equation (2) when you put the determinism of the previous classical problem of getting some physical electromagnetic potentials A equal mechanics. That was a feature which a interpretation for the results that were to zero. So far as I know it was guess­ lot of physicists found very hard to obtained with the new equations. This work of Schrodinger to obtain this accept, but which turned out to be problem of getting the interpretation equation from de Broglie’s equation. unavoidable when one had more proved to be rather more difficult than Now when Schrodinger had that power for understanding the results of just working out the equations. It was equation, the first thing he did, of calculations with the non-commutative not completely solved until two or algebra. three years after the original idea of course, was to apply it to the electron non-commutation was introduced. in the hydrogen atom. He worked out I was working on this and conside­ I don’t think it had ever happened the energy levels of hydrogen, and ring the problem of getting the proba­ in physics previously, that one had he got a wrong result because his bility for other dynamical variables to equations before one knew the equation did not take into account the have specified values. I worked out a general way to interpret them. But that spin of the electron. He went back to general theory for these probabilities. is what happened in this case. it a few months later and then noticed This general theroy enabled one to In the early examples one just had that if he was less ambitious and just transform the Schrodinger wave func­ special rules for interpretation. For wrote his equation in a non-relativistic tion to other forms. One then had the example, one had a matrix to repre­ way and then applied it, he got results possibility of calculating the probabi­ sent the energy, a diagonal matrix, in agreement with observation apart lity of any dynamical variable having and one said its diagonal elements from the fine structure of the hydrogen a specified value, or of several vari­ were the energy levels. That was just spectrum, which depends on the rela­ ables simultaneously having specified a special assumption giving us the tivistic corrections. In the non-relati­ values, provided they commute with energy levels, and it worked. vistic approximation Schrodinger’s each other. The method was to trans­ To get a general interpretation one equation reads like this, in the form the Schrödinger function to refer was helped by some other work that absence of a magnetic field : to these variables that one is intere­ was done independently by Schro- sted in, and again, to form the square dinger. Schrodinger was working quite of its modulus. independently of Heisenberg, and to I was able to work out this general begin with he knew nothing about (4) transformation theory and I felt very Heisenberg’s work.
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