All rights reserved under Pan American and InternatiOnal CLlP) right ('onven tions. Bihliographical No/I' This Dover edition, first publi5hed in 200L is an unabridged reprint Llf the work originally published by the Belfer Graduate School of Science, Yeshiva University, New York, in 1964. Library 0/ Congress Cataloging-in-Publication Data Dirac, P. (Paul Adrien Maurice), 1902 Lectures on quantum mechanics I by Paul A.
Originally published: New York: Belfer Graduate School Llf Science, Yeshiva University, 1964.12-dc21 00-065608 Manufactured in the United States of America ()l1ver Publications, Inc. 31 East 2nd Street, Mineola, N.com CONTENTS Lecture 1Vo. The Hamilton Method 1 2. The Problem of Quantization 25 3.
Quantization on Curved Surfaces 44 4. Quantization on Flat Surfaces 67 [ v ] www. DIRAC Lecture No.1 THE HAMILTONIAN METHOD I'm very happy to be here at Yeshiva and to have this chance to talk to you about some mathematical methods that I have been working on for a number of years. I would like first to describe in a few words the general object of these methods.
In atomic theory we have tv deal with various fields. There are some fields which are very familiar, like the electromagnetic and the gravitational fields; but in recent times we have a number of other fields also to concern ourselves with, because according to the general ideas of De Broglie and Schrodinger every particle is associated with waves and these waves may be considered as a field. So we have in atomic physics the general problem of setting up a theory of various fields in inter- action with each other. We need a theory conforming to the principles of quantum mechanics, but it is quite a difficult matter to get such a theory.
One can get a much simpler theory if one goes over to the corresponding classical mechanics, which is the form which quantum mechanics takes when one makes Planck's constant Ii tend to zero. It is very much easier to visualize what one is doing in terms of classical www.com LECTURES ON QUANTUM MECHANICS mechanics. It will be mainly about classical mechanics that I shall be talking in these lectures. N ow you may think that that is really not good enough, because classical mechanics is not good enough to describe Nature.
Nature is described by quantum mechanics. Why should one, therefore, bother so much about classical mechanics? Well, the quantum field theories are, as I said, quite difficult and so far, people have been able to build up quantum field theories only for fairly simple kinds of fields with simple interactions between them. It is quite possible that these simple fields with the simple interactions between them are not adequate for a description of Nature. The successes which we get with quantum field theories are rather limited.
One is continually running into difficulties and one would like to broaden one's basis and have some possibility of bringing more general fields into account. For example, one would like to take into account the possibility that Maxwell's equations are not accurately valid. When one goes to distances very close to the charges that are producing the fields, one may have to modify Maxwell's field theory so as to make it into a non- linear electrodynamics. This is only one example of the kind of generalization which it is profitable to consider in our present state of ignorance of the basic ideas, the basic forces and the basic character of the fields of atomic theory.
In order to be able to start on this problem of dealing with more general fields, we must go over the classical theory. Now, if we can put the classical theory into the Hamiltonian form, then we can always apply certain standard rules so as to get a first approximation to a quantum theory. My talks will be mainly concerned with [2 ] www.com THE HAMILTONIAN METHOD this problem of putting a general classical theory into the Hamiltonian form. When one has done that, one is well launched onto the path of getting an accurate quantum theory.
One has, in any case, a first approximation. Of course, this work is to be considered as a prelimin- ary piece of work. The final conclusion of this piece of work must be to set up an accurate quantum theory, and that involves quite serious difficulties, difficulties of a fundamental character which people have been worrying over for quite a number of years. Some people are so much impressed by the difficulties of passing over from Hamiltonian classical mechanics to quantum mechanics that they think that maybe the whole method of working from Hamiltonian classical theory is a bad method.
Particularly in the last few years people have been trying to set up alternative methods for getting quantum field theories. They have made quite considerable progress on these lines. They have obtained a number of conditions which have to be satisfied. Still I feel that these alterna- tive methods, although they go quite a long way towards accounting for experimental results, will not lead to a final solution to the problem.
I feel that there will always be something missing from them which we can only get by working from a Hamiltonian, or maybe from some generalization of the concept of a Hamiltonian. So I take the point of view that the Hamiltonian is really very important for quantum theory. In fact, without using Hamiltonian methods one cannot solve some of the simplest problems in quantum theory, for example the problem of getting the Balmer formula for hydrogen, which was the very beginning of quantum mechanics. A Hamiltonian comes in therefore in very elementary ways and it seems to me that it is really quite www.com UTTURES ON QUANTUM MECHANICS ,.
,('Itlial to work from a Hamiltonian; so I want to talk I () YOl! about how far one can develop Hamiltonian Ilwt hods. I would like to begin in an elementary way and I take ;h Illy starting point an action principle. That is to say, I ;1~;~l!lIIe that there is an action integral which depends on I Ill' llIotion, such that, when one varies the motion, and Pllts down the conditions for the action integral to be lationar)" one gets the equations of motion. The method t d starting from an action principle has the one great :Id vantage, that one can easily make the theory conform :t) the principle of relativity.
We need our atomic theory t () conform to relativity because in general we are dealing \\ ith particles moving with high velocities. If we want to bring in the gravitational field, then we han: to make our theory conform to the general principle of relativity, which mean5 working with a space-time which is not fiat. Now the gravitational field is not very important in atomic physics, because gravitational forces arL' extremely weak compared with the other kinds of forces which are present in atomic processes, and for practical purposes one can neglect the gravitational field. People have in recent years worked to some extent on bringing the gravitational field into the quantum theory, but I think that the main object of this work was the hope that bringing in the gravitational field might help to solve some of the difficulties.
As far as one can see at present, that hope is not realized, and bringing in the gravitational field seems to add to the difficulties rather than remove them. So that there is not very much point at present in bringing gravitational fields into atomic theory. However, the methods which I am going to describe are powerful mathematical methods which www.com THE HAMILTONIAN METHOD would be available whether one brings in the gravita- tional field or not. We start off with an action integral which I denote by I = J L dt.
(1-1) It is expressed as a time integral, the integrand L being the Lagrangian. So with an action principle we have a Lagrangian. We have to consider how to pass from that Lagrangian to a Hamiltonian. When we have got the Hamiltonian, we have made the first step toward getting a quantum theory.
You might wonder whether one could not take the Hamiltonian as the starting point and short-circuit this work of beginning with an action integral, getting a Lagrangian from it and passing from the Lagrangian to the Hamiltonian. The objection to trying to make this short-circuit is that it is not at all easy to formulate the conditions for a theory to be relativistic in terms of the Hamiltonian. In terms of the action integral, it is very easy to formulate the conditions for the theory to be relativistic: one simply has to require that the action integral shall be invariant. One can easily construct innumerable examples of action integrals which are invariant.
They will automatically lead to equations of motion agreeing with relativity, and any developments from this action integral will therefore also be in agree- ment with relativity. When we have the Hamiltonian, we can apply a standard method which gives us a first approximation to a quantum theory, and if we are lucky we might be able to go on and get an accurate quantum theory. You might [ 5] www.com LECTURES ON QUANTUM MECHANICS again wonder whether one could not short-circuit that work to some extent. Could one not perhaps pass directly from the Lagrangian to the quantum theory, and short- circuit altogether the Hamiltonian? Well, for some simple examples one can do that.
For some of the simple fields which are used in physics the Lagrangian is quadratic in the velocities, and is like the Lagrangian which one has in the non-relativistic dynamics of particles. For these examples for which the Lagrangian is quadratic in the velocities, people have devised some methods for passing directly from the Lagrangian to the quantum theory. Still, this limitation of the Lagrangian's being quadratic in the velocities is quite a severe one. I want to avoid this limitation and to work with a Lagrangian which can be quite a general function of the velocities.
To get a general formalism which will be applicable, for example, to the non-linear electrodynamics which I mentioned previously, I don't think one can in any way short- circuit the route of starting with an action integral, getting a Lagrangian, passing from the Langrangian to the Hamiltonian, and then passing from the Hamiltonian to the quantum theory. That is the route which I want to discuss in this course of lectures. In order to express things in a simple way to begin with, I would like to start with a dynamical theory involving only a finite number of degrees of freedom, such as you are familiar with in particle dynamics. It is then merely a formal matter to pass from this finite number of degrees of freedom to the infinite num- ber of degrees of freedom which we need for a field theory.
Starting with a finite number of degrees of freedom, we have dynamical coordinates which I denote by q.com THE HAMILTONIAN METHOD The general one is qn, n = 1,···, N, N being the num- her of degrees of freedom. Then we have the velocities dqnldt = qn" The Lagrangian is a function L = L(q, q) of the coordinates and the velocities. You may be a little disturbed at this stage by the importance that the time variable plays in the formalism. We have a time variable t occurring already as soon as we introduce the Lagrangian.
It occurs again in the velocities, and all the work of passing from Lagrangian to Hamiltonian involves one particular time variable. From the relativistic point of view we are thus singling out one particular observer and making our whole formalism refer to the time for this observer. That, of course, is not really very pleasant to a relativist, who would like to treat all observers on the same footing.