The Dreams That Stuff is Made of

by Stephen Hawking

  Rutherford found experimentally that the α-particles emitted by various radioactive elements, such as uranium and radium, have much smaller kinetic energy than that needed to get out over the top of the barrier. It was also known that when α-particles are shot at the nuclei from outside with less kinetic energy than needed to reach the top of the potential barrier they often penetrate into the nuclei, producing artificial nuclear transformations. According to the basic principles of classical mechanics, both phenomena were absolutely impossible, so that no spontaneous nuclear decay resulting in the emission of α-particles, and no artificial nuclear transformations under the influence of α-bombardment could possibly exist. And yet both were experimentally observed!

  If one looks on the situation from the point of view of wave mechanics, it appears quite different, since the motion of the particles is governed by de Broglie’s pilot waves. To understand how wave mechanics explains these classically impossible events, one should remember that wave mechanics stands in the same relation to the classical Newtonian mechanics as wave optics to the old geometrical optics. According to Snell’s Law, a light ray falling on a glass surface at a certain incidence angle i (Fig. 23a) is refracted at a smaller angle r, satisfying the condition sin i/sin r = n where n is the refractive index of glass. If we reverse the situation (Fig. 23b), and let a light ray propagating through glass exit into the air, the angle of refraction will be larger than that of incidence and we will have sin i/sin r = 1/n. Thus a light ray falling on the interface between the glass and air at an angle of incidence greater than a certain critical value will not enter into the air at all but will be totally reflected back into the glass. According to the wave theory of light the situation is different. Light waves undergoing total internal reflection are not reflected from the mathematical boundary between the two substances, but penetrate into the second medium (in this case air) to the depth of several wavelengths λ and then are thrown back into the original medium (Fig. 23c). Therefore, if we place another plate of glass a few wavelengths away (a few microns, in the case of visible light), some amount of light coming into the air will reach the surface of that glass and continue to propagate in the original direction (Fig. 23d). The theory of this phenomenon can be found in the books on optics published a century ago and represents a standard demonstration in many university courses on optics.

  Similarly, de Broglie waves which guide the motion of α-particles and other atomic projectiles can penetrate through the regions of space which are prohibited to particles by classical Newtonian mechanics, and α-particles, protons, etc., can cross the potential barriers whose height is greater than the energy of the incident particle. But the probability of penetration is of physical importance only for particles of atomic mass, and for barriers not more than 10−12 or 10−13 cm wide. Let us take, for example, a uranium nucleus which emits an α-particle after an interval of about 1010 years. An α-particle imprisoned within the uranium potential barrier hits the barrier wall some 1021 times per second, which means that the chance of escape after a simple hit is one out of 1010 × 3 · 107 × 1021 ≅ 3 · 1038 hits (here 3 · 107 is the number of seconds in a year). Similarly, the chances that an atomic projectile will enter the nucleus are very small for each individual hit, but may become considerable if a very large number of nuclear collisions are involved. It was shown in 1929 by Fritz Houtermans and Robert Atkinson that the nuclear collisions caused by intensive thermal motion, known as thermonuclear reactions , are responsible for the production of energy in the Sun and stars. Physicists are working hard now to produce the so-called “controlled thermonuclear reactions” which would supply us with cheap, inexhaustible, and harmless sources of nuclear energy. All this would have been impossible if Newton’s classical mechanics had not been replaced by de Broglie-Schrödinger wave mechanics.

  FIG. 23 Analogy Between Wave Mechanics and Wave Optics. In (a) we have the familiar picture of refraction of light entering from the rarer into the denser medium. In (b) we have the reverse case when the light entering from the denser into the rarer medium can be completely reflected from the interface if the angle of incidence exceeds a certain critical value. According to the wave theory of light, the reflection takes place not on the mathematical surface separating the two media, but within a certain layer several wavelengths thick. Thus, if the second layer of the denser medium is placed a few wavelengths beyond the first layer, a fraction of the incident light will not be totally reflected but will penetrate into the second dense layer propagating in the original direction. Similarly, according to wave mechanics, some particles can penetrate through the regions prohibited by classical mechanics, where the potential is higher than the original kinetic energy of the particles.

  EXCERPTS FROM LECTURES ON QUANTUM MECHANICS

  BY

  PAUL A.M. DIRAC

  DR. 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 to 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 Schrödinger 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 interaction 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 ħ tend to zero. It is very much easier to visualize what one is doing in terms of classical mechanics. It will be mainly about classical mechanics that I shall be talking in these lectures.

  Now you may think that that is really not good enough, because classical mechanics is not good enough to describe Nature. Nature isReprinted courtesy of Dover Publications.

  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 nonlinear 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 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 preliminary piece of work. The final conclusion of this piece of work must be to set up an accurate quant
um 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 alternative 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 essential to work from a Hamiltonian; so I want to talk to you about how far one can develop Hamiltonian methods.

  I would like to begin in an elementary way and I take as my starting point an action principle. That is to say, I assume that there is an action integral which depends on the motion, such that, when one varies the motion, and puts down the conditions for the action integral to be stationary, one gets the equations of motion. The method of starting from an action principle has the one great advantage, that one can easily make the theory conform to the principle of relativity. We need our atomic theory to conform to relativity because in general we are dealing with particles moving with high velocities.

  If we want to bring in the gravitational field, then we have to make our theory conform to the general principle of relativity, which means working with a space-time which is not flat. Now the gravitational field is not very important in atomic physics, because gravitational forces are 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 would be available whether one brings in the gravitational field or not.

  We start off with an action integral which I denote by

  (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 agreement 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 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 number 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. The general one is qn, n = 1, . . . , N, N being the number of degrees of freedom. Then we have the velocities dqn/dt = 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. However, it is a feature of the present formalism which I do not see how one can avoid if one wants to keep to the generality of allowing the Lagrangian to be any function of the coordinates and velocities. We can be sure that the contents of the theory are relativistic, even though the form of the equations is not manifestly relativistic on account of the appearance of one particular time in a dominant place in the theory.

  Let us now develop this Lagrangian dynamics and pass over to Hamiltonian dynamics, following as closely as we can the ideas which one learns about as soon as one deals with dynamics from the point of view of working with general coordinates. We have the Lagrangian equations of motion which follow from the variation of the action integral:

  (1-2)

  To go over to the Hamiltonian formalism, we introduce the momentum variables pn , which are defined by

  (1-3)

  Now in the usual dynamical theory, one makes the assumption that the momenta are independent functions of the velocities, but that assumption is too restrictive for the applications which we are going to make. We want to allow for the possibility of these momenta not being independent functions of the velocities. In that case, there exist certain relations connecting the momentum variables, of the type φ(q , p ) = 0.

  There may be several independent relations of this type, and if there are, we distinguish them one from another by a suffix m = 1,, .
. . , M, so we have

  (1-4)

  The q’s and the p’s are the dynamical variables of the Hamiltonian theory. They are connected by the relations (1-4), which are called the primary constraints of the Hamiltonian formalism. This terminology is due to Bergmann, and I think it is a good one.