《量子力学》The fundamental idea of wave mechanics

ERWIN SCHRODINGER The fundamental idea of wave mechanics Nobel lecture december 12, 1933 On passing through an optical instrument, such as a telescope or a camera lens, a ray of light is subjec reflecting surface. The path of the rays can be constructed if we know the two simple laws which govern the changes in direction: the law of refrac- tion which was discovered by Snellius a few hundred years ago, and the law of reflection with which Archimedes was familiar more than 2,000 years ago As a simple example, Fig. 1 shows a ray A-B which is subjected to refraction at each of the four boundary surfaces of two lenses in accordance with th Fermat defined the total path of a ray of light from a much more general point of view. In different media, light propagates with different velocities, and the radiation path gives the appearance as if the light must arrive at its destination as quickly as possible. (Incidentally, it is permissible here to con- sider any two points along the ray as the starting- and end-points. )The least deviation from the path actually taken would mean a delay this is the fa mous Fermat principle of the shortest light time, which in a marvellous manner determines the entire fate of a ray of light by a single statement and also includes the more general case, when the nature of the medium varies not suddenly at individual surfaces, but gradually from place to place. The at penetrates into it from outside, the more slowly it progresses in an inc oht mosphere of the earth provides an example. The more deeply a ray of lig ingly denser air. Although the differences in the speed of propagation are

ERWIN SCHRÖDINGER The fundamental idea of wave mechanics Nobel Lecture, December 12, 1933 On passing through an optical instrument, such as a telescope or a camera lens, a ray of light is subjected to a change in direction at each refracting or reflecting surface. The path of the rays can be constructed if we know the two simple laws which govern the changes in direction: the law of refrac￾tion which was discovered by Snellius a few hundred years ago, and the law of reflection with which Archimedes was familiar more than 2,000 years ago. As a simple example, Fig. 1 shows a ray A-B which is subjected to refraction at each of the four boundary surfaces of two lenses in accordance with the law of Snellius. Fig. 1. Fermat defined the total path of a ray of light from a much more general point of view. In different media, light propagates with different velocities, and the radiation path gives the appearance as if the light must arrive at its destination as quickly as possible. (Incidentally, it is permissible here to con￾sider any two points along the ray as the starting- and end-points.) The least deviation from the path actually taken would mean a delay. This is the fa￾mous Fermat principle of the shortest light time, which in a marvellous manner determines the entire fate of a ray of light by a single statement and also includes the more general case, when the nature of the medium varies not suddenly at individual surfaces, but gradually from place to place. The at￾mosphere of the earth provides an example. The more deeply a ray of light penetrates into it from outside, the more slowly it progresses in an increas￾ingly denser air. Although the differences in the speed of propagation are

1933 ESCHRODINGER infinitesimal, Fermat's principle in these circumstances demands that the light ray should curve earthward (see Fig. 2), so that it remains a little longer in the higher " faster"layers and reaches its destination more quickly than by the shorter straight path(broken line in the figure; disregard the square, Fig. 2. wWWW for the time being). I think, hardly any of you will have failed to observe that the sun when it is deep on the horizon appears to be not circular but flattened: its vertical diameter looks to be shortened this is a result of the curvature of the rays. According to the wave theory of light, the light rays, strictly speaking, ave only fictitious significance. They are not the physical paths of some particles of light, but are a mathematical device, the so-called orthogonal trajectories of wave surfaces, imaginary guide lines as it were, which point in the direction normal to the wave surface in which the latter advances(cf s.g. 3 which shows the simplest case of concentric spherical wave surfaces and accordingly rectilinear rays, whereas Fig 4 illustrates the case of curved Fig. 4

306 1933 E.SCHRÖDINGER infinitesimal, Fermat’s principle in these circumstances demands that the light ray should curve earthward (see Fig. 2), so that it remains a little longer in the higher "faster" layers and reaches its destination more quickly than by the shorter straight path (broken line in the figure; disregard the square, Fig. 2. WWW1 W1 for the time being). I think, hardly any of you will have failed to observe that the sun when it is deep on the horizon appears to be not circular but flattened: its vertical diameter looks to be shortened. This is a result of the curvature of the rays. According to the wave theory of light, the light rays, strictly speaking, have only fictitious significance. They are not the physical paths of some particles of light, but are a mathematical device, the so-called orthogonal trajectories of wave surfaces, imaginary guide lines as it were, which point in the direction normal to the wave surface in which the latter advances (cf. Fig. 3 which shows the simplest case of concentric spherical wave surfaces and accordingly rectilinear rays, whereas Fig. 4 illustrates the case of curved Fig. 3. Fig. 4

FUNDAMENTAL IDEA OF WAVE MECHANICS 307 rays). It is surprising that a general principle as important as Fermat's relates directly to these mathematical guide lines, and not to the wave surfaces, and one might be inclined for this reason to consider it a mere mathematical curiosity. Far from it. It becomes properly understandable only from the point of view of wave theory and ceases to be a divine miracle. From the wave point of view, the so-called curvature of the light ray is far more readily understandable as a swerving of the wave surface, which must obviously oc- cur when neighbouring parts of a wave surface advance at different speeds, in exactly the same manner as a company of soldiers marching forward will carry out the order"right incline" by the men taking steps ofvarying lengths the right-wing man the smallest, and the left-wing man the longest In at- Aa ospheric refraction of radiation for example( Fig. 2 )the section of wave arface Ww must necessarily swerve to the right towards Ww because its left half is located in slightly higher, thinner air and thus advances more apidly than the right part at lower point. (In passing, I wish to refer to one point at which the Snellius view fails. A horizontally emitted light ray should remain horizontal because the refraction index does not vary in the horizon- al direction. In truth, a horizontal ray curves more strongly than any other which is an obvious consequence of the theory of a swerving wave front. On detailed examination the Fermat principle is found to be completely tantamount to the trivial and obvious statement that-given local distribution of light velocities- the wave front must swerve in the manner indicated cannot prove this here, but shall attempt to make it plausible. I would again ask you to visualize a rank of soldiers marching forward. To ensure that the line remains dressed, let the men be connected by a long rod which each holds firmly in his hand. No orders as to direction are given; the only ord is: let each man march or run as fast as he can. If the nature of the ground varies slowly from place to place, it will be now the right wing, now the left that advances more quickly, and changes in direction will occur spon- taneously. After some time has elapsed, it will be seen that the entire path travelled is not rectilinear, but somehow curved. That this curved path is exactly that by which the destination attained at any moment could be at- tained most rapidly according to the nature of the terrain, is at least quite plausible, since each of the men did his best. It will also be seen that the swerv ing also occurs invariably in the direction in which the terrain is worse, so that it will come to look in the end as if the men had intentionally"by- passed"a place where they would advance slowly The Fermat principle thus appears to be the trivial quintessence of the wave

FUNDAMENTAL IDEA OF WAVE MECHANIC S 307 rays). It is surprising that a general principle as important as Fermat’s relates directly to these mathematical guide lines, and not to the wave surfaces, and one might be inclined for this reason to consider it a mere mathematical curiosity. Far from it. It becomes properly understandable only from the point of view of wave theory and ceases to be a divine miracle. From the wave point of view, the so-called curvature of the light ray is far more readily understandable as a swerving of the wave surface, which must obviously oc￾cur when neighbouring parts of a wave surface advance at different speeds; in exactly the same manner as a company of soldiers marching forward will carry out the order "right incline" by the men taking steps ofvarying lengths, the right-wing man the smallest, and the left-wing man the longest. In at￾mospheric refraction of radiation for example (Fig. 2) the section of wave surface WW must necessarily swerve to the right towards W1W1 because its left half is located in slightly higher, thinner air and thus advances more rapidly than the right part at lower point. (In passing, I wish to refer to one point at which the Snellius’ view fails. A horizontally emitted light ray should remain horizontal because the refraction index does not vary in the horizon￾tal direction. In truth, a horizontal ray curves more strongly than any other, which is an obvious consequence of the theory of a swerving wave front.) On detailed examination the Fermat principle is found to be completely tantamount to the trivial and obvious statement that - given local distribution of light velocities - the wave front must swerve in the manner indicated. I cannot prove this here, but shall attempt to make it plausible. I would again ask you to visualize a rank of soldiers marching forward. To ensure that the line remains dressed, let the men be connected by a long rod which each holds firmly in his hand. No orders as to direction are given; the only order is: let each man march or run as fast as he can. If the nature of the ground varies slowly from place to place, it will be now the right wing, now the left that advances more quickly, and changes in direction will occur spon￾taneously. After some time has elapsed, it will be seen that the entire path travelled is not rectilinear, but somehow curved. That this curved path is exactly that by which the destination attained at any moment could be at￾tained most rapidly according to the nature of the terrain, is at least quite plausible, since each of the men did his best. It will also be seen that the swerv￾ing also occurs invariably in the direction in which the terrain is worse, so that it will come to look in the end as if the men had intentionally "by￾passed" a place where they would advance slowly. The Fermat principle thus appears to be the trivial quintessence of the wave

1933 E SCHRODINGER theory. It was therefore a memorable occasion when Hamilton made the discovery that the true movement of mass points in a field of forces(e.g. of a planet on its orbit around the sun or of a stone thrown in the gravitational field of the earth) is also governed by a very similar general principle, which carries and has made famous the name of its discoverer since then Admittedly, the Hamilton principle does not say exactly that the mass point chooses the quickest way, but it does say something so similar -the analogy with the principle of the shortest travelling time of light is so close, that one was faced with a puzzle. It seemed as if Nature had realized one and the same law twice by entirely different means: first in the case of light, b means of a fairly obvious play of rays; and again in the case of the mass points, which was anything but obvious, unless somehow wave nature were to be attributed to them also. And this, it seemed impossible to do. Becaus the"mass points"on which the laws of mechanics had really been confirmed experimentally at that time were only the large, visible, sometimes very large bodies, the planets, for which a thing like"wave nature"appeared to be out of the question. The smallest, elementary components of matter which we today, much more specifically, call"mass points", were purely hypothetical at the time It was only after the discovery of radioactivity that constant refinements of methods of measurement permitted the properties of these particles to be studied in detail, and now permit the paths of such particles to be photo- graphed and to be measured very exactly (stereophotogrammetrically)by the brilliant method of C.t. R. Wilson, As far as the measurements extend they confirm that the same mechanical laws are valid for particles as for large bodies, planets, etc. However, it was found that neither the molecule nor the individual atom can be considered as the "ultimate component": but even the atom is a system of highly complex structure. Images are formed in our minds of the structure of atoms consisting of particles, images which seem to have a certain similarity with the planetary system. It was only natural that the attempt should at first be made to consider as valid the same laws of motion that had proved themselves so amazingly satisfactory on a large scale. In other words, Hamiltons mechanics, which, as I said above, culminates in the Hamilton principle, were applied also to the "inner life of the atom. That there is a very close analogy between Hamilton's principle and Fermat's optical principle had meanwhile become all but forgotten. If it was remembered, it was considered to be nothing more than a curious trait of the mathematical theory

308 1933 E. SCHRÖDINGER theory. It was therefore a memorable occasion when Hamilton made the discovery that the true movement of mass points in a field of forces (e.g. of a planet on its orbit around the sun or of a stone thrown in the gravitational field of the earth) is also governed by a very similar general principle, which carries and has made famous the name of its discoverer since then. Admittedly, the Hamilton principle does not say exactly that the mass point chooses the quickest way, but it does say something so similar - the analogy with the principle of the shortest travelling time of light is so close, that one was faced with a puzzle. It seemed as if Nature had realized one and the same law twice by entirely different means: first in the case of light, by means of a fairly obvious play of rays; and again in the case of the mass points, which was anything but obvious, unless somehow wave nature were to be attributed to them also. And this, it seemed impossible to do. Because the "mass points" on which the laws of mechanics had really been confirmed experimentally at that time were only the large, visible, sometimes very large bodies, the planets, for which a thing like "wave nature" appeared to be out of the question. The smallest, elementary components of matter which we today, much more specifically, call "mass points", were purely hypothetical at the time. It was only after the discovery of radioactivity that constant refinements of methods of measurement permitted the properties of these particles to be studied in detail, and now permit the paths of such particles to be photo￾graphed and to be measured very exactly (stereophotogrammetrically) by the brilliant method of C. T. R. Wilson. As far as the measurements extend they confirm that the same mechanical laws are valid for particles as for large bodies, planets, etc. However, it was found that neither the molecule nor the individual atom can be considered as the "ultimate component": but even the atom is a system of highly complex structure. Images are formed in our minds of the structure of atoms consisting of particles, images which seem to have a certain similarity with the planetary system. It was only natural that the attempt should at first be made to consider as valid the same laws of motion that had proved themselves so amazingly satisfactory on a large scale. In other words, Hamilton’s mechanics, which, as I said above, culminates in the Hamilton principle, were applied also to the "inner life" of the atom. That there is a very close analogy between Hamilton’s principle and Fermat’s optical principle had meanwhile become all but forgotten. If it was remembered, it was considered to be nothing more than a curious trait of the mathematical theory

FUNDAMENTAL IDEA OF WAVE MECHANICS 309 low, it is very difficult, without further going into details, to convey a proper conception of the success or failure of these classical-mechanical im- ages of the atom On the one hand, Hamiltons principle in particular proved to be the most faithful and reliable guide, which was simply indispensable on the other hand one had to suffer, to do justice to the facts, the rough interference of entirely new incomprehensible postulates, of the so-called quantum conditions and quantum postulates Strident disharmony in the symphony of classical mechanics- yet strangely familiar -played as it were on the same instrument. In mathematical terms we can formulate this as fol- lows: whereas the Hamilton principle merely postulates that a given integral must be a minimum, without the numerical value of the minimum being established by this postulate, it is now demanded that the numerical value of the minimum should be restricted to integral multiples of a universal natu- ral constant, Planck's quantum of action. This incidentally. The situation was fairly desperate. Had the old mechanics failed completely, it would not have been so bad. The way would then have been free to the development of a new system ofmechanics. As it was, one was faced with the difficult task of saving the soul of the old system, whose inspiration clearly held sway in this microcosm, while at the same time flattering it as it were into accepting th quantum conditions not as gross interference but as issuing from its own innermost essence The way out lay just in the possibility, already indicated above, of attrib uting to the Hamilton principle, also, the operation of a wave mechanism on which the point-mechanical processes are essentially based, just as one had long become accustomed to doing in the case of phenomena relating to light and of the Fermat principle which governs them. Admittedly, the in- dividual path of a mass point loses its proper physical significance and be comes as fictitious as the individual isolated ray of light. The essence of the theory, the minimum principle, however, remains not only intact, but reveals its true and simple meaning only under the wave-like aspect, as already ex- plained. Strictly speaking, the new theory is in fact not new, it is a completely organic development, one might almost be tempted to say a more elaborate exposition, of the old theory How was it then that this new more "elaborate"exposition led to notably different results; what enabled it, when applied to the atom, to obviate diffi- culties which the old theory could not solve? What enabled it to render gross interference acceptable or even to make it its own? Again, these matters can best be illustrated by analogy with optics. Quite

FUNDAMENTAL IDEA OF WAVE MECHANIC S 309 Now, it is very difficult, without further going into details, to convey a proper conception of the success or failure of these classical-mechanical im￾ages of the atom. On the one hand, Hamilton’s principle in particular proved to be the most faithful and reliable guide, which was simply indispensable; on the other hand one had to suffer, to do justice to the facts, the rough interference of entirely new incomprehensible postulates, of the so-called quantum conditions and quantum postulates. Strident disharmony in the symphony of classical mechanics - yet strangely familiar - played as it were on the same instrument. In mathematical terms we can formulate this as fol￾lows: whereas the Hamilton principle merely postulates that a given integral must be a minimum, without the numerical value of the minimum being established by this postulate, it is now demanded that the numerical value of the minimum should be restricted to integral multiples of a universal natu￾ral constant, Planck’s quantum of action. This incidentally. The situation was fairly desperate. Had the old mechanics failed completely, it would not have been so bad. The way would then have been free to the development of a new system ofmechanics. As it was, one was faced with the difficult task of saving the soul of the old system, whose inspiration clearly held sway in this microcosm, while at the same time flattering it as it were into accepting the quantum conditions not as gross interference but as issuing from its own innermost essence. The way out lay just in the possibility, already indicated above, of attrib￾uting to the Hamilton principle, also, the operation of a wave mechanism on which the point-mechanical processes are essentially based, just as one had long become accustomed to doing in the case of phenomena relating to light and of the Fermat principle which governs them. Admittedly, the in￾dividual path of a mass point loses its proper physical significance and be￾comes as fictitious as the individual isolated ray of light. The essence of the theory, the minimum principle, however, remains not only intact, but reveals its true and simple meaning only under the wave-like aspect, as already ex￾plained. Strictly speaking, the new theory is in fact not new, it is a completely organic development, one might almost be tempted to say a more elaborate exposition, of the old theory. How was it then that this new more "elaborate" exposition led to notably different results; what enabled it, when applied to the atom, to obviate diffi￾culties which the old theory could not solve? What enabled it to render gross interference acceptable or even to make it its own? Again, these matters can best be illustrated by analogy with optics. Quite