《量子力学》The wave nature of the electron

LOUIS DE BROGLIE The wave nature of the electron Nobel Lecture. December 12. 1929 When in 1920 I resumed my studies of theoretical physics which had long been interrupted by circumstances beyond my control, I was far from the idea that my studies would bring me several years later to receive such a higl and envied prize as that awarded by the Swedish Academy of Sciences each year to a scientist: the Nobel Prize for Physics. What at that time drew me towards theoretical physics was not the hope that such a high distinction would ever crown my work; I was attracted to theoretical physics by the mystery enshrouding the structure of matter and the structure of radiations, a mystery which deepened as the strange quantum concept introduced by Planck in 1900 in his research on black-body radiation continued to encroach on the whole domain of physics To assist you to understand how my studies developed, I must first depict for you the crisis which physics had then been passing through for some twenty years For a long time physicists had been wondering whether light was composed of small, rapidly moving corpuscles. This idea was put forward by the phi- losophers of antiquity and upheld by Newton in the 18th century. After Thomas Young s discovery of interference phenomena and following the admirable work of Augustin Fresnel, the hypothesis of a granular structure of light was entirely abandoned and the wave theory unanimously adopted Thus the physicists of last century spurned absolutely the idea of an atomic structure of light. Although rejected by optics, the atomic theories began making great head way not only in chemistry, where they provided a simple interpretation of the laws of definite proportions, but also in the physics of matter where they made possible an interpretation of a large number of prop- erties of solids, liquids, and gases. In particular they were instrumental in the elaboration of that admirable kinetic theory of gases which, generalized un- der the name of statistical mechanics, enables a clear meaning to be given to the abstract concepts of thermodynamics. Experiment also yielded decisive proof in favour of an atomic constitution of electricity; the concept of the

L OUIS DE B ROGLI E The wave nature of the electron Nobel Lecture, December 12, 1929 When in 1920 I resumed my studies of theoretical physics which had long been interrupted by circumstances beyond my control, I was far from the idea that my studies would bring me several years later to receive such a high and envied prize as that awarded by the Swedish Academy of Sciences each year to a scientist: the Nobel Prize for Physics. What at that time drew me towards theoretical physics was not the hope that such a high distinction would ever crown my work; I was attracted to theoretical physics by the mystery enshrouding the structure of matter and the structure of radiations, a mystery which deepened as the strange quantum concept introduced by Planck in 1900 in his research on black-body radiation continued to encroach on the whole domain of physics. To assist you to understand how my studies developed, I must first depict for you the crisis which physics had then been passing through for some twenty years. For a long time physicists had been wondering whether light was composed of small, rapidly moving corpuscles. This idea was put forward by the phi￾losophers of antiquity and upheld by Newton in the 18th century. After Thomas Young’s discovery of interference phenomena and following the admirable work of Augustin Fresnel, the hypothesis of a granular structure of light was entirely abandoned and the wave theory unanimously adopted. Thus the physicists of last century spurned absolutely the idea of an atomic structure of light. Although rejected by optics, the atomic theories began making great headway not only in chemistry, where they provided a simple interpretation of the laws of definite proportions, but also in the physics of matter where they made possible an interpretation of a large number of prop￾erties of solids, liquids, and gases. In particular they were instrumental in the elaboration of that admirable kinetic theory of gases which, generalized un￾der the name of statistical mechanics, enables a clear meaning to be given to the abstract concepts of thermodynamics. Experiment also yielded decisive proof in favour of an atomic constitution of electricity; the concept of the

WAVE NATURE OF ELECTRON 245 electricity corpuscle owes its appearance to Sir J. J. Thomson and you will all be familiar with H. A. Lorentz, s use of it in his theory of electrons Some thirty years ago, physics was hence divided into two: firstly the physics of matter based on the concept of corpuscles and atoms which were supposed to obey Newtons classical laws of mechanics, and secondly radia- tion physics based on the concept of wave propagation in a hypothetical continuous medium, i.e. the light ether or electromagnetic ether. But these two physics could not remain alien one to the other; they had to be fused together by devising a theory to explain the energy exchanges between mat ter and radiation-and that is where the difficulties arose. While seeking to link these two physics together, imprecise and even inadmissible conclu- sions were in fact arrived at in respect of the energy equilibrium between matter and radiation in a thermally insulated medium: matter, it came to be said, must yield all its energy to the radiation and so tend of its own accord to absolute zero temperature! This absurd conclusion had at all costs to be avoided. By an intuition of his genius Planck realized the way of avoiding it: instead of assuming, in common with the classical wave theory, that light source emits its radiation continuously, it had to be assumed on the contrary that it emits equal and finite quantities, quanta. The energy of each quantum has, moreover, a value proportional to the frequency u of the ra- diation. It is equal to h, h being a universal constant since referred to as Plancks constant The success of Planck's ideas entailed serious consequences. If light is emit- ted as quanta, ought it not, once emitted, to have a granular structure? The existence of radiation quanta thus implies the corpuscular concept of light On the other hand, as shown by Jeans and H. Poincare, it is demonstrable that if the motion of the material particles in light sources obeyed the laws of classical mechanics it would be impossible to derive the exact law of black body radiation, Plancks law. It must therefore be assumed that traditiona dynamics, even as modified by Einstein s theory of relativity, is incapable of accounting for motion on a very small scale The existence of a granular structure of light and of other radiations was confirmed by the discovery of the photoelectric effect. If a beam of light or of X-rays falls on a piece of matter, the latter will emit rapidly moving elec trons. The kinetic energy of these electrons increases linearly with the fre- quency of the incident radiation and is independent of its intensity. This phenomenon can be explained simply by assuming that the radiation is com- posed of quanta lo capable of yielding all their energy to an electron of the

WAVE NATURE OF ELECTRON 245 electricity corpuscle owes its appearance to Sir J. J. Thomson and you will all be familiar with H. A. Lorentz’s use of it in his theory of electrons. Some thirty years ago, physics was hence divided into two: firstly the physics of matter based on the concept of corpuscles and atoms which were supposed to obey Newton’s classical laws of mechanics, and secondly radia￾tion physics based on the concept of wave propagation in a hypothetical continuous medium, i.e. the light ether or electromagnetic ether. But these two physics could not remain alien one to the other; they had to be fused together by devising a theory to explain the energy exchanges between mat￾ter and radiation - and that is where the difficulties arose. While seeking to link these two physics together, imprecise and even inadmissible conclu￾sions were in fact arrived at in respect of the energy equilibrium between matter and radiation in a thermally insulated medium: matter, it came to be said, must yield all its energy to the radiation and so tend of its own accord to absolute zero temperature! This absurd conclusion had at all costs to be avoided. By an intuition of his genius Planck realized the way of avoiding it: instead of assuming, in common with the classical wave theory, that a light source emits its radiation continuously, it had to be assumed on the contrary that it emits equal and finite quantities, quanta. The energy of each quantum has, moreover, a value proportional to the frequency v of the ra￾diation. It is equal to hv, h being a universal constant since referred to as Planck’s constant. The success of Planck’s ideas entailed serious consequences. If light is emit￾ted as quanta, ought it not, once emitted, to have a granular structure? The existence of radiation quanta thus implies the corpuscular concept of light. On the other hand, as shown by Jeans and H. Poincaré, it is demonstrable that if the motion of the material particles in light sources obeyed the laws of classical mechanics it would be impossible to derive the exact law of black￾body radiation, Planck’s law. It must therefore be assumed that traditional dynamics, even as modified by Einstein’s theory of relativity, is incapable of accounting for motion on a very small scale. The existence of a granular structure of light and of other radiations was confirmed by the discovery of the photoelectric effect. If a beam of light or of X-rays falls on a piece of matter, the latter will emit rapidly moving elec￾trons. The kinetic energy of these electrons increases linearly with the fre￾quency of the incident radiation and is independent of its intensity. This phenomenon can be explained simply by assuming that the radiation is com￾posed of quanta hv capable of yielding all their energy to an electron of the

1929 L DE BROGLIE irradiated body: one is thus led to the theory of light quanta proposed Einstein in 1905 and which is, after all, a reversion to Newtons corpuscular theory, completed by the relation for the proportionality between the en- ergy of the corpuscles and the frequency. A number of arguments were put forward by Einstein in support of his viewpoint and in 1922 the discovery by A H. Compton of the X-ray scattering phenomenon which bears his name confirmed it. Nevertheless, it was still necessary to adopt the wave theory to account for interference and diffraction phenomena and no way whatsoever of reconciling the wave theory with the existence of light cor- puscles could be visualized As stated, Planck's investigations cast doubts on the validity of very small scale mechanics. Let us consider a material point which describes a small tra- jectory which is closed or else turning back on itself. According to classical dynamics there are numberless motions of this type which are possible com- plying with the initial conditions, and the possible values for the energy of the moving body form a continuous sequence On the other hand Planck was led to assume that only certain preferred motions, quantized motions, are possible or at least stable, since energy can only assume values forming a discontinuous sequence. This concept seemed rather strange at first but its value had to be recognized because it was this concept which brought Planck to the correct law of black-body radiation and because it then proved its uitfulness in many other fields. Lastly, it was on the concept of atomic mo- tion quantization that Bohr based his famous theory of the atom; it is so familiar to scientists that i shall not summarize it here The necessity of assuming for light two contradictory theories-that of waves and that of corpuscles-and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of When I started to ponder these difficulties two things struck me in the main. Firstly the light-quantum theory cannot be regarded as satisfactory since it defines the energy of a light corpuscle by the relation W= hu which con- ains a frequency D. Now a purely corpuscular theory does not contain any element permitting the definition of a frequency. This reason alone renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity

246 1929 L.DE BROGLIE irradiated body: one is thus led to the theory of light quanta proposed by Einstein in 1905 and which is, after all, a reversion to Newton’s corpuscular theory, completed by the relation for the proportionality between the en￾ergy of the corpuscles and the frequency. A number of arguments were put forward by Einstein in support of his viewpoint and in 1922 the discovery by A. H. Compton of the X-ray scattering phenomenon which bears his name confirmed it. Nevertheless, it was still necessary to adopt the wave theory to account for interference and diffraction phenomena and no way whatsoever of reconciling the wave theory with the existence of light cor￾puscles could be visualized. As stated, Planck’s investigations cast doubts on the validity of very small scale mechanics. Let us consider a material point which describes a small tra￾jectory which is closed or else turning back on itself. According to classical dynamics there are numberless motions of this type which are possible com￾plying with the initial conditions, and the possible values for the energy of the moving body form a continuous sequence. On the other hand Planck was led to assume that only certain preferred motions, quantized motions, are possible or at least stable, since energy can only assume values forming a discontinuous sequence. This concept seemed rather strange at first but its value had to be recognized because it was this concept which brought Planck to the correct law of black-body radiation and because it then proved its fruitfulness in many other fields. Lastly, it was on the concept of atomic mo￾tion quantization that Bohr based his famous theory of the atom; it is SO familiar to scientists that I shall not summarize it here. The necessity of assuming for light two contradictory theories-that of waves and that of corpuscles - and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics. When I started to ponder these difficulties two things struck me in the main. Firstly the light-quantum theory cannot be regarded as satisfactory since it defines the energy of a light corpuscle by the relation W = hv which con￾tains a frequency v. Now a purely corpuscular theory does not contain any element permitting the definition of a frequency. This reason alone renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity

WAVE NATURE OF ELECTRON 247 On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved physics were ose of interference selves could not be represented as simple corpuscles either, but that a Pen and of eigenvibration. That suggested the idea to me that electrons then odicity had also to be assigned to them too. thus arrived at the following overall concept which guided my studies for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all according to Bohrs expression, they constitute two complementary forces of reality, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. The first objective to achieve had, therefore, to be to establish this correspondence. With that in view I started by considering the simplest case: that of an isolated corpuscle, i.e. a corpuscle free from all outside influence. We wish to associate a wave with it. Let us consider first of all a reference syster oxoyozo in which the corpuscle is immobile: this is the intrinsic" system of the corpuscle in the sense of the relativity theory. In this system the wave will be stationary since the corpuscle is immobile: its phase will be the same at every point; it will be represented by an expression of the form sin 2Zvol(to-To); to being the intrinsic time of the corpuscle and to a constant. In accordance with the principle of inertia in every Galilean system, the orpuscle will have a rectilinear and uniform motion. Let us consider such Galilean system and let v=Bc be the velocity of the corpuscle in this system; we shall not restrict generality by taking the direction of the motion as the x-axis. In compliance with Lorentz' transformation, the time t used by an observer of this new system will be associated with the intrinsic time to by the relation -B2 and hence for this observer the phase of the wave will be given by √1-B( Bx

WAVE NATURE OF ELECTRON 247 On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons them￾selves could not be represented as simple corpuscles either, but that a peri￾odicity had also to be assigned to them too. I thus arrived at the following overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent because, according to Bohr’s expression, they constitute two complementary forces of reality, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. The first objective to achieve had, therefore, to be to establish this correspondence. With that in view I started by considering the simplest case: that of an isolated corpuscle, i.e. a corpuscle free from all outside influence. We wish to associate a wave with it. Let us consider first of all a reference system o~~y~z~ in which the corpuscle is immobile: this is the "intrinsic" system of the corpuscle in the sense of the relativity theory. In this system the wave will be stationary since the corpuscle is immobile: its phase will be the same at every point; it will be represented by an expression of the form sin ZZV~(~~~~) ; to being the intrinsic time of the corpuscle and to a constant. In accordance with the principle of inertia in every Galilean system, the corpuscle will have a rectilinear and uniform motion. Let us consider such a Galilean system and let t, = ,dc be the velocity of the corpuscle in this system; we shall not restrict generality by taking the direction of the motion as the x-axis. In compliance with Lorentz’ transformation, the time t used by an observer of this new system will be associated with the intrinsic time to by the relation: and hence for this observer the phase of the wave will be given by

1929 L DE BROGLIE For him the wave will thus have a frequency: √r-B2 and will propagate in the direction of the x-axis at the phase velocity: By the elimination of B between the two preceding formulae the following elation can readily be derived which defines the refractive index of the vacuum n for the waves considered A <<group velocity>> corresponds to this <claw of dispersion>>. You will be aware that the group velocity is the velocity of the resultant amplitude of a group of waves of very close frequencies. Lord Rayleigh showed that this velocity u satisfies equation I a(nm) Here U=v, that is to say that the group velocity of the waves in the system ryxt is equal to the velocity of the corpuscle in this system. This relation is of very great importance for the development of the theory The corpuscle is thus defined in the system ryzt by the frequenc e phase velocity V of its associated wave. To establish the parallelism of which we have spoken, we must seek to link these parameters to the me. chanical parameters, energy and quantity of motion. Since the proportion- ality between energy and frequency is one of the most characteristic relations of the quantum theory, and since, moreover, the frequency and the energy transform in the same way when the Galilean reference system is changed, we may simply write nergy=h x frequency, or W=hu

248 1929 L.DE BROGLI E For him the wave will thus have a frequency: and will propagate in the direction of the x-axis at the phase velocity: By the elimination of b between the two preceding formulae the following relation can readily be derived which defines the refractive index of the vacuum n for the waves considered: J 2 fi-= p2- V2 A <<group velocity>> corresponds to this <<law of dispersion>>. You will be aware that the group velocity is the velocity of the resultant amplitude of a group of waves of very close frequencies. Lord Rayleigh showed that this velocity U satisfies equation : I -=-- a (4 U &J Here U = v, that is to say that the group velocity of the waves in the system xyxt is equal to the velocity of the corpuscle in this system. This relation is of very great importance for the development of the theory. The corpuscle is thus defined in the system xyzt by the frequency v and the phase velocity V of its associated wave. To establish the parallelism of which we have spoken, we must seek to link these parameters to the me￾chanical parameters, energy and quantity of motion. Since the proportion￾ality between energy and frequency is one of the most characteristic relations of the quantum theory, and since, moreover, the frequency and the energy transform in the same way when the Galilean reference system is changed, we may simply write energy = h x frequency, or W = h v