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The Dreams That Stuff is Made of

Page 21

by Stephen Hawking


  in agreement with the ordinary wave theory.

  The dual characters of both matter and light gave rise to many difficulties before the physical principles involved were clearly comprehended, and the following paradox was often discussed. The forces between a part of the grating and the particle certainly diminish very rapidly with the distance between the two. The direction of reflection should therefore be determined only by those parts of the grating which are in the immediate neighborhood of the incident particle, but none the less it is found that the most widely separated portions of the grating are the important factors in determining the sharpness of the diffraction maxima. The source of this contradiction is the confusion of two different experiments (Cases I and II, p. 61). If no experiment is performed which would permit the determination of the position of the particle before its reflection, there is no contradiction with observation if the whole of the grating does act on it. If, on the other hand, an experiment is performed which determines that the particle will strike on a section of length Δx of the grating, it must render the knowledge of the particle’s momentum essentially uncertain by an amount Δp ∼ h/Δx. The direction of its reflection will therefore become correspondingly uncertain. The numerical value of this uncertainty in direction is precisely that which would be calculated from the resolving power of a grating of Δx/d lines. If Δx << d the interference maxima, disappear entirely; not until this case is reached can the path of the particle properly be compared with that expected on the classical particle theory, for not until then can it be determined whether the particle will impinge on a ruling or on one of the plane parts of the surface, etc.

  § 3. THE EXPERIMENT OF EINSTEIN AND RUPPcq

  Another paradox was thought to be presented by the following experiment: An atom (canal ray) is made to pass a slit S of width d with the velocity v, and emits light while doing so. This light is analyzed by a spectroscope behind S. Since the light can reach the spectroscope only during the time t = d/v, the train of waves to be analyzed has a finite length, and the spectroscope will show it as a line whose width corresponds to a frequency range

  On the other hand, the corpuscular theory seems to prohibit such a broadening. The atom emits monochromatic radiation, the energy of each particle of which is hν, and the diaphragm (because of its great mass) will not be able to change the energy of the particles.

  The fallacy lies in neglecting the Doppler effect and the diffraction of the light at the slit. Those photons which reach P from the atom are not all emitted perpendicularly to the canal ray; the angular aperture of the beam of photons is sin α ∼ λ/d because of the diffraction. The Doppler change of frequency due to this is

  or

  in agreement with the previous result. In this experiment the exact validity of the energy law for corpuscles is thus in conformity with the requirements of classical optics.

  FIG. 17

  § 4. EMISSION, ABSORPTION, AND DISPERSION OF RADIATION

  (a) Application of the conservation laws. The postulate of the existence of stationary states, combined with the theory of photons, is sufficient to give a qualitative explanation of the interaction of atoms and radiation. This was the first decisive success of the Bohr theory. The most important results of this theory may be briefly summarized here. Let the stationary states of the atom be numbered 1, 2, 3 .... n .... (Fig. 17), counting from the normal state. An atom in state 3, for example, can spontaneously perform a transition to state 2, and emit a photon of energy hν32 = E3 − E2. In the same way, an atom in state 1 may absorb a photon of energy hν31 = E3 − E1 and thus be excited to the state 3. It must be emphasized that these statements are to be taken quite literally, and not as having only a symbolic significance, for it is possible (e.g., by a Stern-Gerlach experiment) to determine the stationary state of the atoms both before and after the emission. It therefore follows that the intensity of an emission line is proportional to the number of atoms in the upper of the two states associated with it, while the intensity of an absorption line is proportional to the number of atoms in the lower state. These results, which have certainly been amply confirmed by experiment, are entirely characteristic of the quantum theory and can be deduced from no classical theory, either of the wave or particle representation, since even the existence of discrete energy values can never be explained by the classical theory.

  FIG. 18

  An exactly similar situation is met with in the case of scattering. If an atom in state 1 is excited by a photon hν it can re-emit the same light quantum without change of state (the mass of the nucleus being assumed infinite), or it can send out the light quantum of energy hνʹ = hν − E2 + E1 by transition to state 2 (Smekalcr transition; see Fig. 18). The intensity of both kinds of scattered light is proportional to the number of atoms in state 1. If an atom in state 2 is irradiated with light of frequency ν it can emit a photon of energy hν’ = hν + E2 − E 1 of shorter wave-length by transition to state 1, and again the intensity of this “anti-Stokes” scattered light is proportional to the number of atoms in state 2. This has been confirmed by Raman’scs experiments.

  (b) Correspondence principle and the method of virtual charges. The postulate of stationary states and the theory of photons, because of their very nature, cannot yield any information either regarding the interference of the emitted light or even regarding the a priori probability of the transitions involved. The interference properties can be completely accounted for by the classical wave theory, but it is in turn unable to account for the transitions. To treat these successfully a self-consistent quantum theory of radiation is necessary. It is true that an ingenious combination of arguments based on the correspondence principle can make the quantum theory of matter together with a classical theory of radiation furnish quantitative values for the transition probabilities, i.e., either by the use of Schrödinger’s virtual charge density or its equivalent, the element of the matrix representing the electric dipole moment of the atom. Such a formulation of the radiation problem is far from satisfactory, however, and easily leads to false conclusions. These methods may only be applied with the greatest caution, as the following examples may illustrate.

  Consider first the case of an atom containing a single electron, and whose nucleus has an infinite mass. If x ≡ (x, y,z) be the co-ordinate of the electron, and ψ0(x) the Schrödinger function, then

  (63)

  is the element of the matrix representing the dipole moment of the atom. This matrix can enter, strictly speaking, only into calculations based on the principles of the quantum theory of the electron, which in no way involve radiation. It may none the less be interpreted as the dipole moment of the virtual oscillator producing the radiation which is emitted during the transition n → m. This may be deduced from the correspondence principle by remembering that it has been shown that x nm → xn (n − m) in the limit of large quantum numbers, where xn (n − m) is a Fourier coefficient of the classical motion. It may thus be presumed that xnm will enter into the formulas determining the intensity of the radiation in the same way as xn(n − m), i.e., that |xnm |2 will be the a priori probability of the transition n → m. It must be emphasized that this is a purely formal result; it does not follow from any of the physical principles of quantum theory.

  It may be made plausible by another consideration which brings out its unsatisfactory character more clearly. It has been pointed out that the solutions ψ n of the Schrödinger equation are first approximations to the solutions of the classical matter-wave equations [cf. A(8)]. Denoting by ψc a true solution of the latter, the radiation from the charge distribution thus represented will be determined by its dipole moment

  provided the extension of this distribution is small compared to the wave-length of the radiation emitted. Now

  whence the radiation, calculated by means of this classical distribution, should be determined by

  (64)

  This formula is certainly wrong since it is derived from a purely classical theory; the intensity of the r
adiation of frequency (En − Em)/n depends on the coefficient am of the final state, as well as on an of the initial state. This is in direct contradiction to Bohr’s fundamental postulate. The contradiction may be eliminated by arbitrarily dissecting the sum into its separate terms, omitting the offending factors and relating each term to the upper level. The formula (63) for the moment of the virtual dipole associated with the transition then appears once more.

  (c) The complete treatment of radiation and matter. The consistent treatment of radiation phenomena requires the simultaneous application of the quantum theory to radiation and matter, in which case it is naturally immaterial whether the particle or wave representation is used. Dirac,ct in his radiation theory, employs the language of the particle representation, but makes use of conclusions drawn from the wave theory of radiation in his derivation of the Hamiltonian function. The fundamental ideas of this theory are briefly outlined here.

  The atom will be represented by a single electron moving in an electrostatic force field φ0. The relativistically invariant equation of the one electron problem is, according to Diraccu (φ0 scalar potential, φi [i = 1, 2, 3], electromagnetic potentials),

  (65)

  or

  (66)

  (The usual summation convention is adopted.) Here, as before, the pi’s are the momenta canonically conjugate to the qi, and the α’s are operators which satisfy the equations

  (67)

  From the equations of motion it follows that

  (68)

  Except for a factor (−c) the αi’s are thus identical with the velocity matrices. From (66) it follows that the interaction energy of atoms and radiation field can be written in the simple form

  (69)

  The Hamiltonian function of the complete system atom plus radiation field is thus

  (70)

  The problem is brought into a simple mathematical form by assuming the radiation field to be in an inclosure, thus providing an orthogonal system of functions on solution of the Maxwell equations subject to the appropriate boundary conditions. The φi may be developed in this system, and the coefficients [cf. A(123) and (124)] may be written in the form

  where Nr is the number of light quanta belonging to the rth characteristic vibration. The total energy of the radiation field before considering its interaction with the atom is simply

  (71)

  In the development of the φi in the orthogonal system the individual terms still depend on the position of the atom in the inclosure. Since the dependence averages out in the final result when the inclosure is sufficiently large, it is convenient to introduce a mean-square amplitude obtained by averaging the square of the true amplitude over all possible positions of the atom. This yields the following expression for φi:

  (72)

  Here αir is the angle between the electric vector of the r th characteristic vibration and the qi -axis, and σ r is the number of characteristic vibrations in the frequency interval Δνr and solid angle Δωr divided by Δνr Δωr. Thus the Hamiltonian function for the complete system is

  (73)

  where is the component of the vector in the direction of the electric vector of the rth characteristic vibration.

  From equation (73) all the results obtained above by the use of the conservation laws may immediately be deduced. Thus the constancy of H may be proved as in the Appendix (§ 1, p. 121), and it further follows that for the emission or absorption of a light quantum hνr the essential factor is the matrix element of r corresponding to the transition concerned. Except for certain numerical factors which will not be calculated here the transition probability is given directly by the square of this matrix element. If the calculation is carried out (the interaction terms being regarded as perturbations), emission and absorption processes appear as first-order effects and dispersion phenomena as second order. For the details of the calculation the reader is referred to the papers of Dirac.cv

  The formulation of the Hamiltonian of the radiation problem in equation (73) has the disadvantage that it does not appear to involve the interference and coherence properties of the radiation. This is only the case, however, when mean amplitudes are used, as in the foregoing. If the correct amplitudes resulting from the development of the Φi in the orthogonal functions are retained, then the fact that these functions are solutions of the Maxwell equations assures interference and coherence properties for the radiation that correspond to the Maxwell equations. For example, solutions of the Maxwell equations appear as factors of the quantities ar in A (113) and these factors disappear at the position occupied by the atom when the vector potential vanishes there because of interference. Thus there will be no absorption of light in regions where there would be none according to the classical interference theory. From these considerations it follows at once that the classical wave theory is sufficient for the discussion of all questions of coherence and interference.

  § 5. INTERFERENCE AND THE CONSERVATION LAWS

  It is very difficult for us to conceive the fact that the theory of photons does not conflict with the requirements of the Maxwell equations. There have been attempts to avoid the contradiction by finding solutions of the latter which represent “needle” radiation (unidirectional beams), but the results could not be satisfactorily interpreted until the principles of the quantum theory had been elucidated. These show us that whenever an experiment is capable of furnishing information regarding the direction of emission of a photon, its results are precisely those which would be predicted from a solution of the Maxwell equations of the needle type (cf. the reduction of wave-packets, II, § 2c).

  As an example, the recoil produced by the emission of a photon will be discussed. Let an atom go from stationary state n to m with the emission of a photon, and an appropriate change of its total momentum. As we are only concerned with the coherence properties of the emitted radiation, we use the correspondence-principle method, in which the radiation is calculated classically. As source of the radiation we take a charge distribution which is modeled after the expression which would be given by the classical theory of matter waves. The atom will be supposed to consist of one electron (of mass µ, charge −e, co-ordinates re) and a nucleus (of mass M, charge +e, co-ordinates rn). The Schrödinger function of the nth state, in which the atom has the total momentum P, is

  where rc = (µre + Mrn)/(µ + M) is the vector to the center of gravity of the atom. If the matrix element of the probability density associated to the transition n → m , P → Pʹ, E → Eʹ, be calculated, one obtains

  By averaging over the co-ordinates of the nucleus, one obtains the charge density due to the electron, by averaging over the co-ordinates of the electron, that due to the nucleus; the total charge density is their sum. This density is to be considered as the virtual source of the emitted radiation, at least in so far as its coherence properties are concerned. The two component densities are [the common factor e is omitted, r = re − rn is the variable of integration, dv the volume element, and γ = M/(µ + M)]

  The total density is thus

  in which the value of the constant does not interest us. The current densities are given by analogous expressions. The radiation emitted by these charges is to be calculated from the retarded potentials:

  is the scalar potential and analogous expressions may be obtained for the vector potentials Φi (Rʹ is the distance from the point of integration, r, to the point of observation R). The result is therefore

  If one supposes that an experiment has determined the position of the atom with a given accuracy (the value of the momentum P must then be correspondingly uncertain), then this means that the density ρ is given by the foregoing expression only in a finite volume Δv, and is zero elsewhere. If the radiation at a great distance from Δv is required, Rʹ may be expanded in terms of R (the co-ordinates of the point of observation) and r (the co-ordinates of the point of integration):Rʹ = R − R1 · r,

  where R1 = R/ R . The scalar potential is then given by

  in which
hν = E − Eʹ.

  The integral is appreciably different from zero only in that regions for which the factor of r in the exponential is less in absolute magnitude than the reciprocal of Δl, the linear dimension of Δv. In all other regions, the radiation from different portions of Δv is destroyed by interference. Hence

  and the atom recoils with the momentum hν R1/c (except for the natural uncertainty h/Δl). If the direction of recoil is determined by some experimental procedure, the emitted radiation thus behaves like a unidirectional beam. This is only a special case, however, which is realized only when P and Pʹ are determined with sufficient accuracy, and the co-ordinates of the center of gravity are correspondingly unknown. The other extreme is realized when the experiment fixes the position of the atom more precisely than Δl = h / |P − Pʹ| = c/ν, i.e., more precisely than one wave-length of the emitted radiation. The expression for Φ0 then represents a regular spherical wave and no conclusions can be drawn concerning the recoil, since its uncertainty is greater than its probable value.

 

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