22 SUR 0 CLASSICAL 」一刃 C 〃 / CS [1-3 The average rate of emission 0f radiant energy by such a system is consequently dE 16 レ 4e2 襯 ( 3 ー 4 ) 3C3 inasmuch as average COS2 2 vt over a cycle iS 、 one-half. As a result 0f the emission 0f energy, the amplitude 新 0f the motion will decrease with time ; if the fractional change ⅲ energy during a cycle 0f the motion is small' however' this equa- tion ret ains its validity. The radiation emitted by such a system has the frequency 0f the emitting system. lt is plane-polarized, the plane 0f the electric vector being the plane which includes the の axis and the direction 0f propagation 0f the light. ln case that the particle carries out harmonic oscillations along れⅡ three axes も and 名 , with frequencies 材 , , , and and amplitudes ()t a given time) 新 , , and 名 0 , respectively, the total rate of emission 0f radiant energy will be given as the sum 0f three terms similar t0 the right side 0f Equation 3 ー 4 , one giving the rate 0f emission 0f energy as light 0f frequency , one 0f , Of . If the motion of the particle is not simple harmonic, it can be represented by a Fourier series or Fourier integral a or integral 0f harmonic terms similar t0 that 0f Equation 3 ー 2 ; light 0f frequency characteristic 0f each 0f these terms will then be emitted at a rate given by Equation 3 ー 4 , the coeffcient 0f the Fourier term being introduced ⅲ place of 工 0. The emission of light by a system composed 0f several inter- acting electrically charged particles is conveniently discussed ⅲ the following way. A Fourier analysis is first made 0f the motion Of the system ⅱ 1 a given state tO resolve it intO harmonic terms. For a given term, corresponding tO a given frequency 0f motion the coefficient resulting from the analysis (which is a function 0f the coordinates 0f the particles) is expanded as a 2 れ / X , ⅲ which 工 1 , power series in the quantities 工 1 / 入 , , are the coordinates 0f the particles relative t0 some origin (such as the center Of mass) and 入 = c/v is the wave length of the radiation with frequency 材 . The term 0f zero degree ⅲ this expansion is zero, masmuch as the electric charge 0f the system does 110t change with time. The term 0f first degree involves, in addition tO the harmonic function 0f the time, 0 Ⅱ ly
300 PER'TURBA TION 7 ' ″ 0 ん Y / Ⅳ「 0 ん / Ⅳ G T 〃刃 TIME [XI-40a radiation at temperature it also absorbs radiant energy, the rates 0f absorption and 0f emission being given by the classical laws. These opposing processes might be expected to lead to a state Of equilibrium. The following treatment of the correspond- ing problem for quantized systems (atoms or molecules) was given by Einstein1 ⅲ 1916. Let us consider two non-degenerate stationary states 襯 and れ 0f a system, with energy values 幵 and ー「れ such that ー仏。 is greater than 、 . According t0 the B0hr frequency rule, transi- tion from one state t0 another will be accompanied by the emission or absorption Of radiation Of frequency 0f energy 0f radiation and undergo transition to the upper state being p(v)dv). The probability that it will absorb a quantum Of radiation between frequencies and ート dv in unit volume radiation 0f density p ( レ… ) ⅲ this frequency region (the energy assume that the system is ⅲ the lower state ⅲ a bath of B 。→ .p@冖) ・ in unit time iS u the relative phases of oscillator and wave. could either absorb energy from the 升 01d or lose energy tO it, depending according which an oscillator interacting with an electromagnetic wave 2 This postulate is of course closely analogous to the classical theory , 121 ( 1917 ). 1 A. EINSTEIN, ん . d. のな c ん . 2 んい . Ges. 18 , 318 ( 1916 ) ; P んい . Z. 18 , is が - coe. c - 面 d ce イ ~ em お 0 れ . ス。→れ is 刃 e ' s c c .- ー可 0 佖 00 を - etn な 0 and お ーれ十お。→ ). IS transition t0 the lower state with the emission of radiant energy probability thaC the system ⅲ the upper state 襯 will undergo the other proportional t0 it. We therefore assume that the parts, one 0f which is independent 0f the radiation density and postulate2 that the probability 0f emission is the sum 0f two necessary ⅲ order t0 carry through the following argument to tional t0 the density 0f radiation. On the other hand, it is bility 0f absorption 0f radiation is thus assumed to be propor- Bn →。 is called s coq 0 を可 -- - 可 - 当い 24 毎 . The proba-
28 T 〃刃 0 も D QUANTUM T 〃 EO Y 2 1 [II-5b ( ト 1 ) BOhr ⅲ addition gave a method 0f determining the quantized states 0f motion—the stationary states—of the hydrogen atom ・ His method 0f quantization, involving the restriction 0f the angular momentum 0f circular orbits t0 integral multiples 0f the quantum ん / 2 碼 though leading t0 satisfactory energy levels, was soon superseded by a more powerful method' described in the next section. Prob1em 5 ー 1. Consider an electron movmg in a circular orbit about a nucleus Of charge ze. Sh0W that when the centrifugal force is just balanced by the centripetal force ze2/r2, the total energy is equal t0 one-half the potential energy ー Ze2 を . Evaluate the energy 0f the stationary states for which the angular momentum equals れん / 2 碼 with れ = 1 , 2 , 3 , 5b. The Wi1son-Sommerfe1d Rules 0f Quantization. —ln 1915 W. Wilson and A. Sommerfeld discovered independentlyl a powerful method 0f quantization, which was S00 Ⅱ applied' especially by Sommerfeld and his coworkers, ⅲ the discussion lines Of frequencles ′ 1 and レ 2 occur spectrum Of frequently possible tO 血 also a line with frequency 内十 2 or This led directly t0 the idea that a set Of numbers, called r 襯 0 ん es , can be assigned tO an atom, such that the frequencies Of all the spectral lines can be expressed as differences Of pairs Of values. usually given 1 Ⅱ wave numbers, WhiCh iS reCiprOCal Of the wave length expressed in centimeters, iS a for spectro— scopic use. ア e shall use the symbol ラ for term values m wave numbers' reserving the simpler symbol ′ for frequencies in sec¯l. The normal state of the ionized atom is usually chosen as the arbitrary zerO' and the term values which represent states Of the atom with lower energy than the ion are given the positive sign, SO that the relation between } / and ラ is hc The modern student, to whorn the Bohr frequency rule has become common- place, might consider that this rule is clearly evident in the work Of Planck and Einstein. This is not so, however; the confusing identity 0f the mechanical frequencies Of the harmonic oscillator (the only system discussed) and the frequency of the radiation absorbed and emitted by this quantized system delayed recognition of the fact that a fundamental violation 0 「 electromagnetic theory was imperative. W. WILSON, P ん M 叩 . 29 , 795 ( 1915 ) ; A. SOMMERFELD, スれル d. ~ ん . 51 , 1 ( 1916 ).
CHAPTER II THE OLD QUANTUM THEORY 5. THE ORIGIN OF THE OLD QUANTUM THEORY The 01d quantum theory was born ⅲ 1900 , when Max P1anck1 announced his theoretical derivation of the distribution law for black-body radiation which he had previously formulated from empirical considerations. He showed that the results of experl- ment 0 Ⅱ the distribution of energy with frequency of radiation ⅲ equilibrium with matter at a given temperature can be accounted for by postulating that the vibrating particles of matter (considered to act as harmonic oscillators) do not emit or absorb light continuously but instead only ⅲ discrete quanti- ties 0f magnitude ん proportional to the frequency 材 of the light. The constant of proportionality, ん , is a new constant of nature ; it is called れ c ん ' s co れ s 厄 and has the magnitude 6.547 x 10 ー 27 erg sec. lts dimensions (energy X time) are those of the 01d dynamical quantity called ac 0 れ , ・ they are such that the product 0f ん and frequency ′ (with dimensions sec—l) has the dimensions 0f energy. The dimensions of ん are also those of angular momen- tum, and we shall see later that just as ん′ is a れ 0 れル襯 of radiant energy 0f frequency so is ん / 2T a natural unit or quantum of angular Ⅱ 101 Ⅱ e Ⅱ tu Ⅱ 1. The development of the quantum theory was at first slow. lt was not until 1905 that Einstein2 suggested that the quantity Of radiant energy ん was sent out in the process Of emission Of light not ⅲ all directions but instead unidirectionally, like a particle. The name I ん田れ襯 or をん 0 んれ is applied to such a portion 0f radiant energy. Einstein also discussed the photo- electric effect, the fundamental processes of photochemistry, and the heat capacities of solid bodies ⅲ terms of ・ the quantum theory. 市 e Ⅱ light falls on a metal plate, electrons are emitted from it. The maxnnum speed of these photoelectrons, however, 1 M. PLANCK, スれル d. ん . ( 4 ) 4 , 553 ( 1g1 ). 2 A. EINSTEIN, スれれ . d. んい . ( 4 ) 17 , 132 ( 1 5 ). 25
XI-40bl 刃」ー SS ー 0 ⅣÄND ABSORPTION 0 RADIATION 303 taken over all particles ⅲ the system) is called the 00 襯 0 e 可 て . 市 20 0m0 of the s.ystem a10 具 g - the axis and is often represented by the symbol ェ . We now make the approximation that the dimensions of the entire system (a molecule, say) are small compared with the wave length 0f the radiation, SO that the electric field Of the radiation may be considered constant over the system. ln the case under consideration the field strength 刃ェ is given by the expression 蟻 = 工瓔 ( の ( e2 , 翫十 e ー 2 , 心 . Let us temporarily consider the perturbation as due t0 a single 1 ⅲ the right frequency 久 lntroducing ( 0 ) = 0 and 0 れ ( 0 ) side 0f Equation 39 ー 6 ( 0 。 being the coeffcient 0f a particular state and all the Other coeffcients ⅲ the sum being zero), this equation b ec omes 平 0 * 〃′ 0 2 V0m*e ん瓔 ( の ( e2 , 十 If we now introduce the symbol 。 t0 represent the integral ( 4g11 ) こ工 m れ we obtain the equation 2 面の ェ ~ ェの e which gves, 0 Ⅱ integration' 2 2 2 す一 2 第一元 2 2 籠ー ーー ( m ールーんの t - ー ( Ⅳ m ールれ十ん材 十 e ん ( Ⅳーれ十んのま ルールれ十 am(t) = ー瓔 ( の 十 ルールれ一 Of the two terms 0f Equation 40 ー 12 , only one is important' and that one only if the frequency happens tO lie close t0 ( ル。ール。 ) / ん . The Ⅱ merator ⅲ each fraction can vary ⅲ 0 010t0 00g0it0d0 001Y b 、毳 0-0 00d 2 , 00d , ⅲ " 000h " for a single frequency the term , …刃 ! ( の is always small' the ( 4g12 )
324 PE U BA / 0 Ⅳ 7 ' ″刃 0 必ーⅣ 0 ん / ℃ T 〃刃 77 刃 [XI-41c than だ , investigate the system to find what the values Of the quantum numbers れ 1 and 鉈 2 are. The result 0f this investigation will be the same, ⅲ a given case, no matter at what time later than だ the set 0f experiments is carried out, inasmuch as the two oscillators will remain in the definite stationary states ⅲ which they were left at time だ so 10 Ⅱ g as the system is left unperturbed. ThiS sequence Of experiments can be repeated over and over, each time starting with the system ⅲ the state 1 1 , れ 2 = 0 and allowing the coupling t0 be operative for the length 0f time ln this way we can find experimentally the probability of finding the system in the various states 1 1 , れ 2 = 0 ; れ 1 0 ; etc. ; after the perturbation has been operative 0 , 2 for the length of time だ . The same probabilities are given directly by our application of the method of variation 0f constants. The probability of transition t0 states of considerably different energy as the result Of a small perturbation acting for a short time is very small, and we have neglected these transitions. Our calculation shows that the probability 0f finding the system ⅲ the state B depends 0 Ⅱ the value of だⅲ the way given by Equation 41 ー 12 , varying harmonically between the limits 0 and 1. Now in case that we allow the coupling to be operative con- tinuously, the complete system can exist in various stationary states, which we can distinguish from one another by the measure- ment 0f the energy of the system. Two 0f these stationary states have energy values very close to the energy for the 1 of the system with 0 and れ 1 = 0 , states れ 1 = 2 the coupling removed. lt is consequently natural for us t0 draw 0 Ⅱ the foregoing argument and t0 describe the coupled system in these stationary states as resonating between states and お , with the resonance frequency 2 〃 / ん . Even when it is not possible to remove the coupling inter- action, it may be convenient tO use this description. Thus in our discussion of the helium atom we found certain stationary states to be approximately represented by wave functions formed by linear combination of the wave functions ls ( 1 ) 2s ( 2 ) and 2s ( 1 ) ls ( 2 ). These we identify with states and B above, saying that each electron resonates between a ls and a 2s orbit the two electrons changing places with the frequency 1 / ん times 0 , れ 2
320 ~ 刃ん TU BATIO ル T 〃刃 0 ん Y 1 Ⅳ / 0 も一Ⅳ G ア刃 TIME [XI-41b hX ん入 with = 1 are found to be The correct zeroth-order wave functions for the two levels of Equation 41 ー 7. leads t0 values for the energy expressed by the first two terms degenerate levels shows that the first-order perturbation theory giving ″ = 士ん入 / 2 レ 0. A similar treatment Of the succeeding and 1 { 羇@1ル 2 ) 十矚@1) 叫@2) } 2 1 2 & corresponding t0 the lower 0f the two levels and 協 tO the upper. The subscripts and are used t0 indicate that the functions are respe ctively symmetri C an d the coordinates 工 1 and 工 2. We see that we are not justified ⅲ de- scribing the system ⅲ either one 0f these stationary states as con- 1 and the second ⅲ sisting 0f the first oscillator ⅲ the state 1 0 , or the reverse. lnstead, the wave functions the state 2 1 contribute equally t0 each 0f 0 andnl = 0 , れ 2 1 = 1 , 2 the stationary states. lt will be shown ⅲ Section 41C that if the perturbation is small we are justified ⅲ saying that there is reso- nance between these two states 0f motion analogous t0 classical resonance, one oscillator at a given time oscillating with large amplitude, corresponding t0 % 1 = 1 , and at a later time with 0. The frequency with small amplitude, corresponding t0 れ 1 which the oscillators interchange their oscillational states' that is, the frequency of the resonance, is found t0 be 入 / 物 , which is just equal t0 the separation 0f the two energy levels divided by ん This is also the frequency of the classical resonance (Eq. 41 ー 5 ). ln discussing the stationary states of the system 0f two inter- acting harmonic oscillators we have seen above that it is venient tO make use Of certain wave functions 。 ( 工 1 ) , etc. WhiCh are not correct wave functions for the system, the latter being
304 刃 TU お 4770 Ⅳ T 〃 0 必 1 Ⅳ 0 ん / / ル C 7 ' ″刃 77M 霽 [XI-40b expression will b e small unless the denominator is also very small ; that is, unless ん is approximately equal to ルール。 . ln Other words, the presence Of the so-called so れ 0 れ ce de れ 0 襯乞れ佖 - tor Wm ーー仏れ一ん′ causes the influence of the perturbation ⅲ changing the system from the state れ tO the higher state to be large only when the frequency Of the light is close to that given by the B0hr frequency rule. ln this case は a. bsorp い 9 ー is. - t second term whi&isjmportant ; 「 0 ら新 d は c ed smission 0f radiation (with ル。ール。 negative) , the first term woyldplay the same - 工 01e ー - ・一 Neglecting the first term, we obtain for a の佖の , after slight rearrangement, the expression sin2 ー ( ル。ールーんのこ ( の。 ) = 4 ( 朝 2 瓔 2 ( の ( ルール。ーんの 2 ()f , … complex, the square 0f its absolute value is t0 be used ⅲ this equation. ) This expression, however, includes only the terms due t0 a single frequency. ln practice we deal always with a range 0f frequencies. lt is found, 0 Ⅱ carrying through the treatment, that the effects of light of different frequencies are additive, so that we now need only to integrate the above expression over the range Of frequencies concerned. The integrand is seen t0 make a significant contribution only over the so that we are justified in replacing 刃呈 ( の regIOn Of ′ near ′。。 , by the constant 刃!@冖) , obtaining sin2 ー ( ルールれ一ン ( の ( の = 4 ( 2 瓔 20 ( ルールれ一んの 2 This integral can be taken from ー tO 十 , inasmuch the value-of the integrand is very small except ⅲ one 、 region ; and SI Ⅱ 2 工 dx = we can obtain the making use Of the relation equ ation ( 4g13 ) lt 沁 seen that, as the result of the integration over a range of values Of the probability Of transition to the state ⅲ time proporti onal tO ち the coeffcient being the transition prob ability
1-4] SU]. れユ 0 C 戸ア E / 23 a function Of the coordinates. The aggregate of these first- degree terms ⅲ the coordinates with their associated time factors, summed over all frequency values occurring ⅱ 1 the original Fourier analysis) represents a dynamical quantity known as the e ん c な襯 0 襯 e れ t of the system, a vector quantity p defined as ( 3 ー 5 ) equations ⅲ Cartesian coordinates and then altered their form by ln carrying out the first purpose, we have discussed Newton's mechanical methods. methods t0 problems which are later discussed by quantum- of Newton ; and second, t0 illustrate the application of these and 0f Hamilton, have been developed from the original equations 0f classical dynamics, such as the equations of motion of Lagrange first, to indicate the path whereby the more general formulations The purpose of this survey of classical mechanics is twofold : 4. SUMMARY OF CHAPTER I can be detected, the process of quadrupole emission is important. dipole radiation is zero and the presence of very weak radiation Under some circumstances, however, as when the intensity of and in consequence (lipole radiation alone iS ordinarily discussed. negligibly small ⅲ comparison with the rate of dipole emission, pole and higher moments 0f an atom or molecule is usually 0f emission of radiant energy as a result of the change of quadru- 0f the system, and higher powers form higher moments. The rate , / 入 form a quantity Q called the 臾田 d ロ 0 ん襯 0 e The quadratic terms in the expansions ⅲ powers of / 入 , being described as d ゆ 0 ん 0 市襯れ . is usually called d ゆ 0 ん e 襯な s れ , the radiation itself sometimes moment expansion. The ernission Of radiation by thiS mechanism 3 ー 4 , with ea;o replaced by the Fourier coefflcient in the electric- frequency at a rate given by a11 equation similar to Equation ⅲ this representation of P, there will be emitted radiation of electric moment P. Corresponding to each term of frequency 材 particles can be discussed by making a Fourier analysis of the approximation the radiation emitted by a system 0f several the ith particle, with charge . Consequently to this degree of i11 which 新 denotes the vector from the origin to the position of
302 ~ 刃 TU BA ー 0 Ⅳ″ EO ーⅣ 0 石一Ⅳ G 社刃 T 刃 [ 刈 40b 40b. The CaIcuIation of the Einstein Transition ProbabiIities by Perturbation Theory. —According to classical electromagnetic theory, the density 0f energy 0f radiation of frequency ⅲ space, with unit dielectric constant and magnetic permeability, given by the expression 1 ーー刃 2 ′ 4T ( 4g6 ) ⅲ which represents the average value of the square of the electric field strength corresponding to this radiation. The distribution 0f radiation being isotropic, we can write ( 4g7 ) calculation. Since the average value 0f cos2 2 元岬 -. 】 -- we see the complex exponential form being particularly convenient for 刃 , ( の = 2 瓔 ( の cos 元 = 瓔 ( の ( e2 , 十 e ー 2 , ) , ( 4g8 ) variation Of the radiation by writing の direction, etc. We may conveniently introduce the time 刃 , ( の representing the component of the electric field ⅲ the that 1 ー塒 ( の 6 3 ー刃 2 ( の ー瓔 2 ( の . 4 4 4 ( 4g9 ) Let us now consider two stationary states 襯 and of an unperturbed system, represented by the wave functions 平 ! and 重 % and such that Ⅱ is greater than T 「れ . Let us assume that at the time 0 the system is ⅲ the state and that at this time the system comes under the perturbing influence of radiation Of a range 0f frequencies ⅲ the neighborhood Of 。。 , the electric field strength for each frequency being gwen by Equation 4g8. We shall calculate the probability of transition to the state 襯 as a result 0f this perturbation, using the method 0f Section 39. The perturbation energy for a system of electrically charged particles in an electric field 刃 , parallel to the 工 axis is ( 4g10 ) jth particle of the system. The expression (the sum being ⅲ which の represents the charge and the の coordinate of the