The Dreams That Stuff is Made of

by Stephen Hawking

  This model is applied to the explanation of the band spectra of molecules. If a molecule rotates about a fixed axis, the emitted frequencies, according to Bohr, are given by the relation , but, as in the case of the oscillator, the number of different frequencies given by this formula is too large. Out of these frequencies we must choose certain ones by the Principle of Correspondence. For this purpose we consider a component of the electric moment of the rotator. Evidently, in this case the motion is also given by a simple harmonic oscillation, whence we conclude, as above, that there are no other jumps of n than those where n changes by ± 1, i.e., that n–m= ± 1. Introducing this restriction, we obtain for the emitted frequencies (placing m–n= 1, or m = n + 1):

  The rotation frequency of the rotator itself is given by

  Therefore, as n increases, the relative difference between the rotation and the emitted frequency becomes smaller. In both cases we have an equidistant series of frequencies and, indeed, the band spectrum emitted by a rotating molecule appears as a first approximation to consist of such a series.

  We shall not go further into this problem, but instead will now consider the general relation which exists, according to the Principle of Correspondence, between the frequencies and the intensities of the spectral lines calculated classically and the corresponding quantities calculated according to the quantum theory. We consider the electric moment of the system having a Fourier expansion analogous to that of the coördinates

  (4)

  The frequencies can be written

  (5)

  Let a stationary state be determined by

  and another by

  then we can consider in the Ik -space of f dimensions the two points connected by the straight line

  where

  Then

  and

  On the other hand the quantum frequency is

  (6)

  and the relation between the frequencies in the classical and in the quantum theory is the same as that between derivative and difference-ratio. It is also possible to consider the quantum-theory frequencies as the straight-line mean of the classical frequencies, as follows,

  (7)

  If the changes of the quantum numbers are small compared with the numbers themselves, the two expressions for νq u and νcl respectively differ very little. As to the intensities, we expect that they vary approximately in the same way as the quantities |Cτ |2, where Cτ is a function of the Ik’s and of . It is seen that this statement has a definite meaning only if nk is large, because only in this case is it immaterial whether we place in for n the initial value n(1) or the final value n(2). On the other hand, this statement has a unique meaning if Cτ(I) is identically zero for all I’s, for then we expect that a jump of τ does not occur. In other cases the difficulty has been evaded by taking a suitable mean of Cτ(I) over the values of I between the initial and the final states. By this method, Kramers has succeeded in representing satisfactorily the results of observations in certain cases. It is not satisfactory in principle that we should not find in the quantum theory, in the form here presented, a unique determination of the intensities. This is one of the main reasons which led us to formulate our new quantum theory, where this difficulty is overcome.

  LECTURE 5

  Degenerate systems—Secular perturbations—The quantum integrals.

  We now say a few words about the case, left aside so far, of degeneration, that is, that in which there exist identical relations in the Ik’s of the form

  (1)

  Then our theorem of uniqueness no longer holds, and it is no longer possible to formulate the quantum conditions in the formIk = nkh.

  Such is the case, for instance, in the harmonic oscillator of two degrees of freedom.

  The solution of the equations of motion can be written down immediately, because the two coördinates are separable. We obtain

  Here Ix , wx; Iy , wy are two conjugate pairs of action and angle variables. If now wx and wy are not commensurable the motion given by placing (Fig. 3)wx = ωx t + δx, wy = ωy t + δy

  is a so-called Lissajous figure, in which the path comes as near as desired to any point within a rectangle. But if a relation of the formτxωx + τyωy = 0

  exists, for instance if (Fig. 4.)ωx = ωy = ω0 = 2π ν0,

  then the orbit is simple periodic (ellipse). We can now rotate the system of coördinates arbitrarily without changing the form of the solution. But in doing so the sides of the rectangle change continuously and so do the magnitudes √Iz, √Iy, which differ from them only by the constant factor . It is therefore impossible to place Ix and Iy proportional to whole numbers nx, ny. The diagonal of the rectangle, however, that is the square root of the quantity

  remains invariant for such a rotation. We can therefore setIx + Iy = nh

  whereby the total energy

  is uniquely determined. W = nhν0 has therefore exactly the same value as for the linear oscillator. We can describe this behavior as follows: If we introduce, instead of Ix and Iy, two new variables, Ix + Iy = I and Ix–Iy = I’, then two new conjugate angle variables, w, w’, correspond to the latter, with the frequencies,

  and we can only quantize the variable I which alone appears in W and which therefore alone corresponds to a frequency different from zero.

  FIG. 3

  FIG. 4

  The following rule holds in general: In cases of degeneration it is always possible to attain, by means of a linear whole-number transformation of determinant ±1, that H = W shall depend only on a number s of I-variables, among which there are no commensurability relations. We call such variables Iα. To these variables correspond s frequencies να different from zero, while the other f - s frequencies νρ vanish. Only such variables Iα are to be equated to multiples of h. Bohr calls s the degree of periodicity of the system.

  Evidently we can increase the degree of periodicity of a system by introducing perturbing forces, for instance by placing the system in an electric or in a magnetic field. Then the original energy-function, which we shall call H0, is increased by an additional energy which we shall call “perturbation energy” and denote by λH1, where λ is a measure of the magnitude of the additional energy. If the perturbation is small, that is, if λ is small, then there is a simple process whereby the new motions which are to be added to the system which was originally degenerate can be calculated. The influence of the perturbation energy is to change slightly all the magnitudes w, I, but the influence is different for these two kinds of variables. Those angle variables wρ, which belong to the zero frequencies of the unperturbed system, and which therefore were constant, now change slowly with frequencies which are proportional to λ. The other angular variables wα will only undergo small variations of their frequencies. If we take the w0 , I0’s of the unperturbed system as initial variables for the perturbation problem, then we have

  (2)

  and will complete its period many times during a period of . Therefore an approxination can be made by taking an average over

  (3)

  This function can be considered as the energy-function of a new problem of motion for the formerly degenerate variables , . It is required to solve the equations of motion

  (4)

  that is, to find a canonical substitution

  such that H1 is transformed into a function W1 which depends only on I , i.e., (Iα is written for .H = W0 ( Iα ) + λW1( Iα , Iρ ).

  The perturbation frequencies are then

  The name “secular perturbations” is given in celestial mechanics to the slow motions of frequencies νρ.

  We shall only touch upon the question of how the angle and action variables can be actually found in given cases. The method of separation of variables is often used: It is applicable if it is possible to find canonic variables pk, qk for which the Hamilton-Jacobi differential equation can be solved by setting

  (5)

  Then

  (6)

  is a function of qk only and we can show that the integrals taken
over a period

  (7)

  are the action variables. The functions Sk(qk) depend also on these constants Ik. Applying a canonic transformation, in which the Ik’s enter as new action variables, the corresponding new angle variables are defined by

  (8)

  Since S satisfies the Hamilton-Jacobi differential equation, H is transformed into a function W(I1 ··· If) and the condition (A) is satisfied.

  If any coördinate qk varies once between its limits, while the other coördinates qk are kept constant, the change of any variable w k is

  Now,

  whence

  (9)

  If any point in q-space, to which corresponds the point in w-space, describes a closed curve, then the pointwneed not return to its original position, but the end point is given by an expression of the form where the τ’s are whole numbers. The q’s are therefore periodic functions of the w’s, with the fundamental period 1. Condition (B) is thus satisfied.

  According to the definition of Ik , S increases by the amount Ik every time that qk varies over a cycle, the other variables being held constant. As wk increases by 1 at the same time, the function remains unchanged. Therefore, it is periodic, and the condition (C) is satisfied, whence it is proved that w, I are the angle and action variables.

  Many authors introduce the quanta by this integral definition, but it appears to me, as to Bohr, better to define them generally by the properties of periodicity, that is, by the three conditions (A), (B), (C).

  LECTURE 6

  Bohr’s theory of the hydrogen atom—Relativity effect and fine structure—Stark and Zeeman effects.

  After these general considerations we now take up the applications to the theory of atomic structure. As you know it was with the hydrogen atom that Bohr first developed his ideas. We have in this case one nucleus and one electron, that is, a two-body problem which can be reduced, as you know, to a onebody problem: the motion of a point around a fixed center of attraction. If r, φ, and θ are the polar coördinates of the electron relative to the nucleus, and if we place

  where M is the mass of the nucleus, and m that of the electron, we have

  The potential energy of the Coulomb force between a nucleus carrying a Z-fold charge and an electron is

  but we shall also consider general central forces with an arbitrary function U(r).

  Introducing the momenta we obtain

  (1)

  The corresponding Hamilton-Jacobi differential equation can be easily solved by separation of variables. In the case of Coulomb’s law

  FIG. 5

  we obtain the well-known Keplerian motions; of these only periodic orbits, i.e., ellipses, come into consideration in the quantum theory. It is seen at once that the motion is doubly degenerate for it has three degrees of freedom but is only simple-periodic. There is therefore only one action quantity I and one quantum condition. Calculation shows that I is related to the major axis a of the ellipse by the formula

  and for the energy we obtain

  (2)

  Referred to a system of axes directed along the axes of the ellipse, the motion is represented by simple Fourier series

  (3)

  the coefficients of which are continuous functions of the eccentricity ε. The angle variable w is, except for the factor 2π , the “mean anomaly” of astronomers.

  These were the starting formulas for Bohr’s theory of the hydrogen atom. PlacingI = nh

  and

  (4)

  he found

  (5)

  and obtained for the frequencies of the emitted light

  (6)

  For the hydrogen atom Z = 1 and this formula gives in fact all the known lines of hydrogen, in particular the Balmer series (n2 = 2),

  The formula gives not only the dependence on n1 but, what is more important, the correct value of RH. To calculate the latter we have to replace µ by the expression

  We may therefore write

  in which e,m and h are replaced by the best experimental values. Neglecting the small fraction m / M, which is about 1 / 1830, we obtain, dividing by the velocity of light, c = 3 × 1010 cm./sec.,

  while spectroscopic measurements give 109678 cm.−1.

  The series given by n2 = 1, n2 = 2, n2 = 3, n2 = 4, n2 = 5, have also been measured (by Lyman, Paschen, Brackett). Moreover Bohr was justified in maintaining that the series which is obtained by putting Z = 2, and which had until then been ascribed to hydrogen, must belong to ionized helium,

  The fraction m/M is now four times smaller than for the H-atom, because the He-atom is four times heavier. Therefore, the lines for same n1 and n2 do not exactly coincide with the hydrogen lines. This separation is observed experimentally and now we are certain that the spectrum is that of ionized helium, to be sure the most beautiful result of Bohr’s theory.

  Bohr’s theory of all other spectra may be briefly described as an attempt to consider them as modifications of the hydrogen spectrum. Two lines of attack are to be distinguished here; the first is to calculate the influence of secondary effects on the hydrogen atom: The dependence of mass on velocity is taken into consideration and gives the fine structure of the lines, then the influence of external electric and magnetic fields (Stark and Zeeman effects). The second line of attack leads to the study of other atoms and, together with it, to a theoretical systematic study of relations among the atoms and of the periodic system of the elements. Let me speak about the first line of attack.

  Sommerfeld was the first to point out and carry through the idea that the variation of mass demanded by the theory of relativity must have an effect on the spectrum. He replaced the classical energy-function by the relativistic one given in the first lecture:

  (7)

  On account of the smallness of the effect it is sufficient to take into account the first term in the expansion in powers of and write:H = H0 + H1

  where H0 is the classical energy-function and

  is the perturbation function.

  The law of areas holds also in relativistic mechanics. Therefore the orbit is plane, but the plane orbit is no longer a simple periodic ellipse, but is transformed into a “rose-shaped” figure. The motion can also be described as an elliptic motion the major axis of which is rotating uniformly. The law of precession of the perihelion is found, following the method of secular perturbations, by taking the average of the function H1 over the unperturbed motion

  (8)

  where

  is a numerical constant, I’ is the angle variable conjugate to the azimuth w’ of the major axis and depends on the eccentricity ε by the simple relation

  FIG. 6

  Since w’ does not appear in H1, therefore it is a cyclic variable and we have the new quantum condition

  (9)

  k is called the azimuthal quantum number to distinguish it from the main quantum number n. k is always less than or equal to n. The total energy becomes

  (10)

  This formula expresses that every term of the unperturbed spectrum is separated into a number of terms which correspond to the values k = 1, 2 · · ·n. From this arises a splitting of the spectral lines,

  in such a way that k is changed only by ±1, for the rotation of the perihelion determined by I’ = kh is a simple harmonic motion. This split of spectral lines was predicted by Sommerfeld and experimentally verified for hydrogen and ionised helium, not only for the number of lines but also for the absolute value of the separation.ot Kramers has also calculated the intensity of the lines by means of the principle of correspondence and found good agreement with observations.

  FIG. 7

  The influence of an external electric field, i.e., the Stark effect, can be treated in quite an analogous way. The perturbation energy is in this case,

  (11)

  where z is the coördinate of the electron along the z-axis taken parallel to the field E. It is therefore simply required to calculate the average of z. This depends not only on the position of the major axis of the ellipse in the orbital pl
ane, but also on the orientation of this plane in space. It can be shown, however, that the problem of the secular perturbation can be reduced to one of one degree of freedom. The calculation gives for the energy,