we can reduce the problem of integrating the canonical equations for an energy function H(pq) to the following one: A function S is to be determined, such that
(11)
becomes a diagonal matrix. Then the solution of the canonical equations has the formp = Sp0S−1, q = Sq0S−1.
We have therefore a complete analogue of Hamilton-Jacobi’s differential equation. S corresponds to the action-function.
LECTURE 13
The example of the harmonic oscillator—Perturbation theory.
Let us now illustrate these abstract considerations by an example. For this purpose we choose the harmonic oscillator, for which
(1)
The canonical equations
(2)
give by elimination of p and placing
(3)
or, more explicitly,
(4)
To this is added the commutation relation which gives
(5)
There follows from the equation of motion that q (nm) can differ from zero only if
(6)
In the row m of the matrix there are therefore at most two non-vanishing elements, i.e., those for whichWn = Wm + hν0 or Wn = Wm + hν0.
Evidently the order of the elements in the diagonal of a matrix is of no importance. If we perform the same permutation on the rows and columns, all matrix equations are unaltered. We can therefore choose Wm = W0 arbitrarily and denote the “neighboring values” W0 + hν0 and W0–hν0 of W0 by the symbols W1 and W− 1. Each of these has again neighboring values which differ from it by hν0, etc. In this way we obtain an arithmetical series of energy levels,
(7)
The diagonal elements of the commutation relation (5) give
(8)
Whence it follows that |q (n, n + 1)|2 also form an arithmetical series with the difference h/8 π2 µν0. Since all these terms are positive, the series must stop somewhere. We have therefore
therefore
(9)
It is apparent at once that all other elements of the matrix pq—qp are actually zero. We verify further the conservation of energy:
(10)
This vanishes for n ≠ m and we have
(11)
The quantity W0 introduced above has therefore the value . The energy at absolute zero, which has been considered already by Planck and Nernst in statistical problems of the quantum theory, appears here quite naturally.
The formula for the complex amplitudes
(12)
involves arbitrary phases φn which are of great importance for the statistical behavior of the resonator. Besides, Equation (12) goes over into the classical formula
(13)
for large values of n.
The theory of the harmonic oscillator can be used as a starting point for the calculation of more general systems, if we consider these as derived from the former by variation of one of its parameters. The required process can be developed in a way closely analogous to the classical perturbation theory.
We assume the energy given as a power series in the parameter λ,
(14)
Let the mechanical problem defined by H0 (pq) be solved. We know the solution p0, q0 which satisfies the condition
and for which H0(p0q0) becomes a diagonal matrix W0. Now we try to determine a transformation S such that if
(15)
H(pq) is transformed into a diagonal matrix W. This means that S satisfies the Hamilton-Jacobi equation
(16)
To solve this equation we place
(17)
Then we haveS−1 = 1 − λS1 + λ2 (S21–S2 ) − · · · + · · · .
Substituting in Equation (16),
and equating the coefficients of like powers of λ we obtain the following system of approximate equations:
(18)
where H0, H1 . . . are to be considered as functions of p0, q0.
The first equation is satisfied. The others can be solved successively in a way quite analogous to that used in classical theory: The average of the energy is first formed in order to fix the energy constant, forSr H0 − H0 Sr = − (W0Sr − Sr W0)
has no diagonal terms. There follows, in general,
We have, further,
or
(19)
where ν0(mn) are the frequencies of the unperturbed motion.
This solution satisfies the condition
(20)
where the symbol ∼ denotes transposition of the rows and columns and the symbol* the substitution of conjugate complex quantities. As S is only obtained by successive calculation of the approximations S1,S2 ... this relation can be proved only by successive steps. We shall restrict ourselves to the first step. If we must have
then,
but our general formula (19) gives
therefore
(21)
As H1 is an Hermitian matrix, that is since
it follows that
The importance of the relation arises from the fact that the Hermitianness of the matrices p, q is a consequence of this relation. The rule
(22)
holds, as can be easily deduced from the definition of the products:
From this follows that
(23)
and similarly for p.
If we place,
then we have, as a first approximation,
Or more explicitly,
(24)
For the energy we obtain, as a second approximation,
or
(25)
LECTURE 14
The meaning of external forces in the quantum theory and corresponding perturbation formulas—Their application to the theory of dispersion.
Before discussing the significance of these formulas, we consider the more general case where the Hamiltonian function contains the time t explicitly. This can be easily taken account of formally by introducing in H(t , p, q) instead of t a new coördinate q0, to which corresponds a momentum p0, and considering the Hamiltonian function
(1)
To q0 , p0 correspond the canonical equations,
(2)
of which the first says that q0 is the time and the second defines p0.
A closer consideration leads to an important difficulty. The introduction of a function H depending explicitly on t has evidently the physical meaning that the reaction of the system A in question on other systems B which act on A is so small that it can be neglected, and that the quantities depending on these external systems B can be considered to be the same functions of time as they would be without the presence of A. In classical theory, where the interactions of two systems depend only on their instantaneous motion, the condition for this is that the coupling energy be small. But in the quantum theory this is not obviously so. Here the reaction depends, as our perturbation formulas show, not only on the instantaneous state of the system, but on all the states of the system together, for the products occurring in the formulas contain sums over all the states. The perturbation of the system A, due to a motion of the system B given as a function of the time can be taken care of only as long as approximations are restricted to those for which the quantities belonging to B enter only linearly in the perturbation function H1 . Higher approximations have no meaning even in the case of a weak coupling. But if the assumption is made that the system A under consideration is negligible energetically compared with the external systems B, then going over to higher approximations can also be justified in the quantum theory.
We shall restrict ourselves here to the first approximation q1, p1. We consider the special case where the system defined by H0 is acted upon by an electric field E. Then the perturbation function is, to a first approximation,
(3)
According to what has been said above, E can be looked upon as a function of the time. If in particular we are considering a monochromatic light wave of the frequency ν E = E0 cos 2π ν t,
therefore
then we get for the perturbation of the coördinates,
or, as p1 = �
�q1,
(4)
For the diagonal terms we have, in particular,
(5)
The polarization produced by the field E is obtained by multiplying q1 by the charge e and then the index of refraction can be calculated by well-known methods.
This formula for q1 (nn) contains Kramers’ theory of dispersion, which was found by considerations of correspondence. To understand its meaning we recall the relation between the theory of dispersion and the quantum theory of multiple-periodic systems. When a light wave acts on such a system, the electronic orbits perform oscillations. The resonance points of these forced oscillations lie evidently where the Fourier analysis of the orbits leads to a harmonic overtone. Debye attempted to calculate the dispersion formula for the hydrogen molecule using the model shown in Fig. 16, and Sommerfeld extended this process to more general molecular models with electrons arranged in rings. If they found a fairly good agreement with measurements of refractive indices it was only because the range of measurements lay very far from the characteristic resonance points. The incorrectness of the formula follows already from the fact that some of the resonance points have imaginary proper frequencies, which is always a sign of instability of the motion. It is rendered more evident by the fact that the resonance points have no relation to the frequencies which the system would emit according to the quantum theory. It is quite clear, however, that the frequencies actually emitted must determine essentially the resonance or dispersion curve, and not the higher harmonics of the stationary motion which are not optically observable.
FIG. 16
The first step towards a rational change of the theory of dispersion in this sense was made by Ladenburg. His dispersion formula consists essentially of those terms in the above expression (5) for q1(nn) for which n < kand which, therefore, correspond to “upward jumps,” that is, to absorption processes. Ladenburg also discovered a relation between the numerator of the dispersion formula |q0(nk )|2ν0(k n) and the transition probabilities between the states n and k which appear in Einstein’s derivation of Planck’s formula.
Kramers has given the complete expression for q1(nn) in which the emission terms (n > k) are also taken into account. Sinceν0(kn) = −ν0(nk)
these terms give “negative” contributions to the dispersion. Kramers’ formula has the great advantage of reducing in the limit to the classical formula for the influence of an alternating field on a multiple-periodic system. It therefore satisfies the principle of correspondence.
The case of a constant electric field (Stark effect), as represented by our original formula, was used by Pauli to estimate the intensity of the spectral lines of the mercury atom which do not appear in its natural state (for which q0(nm) = 0) and which are first excited by the field (q1 (nm) ≠ 0).
Kramers’ work suggested to me that quite generally the perturbation energy cannot depend on the classical frequencies of the unperturbed system, but rather on the quantic frequencies and this has been recently confirmed by Schrödinger’s considerations on the actual structure of certain line spectra, e.g. aluminum. By correspondence considerations I arrived at the expression (25) Lecture 13 for the perturbation energy W(2). By similar considerations Heisenberg and Kramers also found and discussed the expression q1 (nm ) [Eq. (4), Lecture 14] for a light wave. They correspond to the phenomenon that light of frequency ν is not only scattered as light of the same frequency, as in the classical theory, but also as light of other colors belonging to the combination frequencies ν ± ν0 (nk). This phenomenon had been already postulated by Smekal from considerations on light quanta.
Consider finally the limiting case of very high frequencies of the exciting lightν >> |ν0(mk)| , ν >> |ν(kn)|.
We then obtain
and since
therefore
(6)
Compare this with the excitation of a free electron by the same lectric field E0cos2π νt. Here we must only take that part corresponding to an element of the matrix. We have the differential equation
the solution of which is
Our quantic commutation rule can therefore be interpreted as the condition that, for sufficiently high frequencies, the electron behave as in classical theory, where the scattered light of frequency ν has the correct intensity and the scattered light of the combination frequencies vanishes. Starting from this condition Kuhn and Thomas found a formula equivalent to the commutation rule, as already stated above, and they and Reiche applied it to dispersion problems.
LECTURE 15
Systems o more than one degree of freedom—The commutation rules—The analogue of the Hamilton-Jacobi theory—Degenerate systems.
We now consider systems of f degrees of freedom. By an immediate generalization they may be represented by 2f-dimensional matrices
(1)
This representation is sometimes very convenient and clear, but not at all necessary. We can always imagine the matrix written in two dimensions. Then, as already shown for one degree of freedom, the expression of the stationary states as given by the arrangement of the rows is quite immaterial, as contrasted with the older theory. We can therefore always transform a 2 f-dimensional matrix into a two-dimensional one. We can for instance write a 4-dimensional matrix q (n1 , n2; m1, m2) as follows:
The definitions of addition and of multiplication are quite independent of the order of the indices. The rules of matrix calculus are therefore applicable as before. We can therefore define a Hamiltonian functionH (q1 ... q f, p ... p f)
and have the equations of motion:
(2)
The quantic commutation rules are fundamental. We make the following immediate generalization:
(3)
Whence follows, as before, for any arbitrary function f (q1 . . .q f , p1, . . . p f ),
(4)
Therefore the proof of the conservation of energy and of the frequency condition remains the same as above, as does the concept of canonic transformations
(5)
and the Hamilton-Jacobi equation,
(6)
The large number of commutation rules gives rise to the question whether pk , qk can be at all determined so as to satisfy all the conditions. It is easily seen that all the condition equations are not independent. From the canonical equations of motion alone follows, for instance,
The general proof of the possibility of satisfying all the conditions can be given by means of the theory of perturbations, starting from an unperturbed system with the energy-function
which therefore consists of f uncoupled systems. Let the motions of these be represented by two-dimensional matrices , . If these f uncoupled systems are considered formally as a single system of f degrees of freedom, , are to be represented by 2 f-dimensional matrices for which the relations
hold where
From these there follow in the first place the relations
(7)
Next the relation originally postulated for the 2-dimensional matrices,
(8)
is also correct for 2 f-dimensional matrices.
If now the Hamiltonian function for the coupled system is
(9)
then we have shown that a solution of the unperturbed system exists which satisfies all the commutation relations. If we further suppose that the system H0 is not degenerate, that is, that in the diagonal matrix W0, which results from H0 by the introduction of , , no two diagonal elements are equal, then we can find the motion of the perturbed system by the method of successive approximations discussed above. We place
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