The Dreams That Stuff is Made of
Page 57
We shall now show that the Hamiltonian (30) leads to the correct expressions for Einstein’s A’s and B’s. We must first modify slightly the analysis of § 5 so as to apply to the case when the system has a large number of discrete stationary states instead of a continuous range. Instead of equation (21) we shall now haveih a (α’) = Σa" V (α’α”) a α′′).
If the system is initially in the state α0, we must take the initial value of a (α′) to be δα‘α 0 , which is now correctly normalised. This gives for a first approximation
which leads to
corresponding to (22). If, as before, we transform to the variables W, γ1, γ2 ... γu−1, we obtain (when γ’ ≠ γ0) a (W′γ′) = v (W’, γ ’; W0 , γ 0) [1 − e i (W’−W0 )t/h ] / (W’ − W0).
The probability of the system being in a state for which each γ k equals is ΣW’ |a (W’γ’)|2 . If the stationary states lie close together and if the time t is not too great, we can replace this sum by the integral (ΔW)−1 ∫ |a (W’γ’ )|2 d W’, where ΔW is the separation between the energy levels. Evaluating this integral as before, we obtain for the probability per unit time of a transition to a state for which each γk =
(32)
In applying this result we can take the γ’s to be any set of variables that are independent of the total proper energy W and that together with W define a stationary state.
We now return to the problem defined by the Hamiltonian (30) and consider an absorption process in which the atom jumps from the state J0 to the state J’ with the absorption of a light-quantum from state r. We take the variables γ ’ to be the variables J’ of the atom together with variables that define the direction of motion and state of polarisation of the absorbed quantum, but not its energy. The matrix element v (W0 , γ′; W0 , γ0 ) is now
where Ẋr ( J0 J’) is the ordinary (J0J′) matrix element of Ẋr . Hence from (32) the probability per unit time of the absorption process is
To obtain the probability for the process when the light-quantum comes from any direction in a solid angle dω, we must multiply this expression by the number of possible directions for the light-quantum in the solid angle dω, which is dω σr ΔW/2π h. This gives
with the help of (28). Hence the probability coefficient for the absorption process is . |Ẋr (J0J’)|2, in agreement with the usual value for Einstein’s absorption coefficient in the matrix mechanics. The agreement for the emission coefficients may be verified in the same manner.
The present theory, since it gives a proper account of spontaneous emission, must presumably give the effect of radiation reaction on the emitting system, and enable one to calculate the natural breadths of spectral lines, if one can overcome the mathematical difficulties involved in the general solution of the wave problem corresponding to the Hamiltonian (30). Also the theory enables one to understand how it comes about that there is no violation of the law of the conservation of energy when, say, a photo-electron is emitted from an atom under the action of extremely weak incident radiation. The energy of interaction of the atom and the radiation is a q-number that does not commute with the first integrals of the motion of the atom alone or with the intensity of the radiation. Thus one cannot specify this energy by a c-number at the same time that one specifies the stationary state of the atom and the intensity of the radiation by c-numbers. In particular, one cannot say that the interaction energy tends to zero as the intensity of the incident radiation tends to zero. There is thus always an unspecifiable amount of interaction energy which can supply the energy for the photo-electron.
I would like to express my thanks to Prof. Niels Bohr for his interest in this work and for much friendly discussion about it.
SUMMARY.
The problem is treated of an assembly of similar systems satisfying the Einstein-Bose statistical mechanics, which interact with another different system, a Hamiltonian function being obtained to describe the motion. The theory is applied to the interaction of an assembly of light-quanta with an ordinary atom, and it is shown that it gives Einstein’s laws for the emission and absorption of radiation.
The interaction of an atom with electromagnetic waves is then considered, and it is shown that if one takes the energies and phases of the waves to be q-numbers satisfying the proper quantum conditions instead of c-numbers, the Hamiltonian function takes the same form as in the light-quantum treatment. The theory leads to the correct expressions for Einstein’s A’s and B’s.
THE LAGRANGIAN IN QUANTUM MECHANICS
BY
PAUL A.M. DIRAC
Physikalische Zeitschrift der Sowjetunion, Band 3 Heft 1 (1933)
Quantum mechanics was built up on a foundation of analogy with the Hamiltonian theory of classical mechanics. This is because the classical notion of canonical coordinates and momenta was found to be one with a very simple quantum analogue, as a result of which the whole of the classical Hamiltonian theory, which is just a structure built up on this notion, could be taken over in all its details into quantum mechanics.
Now there is an alternative formulation for classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinates and velocities instead of coordinates and momenta. The two formulations are, of course, closely related, but there are reasons for believing that the Lagrangian one is the more fundamental.
In the first place the Lagrangian method allows one to collect together all the equations of motion and express them as the stationary property of a certain action function. (This action function is just the time-integral of the Lagrangian). There is no corresponding action principle in terms of the coordinates and momenta of the Hamiltonian theory. Secondly the Lagrangian method can easily be expressed relativistically, on account of the action function being a relativistic invariant; while the Hamiltonian method is essentially nonrelativistic in form, since it marks out a particular time variable as the canonical conjugate of the Hamiltonian function.
For these reasons it would seem desirable to take up the question of what corresponds in the quantum theory to the Lagrangian method of the classical theory. A little consideration shows, however, that one cannot expect to be able to take over the classical Lagrangian equations in any very direct way. These equations involve partial derivatives of the Lagrangian with respect to the coordinates and velocities and no meaning can be given to such derivatives in quantum mechanics. The only differentiation process that can be carried out with respect to the dynamical variables of quantum mechanics is that of forming Poisson brackets and this process leads to the Hamiltonian theory.kl
We must therefore seek our quantum Lagrangian theory in an indirect way. We must try to take over the ideas of the classical Lagrangian theory, not the equations of the classical Lagrangian theory.
CONTACT TRANSFORMATIONS.
Lagrangian theory is closely connected with the theory of contact transformations. We shall therefore begin with a discussion of the analogy between classical and quantum contact transformations. Let the two sets of variables be pr, qr and Pr, Qr, (r = 1, 2 ... n) and suppose the q’s and Q’s to be all independent, so that any function of the dynamical variables can be expressed in terms of them. It is well known that in the classical theory the transformation equations for this case can be put in the form
(1)
where S is some function of the q’s and Q’s.
In the quantum theory we may take a representation in which the q’s are diagonal, and a second representation in which the Q’s are diagonal. There will be a transformation function (qʹ| Qʹ) connecting the two representations. We shall now show that this transformation function is the quantum analogue of etS/h.
If α is any function of the dynamical variables in the quantum theory, it will have a “mixed” representative (q’|α|Q’), which may be defined in terms of either of the usual representatives (q‘|α|q"), (Q’|α|Q") by(q’|α|Q") = ∫ (q’|α|q")dq"(q"|Q’) = ∫ (q’|Q") d Q"(Q"|α|Q’).
From the first of these definitions we obtain
(2)
(3)
and from the second
(4)
(5)
Note the difference in sign in (3) and (5).
Equations (2) and (4) may be generalised as follows. Let f (q) be any function of the q’s and g (Q) any function of the Q’s. Then
Further, if fk(q) and gk(Q), (k = 1, 2..., m) denote two sets of functions of the q’s and Q’s respectively,(q’ |Σk f k (q) gk (Q)| Q’) = Σk fk (q’) gk (Q’) · (q’|Q’).
Thus if α is any function of the dynamical variables and we suppose it to be expressed as a function α(qQ) of the q’s and Q’s in a “well-ordered” way, that is, so that it consists of a sum of terms of the form f(q)g (Q), we shall have
(6)
This is a rather remarkable equation, giving us a connection between α(qQ), which is a function of operators, and α(q’Q’), which is a function of numerical variables.
Let us apply this result for α = pr . Putting
(7)
where U is a new function of the q’’s and Q’’s we get from (3)
By comparing this with (6) we obtain
as an equation between operators or dynamical variables, which holds provided ∂ U/∂ qr is well-ordered. Similarly, by applying the result (6) for α = Pr and using (5), we get
provided ∂U/∂Qr is well-ordered. These equations are of the same form as (1) and show that the U defined by (7) is the analogue of the classical function S, which is what we had to prove.
Incidentally, we have obtained another theorem at the same time, namely that equations (1) hold also in the quantum theory provided the right-hand sides are suitably interpreted, the variables being treated classically for the purpose of the differentiations and the derivatives being then well-ordered. This theorem has been previously proved by Jordan by a different method.km
THE LAGRANGIAN AND THE ACTION PRINCIPLE.
The equations of motion of the classical theory cause the dynamical variables to vary in such a way that their values qt, pt at any time t are connected with their values qT , pT . at any other time T by a contact transformation, which may be put into the form (1) with q, p = qt , pt; Q, P = qT , pT and S equal to the time integral of the Lagrangian over the range: T to t. In the quantum theory the qt, pt will still be connected with the qT, pT by a contact transformation and there will be a transformation function (qt |qT) connecting the two representations in which the qt and the qT are diagonal respectively. The work of the preceding section now shows that
(8)
where L is the Lagrangian. If we take T to differ only infinitely little from t, we get the result
(9)
The transformation functions in (8) and (9) are very fundamental things in the quantum theory and it is satisfactory to find that they have their classical analogues, expressible simply in terms of the Lagrangian. We have here the natural extension of the well-known result that the phase of the wave function corresponds to Hamilton’s principle function in classical theory. The analogy (9) suggests that we ought to consider the classical Lagrangian, not as a function of the coordinates and velocities, but rather as a function of the coordinates at time t and the coordinates at time t + dt.
For simplicity in the further discussion in this section we shall take the case of a single degree of freedom, although the argument applies also to the general case. We shall use the notation
so that A(tT) is the classical analogue of (qt |qT).
Suppose we divide up the time interval T → t into a large number of small sections T → t1, t1 → t2, . . . , tm−1 → tm, tm → t by the introduction of a sequence of intermediate times t1, t2, . . . tm. Then
(10)
Now in the quantum theory we have
(11)
where qk denotes q at the intermediate time tk, (k = 1, 2... m). Equation (11) at first sight does not seem to correspond properly to equation (10), since on the right-hand side of (11) we must integrate after doing the multiplication while on the right-hand side of (10) there is no integration.
Let us examine this discrepancy by seeing what becomes of (11) when we regard t as extremely small. From the results (8) and (9) we see that the integrand in (11) must be of the form ei F/h where F is a function of qT , q1, q2 . . . qm, qt which remains finite as h tends to zero. Let us now picture one of the intermediate q’s, say qk, as varying continuously while the others are fixed. Owing to the smallness of h, we shall then in general have F | h varying extremely rapidly. This means that ei F/h will vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the domain of integration of qk is thus that for which a comparatively large variation in qk produces only a very small variation in F. This part is the neighbourhood of a point for which F is stationary with respect to small variations in qk.
We can apply this argument to each of the variables of integration in the right-hand side of (11) and obtain the result that the only important part in the domain of integration is that for which F is stationary for small variations in all the intermediate q’s. But, by applying (8) to each of the small time sections, we see that F has for its classical analogue
which is just the action function which classical mechanics requires to be stationary for small variations in all the intermediate q’s. This shows the way in which equation (11) goes over into classical results when h becomes extremely small.
We now return to the general case when h is not small. We see that, for comparison with the quantum theory, equation (10) must be interpreted in the following way. Each of the quantities A must be considered as a function of the q’s at the two times to which it refers. The right-hand side is then a function, not only of qT and qt, but also of q1, q2, . . . qm, and in order to get from it a function of qT and qt only, which we can equate to the left-hand side, we must substitute for q1, q2 . . . qm their values given by the action principle. This process of substitution for the intermediate q’s then corrésponds to the process of integration over all values of these q’s in (11).
Equation (11) contains the quantum analogue of the action principle, as may be seen more explicitly from the following argument. From equation (11) we can extract the statement (a rather trivial one) that, if we take specified values for qT and qt, then the importance of our considering any set of values for the intermediate q’s is determined by the importance of this set of values in the integration on the righthand side of (11). If we now make h tend to zero, this statement goes over into the classical statement that, if we take specified values for qT and qt, then the importance of our considering any set of values for the intermediate q’s is zero unless these values make the action function stationary. This statement is one way of formulating the classical action principle.
APPLICATION TO FIELD DYNAMICS.
We may treat the problem of a vibrating medium in the classical theory by Lagrangian methods which form a natural generalisation of those for particles. We choose as our coordinates suitable field quantities or potentials. Each coordinate is then a function of the four space-time variables x, y, z, t, corresponding to the fact that in particle theory it is a function of just the one variable t. Thus the one independent variable t of particle theory is to be generalised to four independent variables x, y, z, t.kn
We introduce at each point of space-time a Lagrangian density, which must be a function of the coordinates and their first derivatives with respect to x, y, z and t, corresponding to the Lagrangian in particle theory being a function of coordinates and velocities. The integral of the Lagrangian density over any (four-dimensional) region of space-time must then be stationary for all small variations of the coordinates inside the region, provided the coordinates on the boundary remain invariant.
It is now easy to see what the quantum analogue of all this must be. If S denotes the integral of the classical Lagrangian density over a particular region of space-time, we should expect there to be a quantum analogue of eiS/h corresponding
to the (qt |qT) of particle theory. This (qt|qT) is a function of the values of the coordinates at the ends of the time interval to which it refers and so we should expect the quantum analogue of eiS/h to be a function (really a functional) of the values of the coordinates on the boundary of the space-time region. This quantum analogue will be a sort of, “generalized transformation function”. It cannot in general be interpreted, like (qt |qT), as giving a transformation between one set of dynamical variables and another, but it is a four-dimensional generalization of (qt|qT) in the following sense.
Corresponding to the composition law for (qt|qT)
(12)
the generalized transformation function (g.t.f.) will have the following composition law. Take a given region of space-time and divide it up into two parts. Then the g.t.f. for the whole region will equal the product of the g.t.f.’s for the two parts, integrated over all values for the coordinates on the common boundary of the two parts.