xi is asymptotically equivalent to multiplication by Xbxi/bs, where S SOME REMARKS CONCERNING SCHRODINGER'S WA VE EQ UA TION

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1 (6 eks (VO + vl +... ) ~(1) VOL. 19, 1933 MA THEMA TICS: G. D. BIRKHOFF 339 is necessary due to the weakness that only the approximate frequency distribution ml is known except at the value I = '/2 n. If the data have not been reduced to a zero mean, the shape of the periodogram is not affected but the whole is raised by twice the mean value. The modification of the probability formulae for such data is a routine task. SOME REMARKS CONCERNING SCHRODINGER'S WA VE EQ UA TION By GEORGE D. BIRKHOFF DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY Communicated January 30, 1933 In the present note I propose to approach the wave equation of Schrodinger by a method which, although closely related to methods used by de Broglie and Brillouin,l Schrodinger2 and Dirac,3 is distinct from these and has the advantage of fixing the position of the wave equation from a purely mathematical point of view: namely, the wave equation is the simplest form of linear partial differential equation involving a parameter, for which the classical process determining asymptotic series solutions gives a "multiplier equation" identical with the Hamilton-Jacobi equation, while the Cauchy characteristics of the multiplier equation (along which the asymptotic "wave packet" is always propagated) then become the dynamical trajectories of the corresponding Hamiltonian system. Let L(4,6, X) = 0 be any linear (ordinary or partial) differential equation in if with n - 1 independent variables xi,... x,, and involving a parameter X. This parameter is to be thought of as large in absolute value. In the case which interests us certain of the coefficients of i6 and its derivatives in the above equation become large as X increases. Under these circumstances it has been found in many cases that the operation of differentiation of ii or of its derivatives with respect to any variable xi is asymptotically equivalent to multiplication by Xbxi/bs, where S is a definite function of xi,... x,. Thence one is led to an asymptotic series solution, where vo, vi,... do not involve X. It is with certain general facts concerning this classical, formal process and its relation to the Schrodinger wave equation that the present note deals. If we introduce the modified differential operators

2 340 MA THEMA TICS: G. D. BIRKHOFF PROC. N. A. S..a'i'=la((= l,... n),4 b(i] 1 ( axi X axi the original equation in 4t' may be written as L(s,a A atx', a' 12,. ;Xi,... X.; 0) = (2) ax, X X where L may be expanded in the form of an infinite series in 1/X, Li L _- Lo + L1+...(2') in the case which we are considering. Here Lo may be called the "principal part" of Lo, and may be more explicitly written as follows Lo(+) = + i axj + E ij axj?xj (3) where we assume tij = tji, sijk = tikj = tjki = {jik = = tkjii etc. If now the formal series (1) be substituted in the equation (2), and the coefficients of e)s/xk on the left be equated to 0 for k = 0, 1,..., the first equation determines S and may be written where p (Xl' x"' bxl ' dx") ' (4) p +Et bs + a8sbs +- (5) i )xj ij axs Jxj is a polynomial of degree n in as/axl,... as/a)x, if n is the order of the equation in 4. Conversely any polynomial P determines a corresponding Lo. We shall term (4) the "multiplier equation" for L(4+) = 0. This is a differential equation of the first order in S which does not contain S and is in general non-linear. When we proceed with the further similar equations for k = 1, 2,. we find that the second equation determines vo in (1), the third determines v1, etc. We shall be interested primarily in the equation for vo. Let us observe first that we have, to terms in 1/X, _6114 e),s (-vi _ 8OV+ as)vl axj axi 'X ax xi e.(ax+--ivo +-.aavl)

3 VOL. 19, 1933 MA THEMA TICS: G. D. BIRKHOFF 341 X/Fs as 1 F?bS 1 1 1?'S 'S 'a'ei --IIs s+]vo+---av1 j,etc. bx,bxj ^-e\l ix, X bx,j L x bxj x o X (x,ixj / Here we have used an obvious operational notation. On substituting these expressions in equation (2), the terms in 1/X which involve vt disappear because of the multiplier equation (4), and there remains the following equation for k = 1, bp?-vo +1/ 2P S = Of (6) i byi xi 2 T =yiobyjx(6bxj where we have written yi = bs/ixi for i = 1,... n. Suppose now that we take any complete solution S(xi,... x"; cl,... c) + cn of the multiplier equation. Then x1, * Xn, yl*, Yx may be regarded as independent except for the equation of condition P = 0. If we write dx= P (i = 1,... n) (7) dr ayi and let 4) denote the coefficient of vo in (6), the equation (6) becomes dvo-fd +4vo = 0, whence =Ke. (6') d,r Hence the equation (6) specifies how v0 varies along any curve x, =x,(x), (i = 1,... n), in xi,... xx space representing a solution of (7). But we have also for any i along such a curve dy, a2s dxj _ 2S ap dtr j bxi bxj dr j bxi bxj byjs Furthermore, from the multiplier equation (4) we infer (in xi,... xx space) ')p (2S so that we deduce ap+ E Pa a j byiaxio _of dy _ P (i=1,...n). (8) d-r aixj The equations (7), (8) are the 2n ordinary differential equations in xi,...x, yi,... yn of Hamiltonian type which define the Cauchy characteristics of the multiplier equation (4), provided we restrict attention to those solutions (involving 2n - 1 arbitrary constants) for which P = 0. The general significance of the above results is clear: Consider a regular dimensions so that, for the given function transversal surface T of n - 1 S, the corresponding characteristics intersect T in one and the same sense

4 342 MA THEMA TICS: G. D. BIRKHOFF PROC. N. A. S. throughout. The value of vo may then be given arbitrarily on the transversal surface T, and is then determined elsewhere by integrating along these Cauchy characteristics. For k # 0, the equations analogous to (6') are of tbe form d-+ bvk + 4k = 0 (k = 1, 2, 3,*) (6) di where bk depends linearly and homogeneously upon vo, vi,... vk- 1. Hence each coefficient Vk is determined up to a constant multiple of vo by its value on T. If in particular vo, vi,... are taken as 0 in T except throughout some region V of T, and as vanishing along the boundary of this region, together with their partial derivatives, these functions may be taken to be O except within the corresponding tube of characteristics (on which they are determined to the extent indicated above), and vanishing similarly on the boundary of the tube. Such a region V changes position as the "time" r changes, and the series solution thus gives rise to what we shall term an "asymptotic wave packet." Let us call any function vo which satisfies the linear partial differential equation (6) an "amplitude function," associated with the corresponding "phase function" S. If vo is any assigned amplitude function of S whatsoever, then the integral I = j vdxldx2... dx,, (9) does not vary with r. Here V(r) is the region into which the arbitrary region V at T = 0 is carried along the Cauchy characteristic in "time" r. Conversely if this integral is independent of r for every V, vo will be an amplitude function of S. To prove this result we observe that the integral I may be written J vo(xi, x,,)jdxi... dxn where the transformation from V to VIr is xi = fi(xi,... x,), (i =...), and J is the functional determinant (xl,... x,)/b(xy,... x.). Hence the derivative of I at T = 0 (when xi = xi) is But for I vod vo dxl.. dx,. i axi ayi dt Ar small we have Xi = Xi + Ar (=,...n), ayi and, in consequence,

5 VOL. 19, 1933 MA THEMA TICS: G. D. BIRKHOFF P I.2S ax. byi i (y,6 yy6xix/ whence dj _2 62P 62S dr T0o ie a yi + Eyi a j bx; axj Hence the integral written above vanishes because of the condition (6) upon vo. Consequently di/d-r = 0 for r = 0, and likewise for any r, so that I is necessarily constant. Conversely if the integral I is independent of r for all regions V, it is clear that vo must be an amplitude function. This completes our proof of the above italicized statement. Suppose now that we take n + 1 variables t, xl,.. x", with e _ s +s H(l }x x -a" so that the "multiplier equation" takes the form of the Hamilton-Jacobi partial differential equation witb energy H. The corresponding "principal linear equation" Lo(ik) = 0 is then the usual Schrodinger wave equation 2 ri b /,2r r H l =.X)0 (10) 2 aa H(l XX h bxl * h bx,) provided we take X = 27ri/h. Furthermore the corresponding Hamiltonian equations are together with dt d asa dt ' dro dxi 6H dyi = H (=1,... n) (1 dr y ' dr a-xi so that as/at = const., and r = t + const. The Schr6dinger wave equation is therefore merely the principal equation which has the usual Hamilton-Jacobi partial differential equation H as its multiplier equation. Evidently t = to is a transversal surface T in this special case and the n + 1 dimensional element of volume is dtdxl.. dx,,. Furthermore we may write S = c,t + S*(xi,... X,,; C1,... cn-1) + cn+1 since y, = 6S/I t appears only linearly. Here the Cauchy characteristics become the dynamical trajectories of the corresponding Hamiltonian system (11).

6 344 MA THEMA TICS: G. D. BIRKHOFF PROC. N. A.- S. If we write to a first approximation (vo real,,6 exsvo, y6e)svo X pure imaginary), we see that along any part of an asymptotic wave packet, the integral f Pdtdxi... dx. reduces to fvo2dtdxl... dx, which by our general result remains constant. It follows then that f Wx1...dx. also remains constant. To establish this we begin with the n + 1 dimensional element of volume, taking limits to and t, for t with t - to = At, and then let At tend to zero. Hence we arrive at the following result: In the special case of the Schr6dinger wave equation in 0, the "asymptotic wave packets" follow the corresponding dynamical trajectories, whik the squared.amplitude integral J...dx... dx,, remains constant over any part of the packet, at least to the order of terms in 1/X.5 In conclusion, it is desirable to note what happens un'der any change of independent variables in the general case xi = f1(xi,... x") (i = 1,... n), It is to be observed first of all that because of the identities a"]u 6j. 6[]u ae2l u [xk 2xl a[2u 1 k a[1lu xj bxj bxkbxl k bx EXiXjbXk t. ax, = E xi axj, bxibxj ki Eg the components of L =Lo + LI + do not remain individually invariant in the equation (1), and in particular the principle part Lo is not carried over into the new principal part by the ordinary rules. In fact the coefficients t in the principal part transform by the rules valid for the attached Hamilton.-Jacobi equation. Hence SchrAdinger's wave equation in the form (10) is only maintained (in general) under a linear transformation of the independent variables. This fact indicates that any coordinate system from which we start is to be regarded as a privileged absolute system of reference for the Schr6dinger wave equation, up to an arbitrary linear transformation. 1 L. de Broglie and L. Brillouin, Selected Papers in Wave Mechanics, (translation, London, 1928). 2 E. Schrodinger, Collected Papers on Wave Mechanics, 1-30 (translation, London, 1928). 8 P. L. M. Dirac, Quantum Mechanics, (Oxford, 1930). 4 See my paper, Trans. Am. Math. Soc., 9, (1908). b Dirac asserts incorrectly (loc. cit., p. 121) that the amplitude is constant along a dynamical trajectory. He overlooks the terms in the second partial derivatives in the equation corresponding to (6) in the special case.