ELEMENTS of the QUANTUM THEORY* IX. THE HELZUX ATOH. PART II.
THE VARIATIONAL XETHOD
SAUL DUSHMAN Research Laboratory, The General Electric Co.,Schenectady, New York
of the method were made bv Lamancre and Hamilton. HE variational (or Ritz) method has been used The principle formulated b$ thLlat& is regarded as extensively in classical physics, especially in the one of the most fundamental generalizations in physics, field of dynamics, where its application is at least and may be stated in the following form. If we compare any dynamical path (that is, one as old as d'tllembert's principle of virtual displacewhich proceeds in accordance with the laws of dyment5.t However, the most important applications namics) with naried Fths, which have the same termini This is the twelfth of a series of articles presenting a more (in the configuration space) and which are described in detailed and extended treatment of the subject matter covered in Dr. Dushman's contribution t o the symposium a n Modemizing the same time, then the time integral of the function the Course in General Chemistry conducted by the Division of L = L(qi, &, 1), known as the Lagrangian or kinetic Chemical llducation ar the eighry-eighth m r e t i n ~of the American potential function, is an extremum (that is, either a Chemical Society, Cleveland. Ohio. Seprember 12. 1934. The maximum or minimum) for the dynamical path. nurhor recwves the rirhr to ~uhlicarionin book form This priuriple i s d ~ ~ c uin&AI. C..\ V ~ ~ s 1 ~ . n ' s " T h ~ d y n a m i c s The function L is defined by the relation, of particles and of rigid, elxtic and fluid hodics," B. G. Teuhner. Lcipzig, 1912, p. 63. This is an invaluable reference on classical L=T-V (357) MATHEMATICAL FORMULATION
T
mechanics. The following references also contain a discussion of classical mechanics: L. P a u l w o AND E. B. WILSON, "IntTodnction t o quantum mechanics," McGraw-Hill Book Company.Inc.,NewYork, 1935. J. H. JEANS. "Theoretical mechanics." Ginn and Company, Boston, i907. W. V. HOUSTON,"Principles of mathematical physics," McGraw-Hill Book Company, Inc., New Yark, 1934. S. D u s w m ~Taylor's , "Treatise on physical chemistry," Vol. 2. p. 1264, D. Van Nostrand Co.. New York; 1930.
where T denotes the kinetic, and V the potential energy. If ql, qa. . . . .q, denote the f generalized coijrdinates of the system, then T = T(qi, gi), and V = V(qi), that is, the potential energy is a function of the f coordinates only. In the case of a consemati~e system, E =T V = a constant
+
We can express Hamilton's principle in the variational form
Let us consider first the S. equation in the operator form,
where the symbol S is used to indicate any arbitrary (small) variation, and in any treatise on the calculus of variation* it is shown that the conditions which must be satisfied in order that the integral in (358) shall be an extremum are given by the f equations, m e for each generalized co6rdinute, of the form
where or2 = 8?yZp/hz, and H, known as the Hamiltonian, designates the operator in the brackets. Multiplying both sides by 5 and integrating over the configuration space, the result is
(359)t
where ¶; = dg;/dt, and it will he observed that b L / a p is the partial differential coefficient of L with respect to the particular variable q;. The last equation is known as Euler's equation in the calculus of variation, and also as Legrange's equation in dvnamics, ~twill be observed that in eauation (358) the variation afiplies to a function, L, and not to a co6rdinate (as in ordinary calculus), and it is because of this generality of the criterion (359) that the latter has proved so extremely useful in solving many problems in classical dynamics. I t was therefore not unreasonable to expect that the method of variations would find equally important application in connection with the problems of quantum mechanics. In fact, the most striking feature about the whole history of the development of theoretical physics during the past two hundred years has been the continuous extension of concepts. The experimental discoveries always seem to demand revolutionary changes in ideas and yet, when in the course of time these observations are arranged in a logical frame, it is found that many of the concepts held previously require only a slight modification, or extension, in order to be able to reconcile them with the new facts. In the case of quantum mechanics it is readily shown that the S. equation is essentially the Euler differential equation which must be satisfied in order that a certain definite integral (which corresponds to the total energy) shall be a minimum.$ Hence, instead of attempting to solve the S. equation in a particularly di5cult case, it may be much more feasible to introduce one or more arbitrary parameters into the expression for the corresponding variational integral. By investigating the effect of variations in the values of these parameters, on the value of the integral, it is then possible to determine a minimum value for the latter, that is, a minimum value for the eigenvalue which corresponds to the given S. equation for a particular energy state.
-
' The simplest treatmcot with which the writer is familiar is in the smnll volume by W. E. BWKLY."Introduction ro the calculus of variations." Harvard I.'niv. I'resr. Camhridge, 1920. t In most cases it will he found that the integral which satisfies the dilTerentizl equation (3.59) is a minimum. However, the nrwf of this conclusion would involve nmch more tedious mathematics. f This was pointed out b y E. SCHROEDINOER in his first paper on "Quantization as an eigenvalue problem," Ann. Physik, 141, 79, 361 (1926).
J
J
J
where d~ is the element of volume in this space. Hence, S;H+dr E =
(361)
For normalized functions, the - denominator in (361) is equal to unity, and E =
J-
+H&T, where
+ is an
eigenfunction for the given S. equation. Suppose, however, that we do not know the exact form of which should he used to solve equation (360). We may then make a more or less shrewd guess as to the form of the function. Let designate this new function, and let
+
I
=
JTH*d"
and N =SF$&
1
(362)
Then, as shown in the following section, the minimum value of I, which we shall denote by M, is always greater (more positive) than the true eigenvalue for the corresponding S. equation. That is,
For $ = +, the correct eigenfunction, the expression on the left-hand side of the last equation is equal to zero. To demonstrate the validity of (363) we proceed as follows:** If Eo = M / N = minimum, then
Hence 6M
- E06N = 0
*' A proof of this theorem was first given by C. Ec-T, P h y ~ Rm.. 36, 878 (1930). ,For the proof given in the text the writer
is indebted to Dr. F. Seitz.
That is,
This may also be written in the form
or J(G)(H
- Eo)+dr + f T(H - E36+dr
=0
(364)
Since W is arbitrary, let us replace it by W * Hence, since $4 = 1 J. IZ9 a real function, we must replace 63 by -is+, in order that the product (-is91 (is*) shall be real. Consequently, (364) becomes, J
J
Dividing both terms by i, and comparing the resulting equation with (364), it follows that the latter can he valid only if each integral mnishes identica1ly.t Hence, (H - E 3 + = 0
But +$.+ azx
d
=
a+ '
+a+
z ( z )- ( G )
Hence, designating dzdydz by dn,
J + H G= ~
5 J {(2)'+ ($)I+
(2)'jdv
subject to the condition,
J
where pz = momentum along x-axis, = p i , and similarly for py and A. Hence we can write the expression for the total energy in the Hamiltonian form, E
=
H(P'I
where T = kinetic energy, and V = potential energy, then in terms of rectangular coordinates,
(365)
which shows that Eo is an eigenvalue corresponding to the function J.. These considerations show that the S. equation is the Euler condition which must be satisfied in order that the integral I in (362) shall be a minimum. The latter may also be expressed in another form, in which it has been used quite frequently by writers on quantum mechanics. Referring to equation (360), which is merely the S. equation, let us consider the case in which 4 as well as V are functions of the Cartesian coordinates, x, y, z, and 4 is a real function. (This limitation to real functions and rectangular coordinates simplifies the mathematics hut does not detract from the general validity of the conclusions.) Then,
I
minimum. where a denotes x.. -. v. or Z. Thus the's. equaltion is the differential equation which must be satisfied in order that the internal - in equation (366) shall he a minimum. The form of this integral indicates a method by which it may be derived formally from the Hamiltonian ex. pression for in classical dynamics. If we write the expression for E in the form =
+~+~]dxdyda
.
..
E = H b , P I ) = T k i . &)
+V(d
(367)
under these conditions the element of volume is
dMnby
a,
=
a.&, -~. . . . .&, -.
where .\/;I is known as the "discriminant,"$ and therefore the integral to he miniized is
+2tixdydz = 1.
*This part of the proof is taken from notes on lectures delivered by P. A. M. Dirac at Princeton gniversity in 1931. 6 , su& that +o+ = +@+is known as gnmjt& t of this or self-adjoint. Evidently H or ( H - E) is an type.
