ELEMENTS of the QUANTUM THEORY* II. THE DIFFERENTIAL EQUATION FOR A WAVE iNOTZON SAUL DUSHMAN Research Laboratory, General Electric Co., Schenectady, New York
to certain fundamental mathematical ideas, it has been considered advisable to precede the derivation of the S. T IS the concept of probability that characterizes the new point of view, and quantum mechanics repre- equation and the consideration of some of its applicasents a modification of classical mechanics which tions, by some remarks on these more purely matheenables us to predict the behavior of any system from matical aspects. the point of view of the principle of indeterminism. SOME FUNDAMENTAL DIFFERENTIAL EQUATIONS$ As a consequence of the application of this new meA differential equation is a quantitative expression chanics, certain conclusions have been deduced which of a hypothesis regarding the mechanism of a given are quite different from those expected on the basis of phenomenon. In general, what we measure as "pointerNewtonian mechanics. On the other hand, the new readings" are the results over a period of time, or over theory leads to discrete energy states for atomic sysa finite distance, of certain forces actmg between the tems by a much less artificial mode of derivation than bodies constituting the system under observation. was possible in the Bohr-Sommerfeld theory. FurtherThese forces vary with time and with changes in relative more, the calculation now leads to correct solutions in arrangement of the parts of the system, but from the those cases where the latter theory failed to give a satisintegration of the differential equation, whenever this factory answer, while it yields the same results as the is feasible, it is possible to deduce the total change in older theory wherever the latter did give results in agreethe magnitudes defining the state of the system under ment with observation. any given conditions or a t any instant of time in terms The actual mathematical technic of the new quanof the initial conditions. tum mechanics has evolved from what appeared a t first (1) Let us consider the very simple law governing the to be three different lines of attack. The 6rst one, motion of a body under the action of a constant force. originated by W. Heisenberg, is a purely symbolic type Let F denote the force, p the mass, and s the distance of mathematics and is quite unsuitable for elementary measured along the path of motion. presentation. The second one, developed by E. By definition, the force is equal to the rate of change Schroedinger (we shall use the symbol S. in referring to of momentum. Hence him), has enjoyed greater popularity, probably, as Eddington has suggested, "because it is the only one that is simple enough to be misunderstood." The third line of development is that presented by P. where v = velocity a t any instant. Dirac in his treatise on "The,Principles of Quantum F& velocities which are small compared with that Mechanics"; one which involves, like Heisenberg's of light, p is constant. Therefore, we can write treatment, a symbolism which is apt to repel any who are not "mathematically minded." In the presentation which the writer has attempted in the following sections, only the most essential aspects But velocity is defined as rate of change with time of the S. technic are considered, without regard to the of distance along the path. -That is, particular arguments by which S. actually developed his equation. The reader who desires to obtain an idea of the actual method of derivation used by S. will find this presented in various publications.+ Consequently, we obtain the result that However, since it is impossible to present even the simplest formulation of the S. theorv without recourse THE CONCEPTS OF QUANTUM MECHANICS
I
T h i s is the second of a series of articles presenting a more detailed and extended treatment of the subject matter covered in Dr. Dushman's contribution to the symposium on Modernizing the Course in General Chemistry conducted by the Division of Chemical Education at the eighty-eighthmeeting of the American Chemical Society, Cleveland, Ohio, September 12, 1934. The author reserves the right to publication in book form. Problems correlated with the subject matter of this article will be found on this month's Mathematical Problem Page (p. 294). t See list of "General References on Quantum Mechanics," on p. 2%.
Since F is constant, we obtain the differential tim of the second order, das F d=s d ~ - ; = O o r -dl'
where a is the "acceleration."
-
a =
0
i See list of "References on Mathematics," on p. 284.
equa-
We now proceed to find s as a function of 1. This will give us a "solution" of the differential equation. Integrating once, we have
so that
~ velocity a t t = 0. This is a where no = ( d ~ / d tis) the differential equution of the first order, and integration of this equation gives the result
defines the slope, while d defines the intercept on the y axis, since y = d for x = 0. (3) A very important type of second-order differential equation is that representing a simple h a n o n u motion. This motion is determined by the relation that the restoring force is proportional to the displacement. Hence, if F denotes the force,
where k is a positive constant and the negative sign indicates that for x positive, Facts in the direction of decreasing values of x. Thus equation (8) represents another variation of equation (4) in which F is proportional to x. This equation cannot be solved by integrating twice, as in the case of equations (4) and (6), for the first integration gives
where so is the value of s a t t = 0. Figure 5 shows plots of (1) F/p = d2s/dt2= a, the acceleration, (2) o = ds/dt, the velocity, and (3) s, the distance traversed in time t, all as functions of t .
and unless we know x as a function of t, it is impossible to carry out the integration indicated. The method actually adopted for solving such an equation as (8) is extremely interesting. If we write the equation in the form
and designate the ofierator* d/dt by D, we obtain the algebraic equation
-t FIGGRE 5.-GEOMETRICAL INTERPRETATIONOF A
SINGLEDIFFERENTIAL EQUATION AND
OF ITS
where u2 = k/p, to denote that k / $ i s positive. The left-hand side of equation (9b) may be represented as the product of two factors thus:
SOI.UTION
(D
Equation (5) is designated a general solution of equation (4), and i t will be observed that in the process of integration, huo constants are i n t r o d u p , one of which corresponds to the initial value of ds/dt and the other to the initial value of s. The differential equation (4) expresses the equation of motion of a body under the action of gravity, where Flp = g, the gravitational constant. (2) The special case of equation (1) in which F, and consequently, a,is equal to zero, gives rise to the second-order differential equation
+ iw) (D - iw)x
2
= 6, a constant.
and this in turn, to the result where d is a second integration constant. The last equation is that of a straight line, andIc
(10)
where i = In this manner we obtain the huo kt order differential equations, (D - iw)x = 0 iw)x = 0 and (D
(11) (12)
+
Equation (11) is identical with the equation
-
dl @
- iwr
dx-
.
= 0, that is x = zwdl,
* The designation "operator" may appear quite technical. However, its significance is merely that indicated by the literal interoretation. Thus.. va indicates that we are t o oerform the operation y X X y. and dy that we shall take the square root of y. Other operators are log and sin. The notation sin 8 states the nature of the operation to be performed on the angle 8. Hence, the expression @/dB) sin 0 indicates the following sequence of operations: (1) finding the sineof 8, and (2) determining the variation in sin 8 with change in 0. I t will be observed from this example that the order in which the operators are given is of fundamental importance. Obviously sin (d/&)O has no meaning. From the illustration in the text it will also be observed that operators may be treated as algebraic symbols, that is, we may multiply and (usually) divide by them, as if they were actual magmtude+subject of course to the rule that the operators are not always tommutolive, that is, the order in which they are expressed indicates a definite sequence of operations.
.
which leads to the first-order equation,
= 0
which yields the solution, where in = logarithm to base e, and c is an arbitrary constant. That is, 2
=
&d+c
=
C.id
where C = cC. Similarly, equation (12) yields .the solution,
of the constant w is evident. Putting w = 2rv, it is seen that for t = n/v (where n is an integer), sin ot = sin 2an = 0 , while cos wt = cos 27rn = 1. Hence, r denotes the frequency of the harmonic motion, and l / v = r , is the period of the oscillation. The particular values of the constants A and B depend upon the initial conditions. Thus, if x = 0 for t = 0 , A = 0, and the particular solution is
x = D.-bI
where D is another arbitrary constant. The complete integral of equation (8) or (9a) is, therefore, x = Cti.f
where w =
+ D&d
(13)
dR7;:
Equation (17) may be expressed in still another form, thus :
where
The right-hand side of equation (13) may be expressed in a more familiar form by making use of the series expansion formulas* for &'t and a-"
-
x = B sin wt.
(sin 6 cos wt
a
= sin-'(A/-)
z
- -.
That is,
+ cos 6 sin wt)
sin (wt
+ 6)
(19)
The constant corresponds to the amplitude of the harmonic motion, while 6 is designated as the phase angle. Equation (19) expresses in the form of a single harmonic function the same motion as that expressed by the sum of two harmonic functions in (17). That the equations are equivalent may be demonstrated readily by considering the case A = B. Then the two equations become
Therefore,
x = A(cos wt
+ sin wt)
(20)
and, since sin ?r/4 = Cos ?r/4 =J/G
..
and
In the first of these equations, x is expressed as the sum of two identical harmonic waves which are 90" out of phase, while in the second equation, xis expressed That is, as a single harmonic wave, havink an amplitude &I = ccos wt i sin wt times that of each of the waves in (20) and 45' out of * s-iu~ = cos wt - i sin wt (16) phase with each of these. The significance of equations (19) and (21) is illusIn consequence, equation (13) may be expressed in trated by the plots in Figure 6. Curve I corresponds to the form y = 3 cos wt; curve I1 to y = 4 sin wt, and w e I11 x = (C + D) cos wt + i(C - D) sin at t o y = 5 sin (wt 6) where 6 = 36.8'. I t will be ob= Aeoswt Bsinwt (17) served that each ordinate for curve I11 is the sum of thebrdinates, for the same value of t, for curves I and where 11. This illustrates a generalization known as the A - C + D Principle of Sufirposition, which is of extremely great B = i(C - D) importance in quantum mechanics. I t will be observed that in equation (18) the complex that is, quantities C and D differ only in the sign of the imagiiB A c = -A - iB, , D = (18) nary term, iB. Such expressions which d8er only in 2 the sign of the imaginary part are said to be conjugate. In quantum mechanics problems the form of solution It is customary to designate the conjugate of any given in (13) is much more convenient than that given complex quantity by a bar over the symbol or an in (17). But from the latter the physical significance asterisk. Hence, it is evident from equation (IS), * These expansions are derived from Maclaurin's series, which that C. E D and in turn is a special application of Taylor's theorem. =
2i sin wt
(15)
6
+
+
+
+
-
-
and
Also, since
=21CoI - 2 1 C I
it is observed that the product of a complex quantity by its conjugate is always real. The positive value of the square root of this product, that is, -/2 is known as the modulus of C or D and is usually written in the form I C l = / D l = -/2.
+
Also ( A 2 B 3 / 4 = ( C la = I D la is designated the norm of the complex quantity C or D.
as before, whiie equation (19) assumes the form
-. sin w t (194 The formulation of wave motions in terms of exponential expressions such as (13) or (13a) is of great importance in the consideration of many of the problems of quantum mechanics. (4) Lastly, we shall consider the second-order differential equation of the form x =
z* - may = 0 dZy
(22)
where m is a constant. Such a problem would arise if in the case discussed in equation (a), the force were assumed to be proportional to the displacement and in the same direction, that is, if the sign of k were positive instead of negative. As in the previous cases, it is to be expected that the general solution must contain two arbitrary constants, which are determined in any given case by the particular boundary conditions. Using the operator method, equation (22) becomes (Da - m9)y = 0 FIGUREB.-ILLUSTRATING m E SUPERPOSITION PRINCIPLE OP WAVEMOTIONS
that is,
In the case for which B is zero, that is, C real quantity,
whence follows the general solution,
z = C(&
=
D, a
+ ,-id)
= 2C cos wt =
A sin (wt
=
A cos wt
+ r/2) from (19).
since sin-' ( A / A ) = ~ / 2 . This leads to the result A = 2C, as is evident also from (18). It will be observed that = 2 1 C ( is the amplitude of the wave motion defQd by (13) or (17). ' Another point worth noting is that the quantities C and D in (13) actually define the phase angle, 6, by means of the relation
(D
- m)(D + m)y
y = AF
= 0,
+ Bs-mx
(23)
where A and B are two arbitrary cpnstants. Corresponding to the expon&tial expressions for the sine and cosine functions given in (14) and (15), we have the following relations: Px
+
.mx
-
6-2
=
2 cosh mx
(-mr
=
2 sinh kqx
(24)
where cosh and sinh denote the "hyperbolic" cosine and "hyperbolic" sine functions, respectively. Hence (23) may be written in the form, analogous to (I?), y = (A
+ B) cosh mx + (A - B) sinh mx
(25)
+
where ( A B ) and ( A - B ) -evidently may be replaced by two new constants, thus retaining the general form of solution with two arbitrary constants. We may, however, introduce the phase angle into Figure 7 shows the graphs for cosh x and sinh x, equation (13) and write and it will be observed that the two functions become x = C&(aI-61 + DOc-i(wt-6) (134 more and more nearly equal to each other for large positive values of x, while they diverge more and more This may be expressed in the same form as (I?), with increasingly negative values of x. with, however, the following conditions:
so that
We shall now consider further the significance attached to the expressions A&x and Be-&' in quantum mechanics problems. i sin ax [cf. equation (16)], Since ekx = cos ax
+
it is evident that this expression corresponds to a wave motion of wave-length A, such that a =
%/A,
the argument for deducing this conclusion being the same as that used previously in deriving the physical significance of w in equation (17).
From this point of view, the fact that the amfilitude of the wave is the sum of the moduli of the two complex quantities C and D in equation (13) receives an interesting physical interpretation, for it is evident that in the superposition of wave motions, the amplitude of the resultant wave is the sum of the amplitudes of the individual constituent wave motions. DIPFERENTIAL EQUATION FOR THE VIBRATION OF A STRING
It is a familiar fact that a string stretched between two fixed points, when made to vibrate, exhibits nodes and loops. The nature of the wave pattern is determined by the length of the string, L, in accordance with the relation
FIGURE ~.-=OT
OF
HYPERGEO~TRIC COSINE AND
SINEFUNCTIONS
Hence, the quantity A&%-4 denotes a traveling wave motion of wave-length, A = 2?r/cu, and of frequency v = w/2a. The velocity (phase velocity), u, is evidently given by
+
x, while If x is increased from the v&ue x0 to xo t is increased from the value 6 to do t, where t = m / w , the exponential function remains unaltered. Therefore, & ( a x - w l ) represents a wave motion for which the direction of proNgation i s from left tu right (corresponding to increasing positive values of x). Similarly, it may be shown that the expression c-i(ax+wl) represents a wave motion for which the direction of popagation is from right to left, corresponding to more negative values of %. It follows from these considerations that each of the expressions:
+
y = A(&=
+ ,i ).-id = = 2.4 EOS m ~ c - i w l
y = A(&
- .-im
and
),-id
= 2Ai sin
re~resentsa statiomrv wave which is the resultant of' two wave motion; identical in frequency, wavelength, and amplitude, but traveling in opposite directions.
where n is an integral value and A/2, which is one-half the wave-length, is the distance between two consecutive nodes. Similarly, a stretched membrane, organ pipe, or any other vibrating system exhibits definite wave patterns when vibrating, and in acoustics use is made of this fact to produce notes of definite frequencies or wavelengths. This observation that the vibrations of such systems are characterized, not by a continuously varying range of frequencies, but by a series of discrete values of these frequencies, is analogous to the spectroscopic obsemation that an atomic system may exist only in a series of states corresponding to a discontinuous series of values for its energy. It is this analogy, which is of a purely mathematical nature, that Schroedinger utilized in developing his "wave equation" for interpreting the behavior of electrons and atomic systems, and, therefore, before proceeding with the consideration of the wave equation, it is necessary to understand some aspects of the bathematical treatment of the problems of vibrating systems. The simplest of these problems is that of a stretched string, as for instance, the wire of a piano, or the string of a violin. It is a problem involving only one coordinate in space, which is the distance along the string, and also time as a-second variable, since the amplitude is a function of the time. Let us consider a stretched string, infinitely long, extended along the x-axis. Let T denote the tension along the string (that is, the force which maintains the string in a stretched condition), and p, the mass per unit length. To derive the differential equation for the motion of the strina, we consider the forces acting on an element oC leug& Ax, which is so short that it &ay be regarded as practically straight. When the string is drawn sideways (in the direction of the y-axis), these forces have comDonents along the two axes of coordinates. Let 0 denote the anglgbetween the element and the original position of the string (along the x-axis), as shown in Figure 8.
The mass of the element Ax is PAX, and its acceleration in the direction of the y-axis is b2y/bt2. Hence,
6 ----A_----
ei
9
in mathematical equations of indicating such a condition and will be used quite frequently in the following sections.) To solve the partial differential equation of the form
element the F force = pis( aacting z y / 3 t a ) Aon * . (26) the
I! I 11 U
The symbol a is used in this equation to indicate dx F , ~ ~. - I ~L L ~~ ~ ~ ~that A ~y ~is N a function ~ of both DERIVATION OF PARTIAL DIPx and t. Hence, by/ bt and FERENTIAL EQUATION FOR VIay,'bt2, ~ , t i a differenj BRATION OF A STRING. tial coeficients. The force F is balanced by the force arising from the differencebetween the tensions a t the two ends of the element Ax. That is, F
= T [ s i n (0 = T [ t a n (0
+ AO) - tan 01
But tan 0 = b y j a r
tan (8
where f(x) is a function of x only, and g(t) a function of t only. Hence,
+ AO) - sin 81
since for small values of 0, sin 0 and tan 0 are approximately equal.
and
the classical method of procedure is that known as solution by seeparation of the variables, which is applicable provided u is a constant. Let us assume
+ A@) - tan R
Y" = f%).c(t)
= &?(t)(dY/d+)
and j;
=
f(z) .&)
= f(x)(dPg/dt').
Sincef(x) and g(t) are each functions of only the corresponding variable, the partial differential coefficients may be replaced by ordinary differential coefficients. Substituting in equation (29) and dividing by f(x).g(t), we obtain the relation
= A tan 8
d ( t a n R).Az = -.AX bay = dr axP
Hence.
and, comparing this equation with (26), it follows that
that is.
This relation is usually written in the form
where primes denote differentiation with respect to x, and dots, differentiation with respect to t. Equation (28) is the partial differential equation which represents a wave motion along a stretched string. it is of interest to consider the &nificance of the coef6cient PIT. The constant P has the dimension of mass divided by length, that is d l , while T has the dimensions of a force, that is pl/t2. Thus, p/T has the dimension, t2/12, which is identical with that of l/u2 where u denotes a velocity. Therefore, we may replace p/T by a constant l/u2 which has the additional advantage of indicatingthat this coefficient corresponds to a fisitiue magnitude. (This is the customary method
Since the left-hand side is independent of t, while the right-hand side does not involve x, we may equate each term to an arbitrary constant, which we shall designate by -m2 (to indicate that-it corresponds to a .. negative magnitude). We thus obtain h o ordinary differentialequations,
and
These equations are similar to one of the types considered in a previous section, and therefore the general solutions may be given without further explanation. Expressed in the exponential form, these solutions are as follows:
where A, B, C,and D are four arbitrary constants, the values of which are determined by the particular physical conditions (known as Gsboundary,j and conditions). In this case, y = 0 for both x = 0 and x = L, where L = length of string (which is fastened a t both ends). Therefore, for = (x)
-0 - A + B
Hence, A = -B.
For x = L =
f(*)
- .-imL)
0 = A(&L
= 2Ai sin mL.
(36)
Since A is not equal to zero, sin mL = 0, which means that nx mL = n r ; m = -
(37)
L
where n = 1, 2, 3, etc. Thus the solution of (34) is
It will be observed that the differential equations (32) and (33) have physically significant solutions only for those discrete values of m which are defined by equation (37). These are known as characteristic values or eigenwalues, and the corresponding values of the functions f(x) and g(t), as defined in equation (41),are known as characteristic functions or eigenfunctions. Corresponding to each frequency v,, there will be a definite vibration of an amplitude, defined by the magnitude of the coefficient AC in equation (40), and there will exist n loops (regions of maximum amplitude) and (n - 1) nodes (regions of zero amplitude) between the ends of the wire. SCHROEDINGER EQUATION FOR ONE COORDINATE
(38)
The actual stimulus to the wave mechanics, as it is designated, developed by Schroedingerwas derived from certain theoretical speculations of Louis de Broglie regarding the analogy between the laws of geometrical g(6) = 0 = C + D; and C = -D. o ~ t i c sand those of classical dvnamics. As is well known, the laws of geometrical optics are As in the case of f(x), we thus derive the solution more and more valid, the shorter the wave-length of g(t) = 2Ci sin mul, light. For light waves comparable in length with those and inserting the value of m derived in (37), this be- of the object upon which they impinge, that is, for rays having a radius of curvature comparable with the wavecomes length, we must interpret the observations from the nr g(t) = 2Ci sin y . ut point of view of the undulatorv theory. This suggested " lo de Broglie the possibility chat ~ e k o n i a nda' mics denote the wave-length, and v the fie- is also an approximation which is valid for macroscopic Now if quency, it is evident that f(x) must become zero for systems, but not for atomic systems because the radius and so forth. Therefore, in of curvature of the electronic orbit is of the same magni.x = 0, = X/2, x = equation (38) we can write tude as the wave-length associated with the corpuscular n A. 2 L motion. L = -, that is h. = -, Let us consider, for instance, the Bohr orbit of the 2 n electron in the normal state of the hydrogen atom. which states that L must be an integral multiple of one- The quantum condition, introduced by Bohr for deterhalf the wave-length of the note emitted by the vibratmining the discrete energy states, is given by the ing string. relation Consequently Turning now to the solution in equation (35), we have, as initial condition, that is for t = 0,
f(x) = 2Ai sin
2x1.
A.
.
= 2Aisinrforz = L
Mwr = h/2+
*
(39)
nX. =-. 2
Also it follows, since u = v.L, that g(t) = 2Ci sin 2xv.l.
(40)
In these last two equations, v, designates the frequency of the harmonic or nth normal mode of vibration and A,, designates the corresponding wave-length. Thus the particular solution of equation (29) is of the form 2,s y = f(x).g(t) = -4A C sin - sin 2ru.l A,
.
(41)
where
L
= 2L/n = A/n,
.v
=
u / L = nu/2L = nv.
and X and v the fundamental wave-length and frequency, respe'ctively.
where v denotes the velocity of the electron in the circular orbit of radius, r. If there is a wave-length, X, associated with this motion, the circumference of the orbit must be an integral multiple of A, and for the simplest case this multiple will be unity. Hence,
-
2rr = h
Comparing these two equations, it follows that A = h/M"
which is the famous de Broglie relation. As has been mentioned in a previous section, this suggested association of wave motion with corpuscles in motion has been confirmed by the investigations on diffraction of electrons and protons. We must now consider the manner in which this observed dualism in the behavior of corpuscles has been utilized by Schroedinger in formulating his famous theory of wave mechanics.
Let E denote the total energy of a particle, T,the kinetic energy of the particle, and U, the potential energy a t any point in space. Then, in accordance with the law of conservation of energy and
for the present in the stationary mve patterns, we may postpone for a subsequent section any further consideration of the exact function by which to represent the variation in +(x, t) with t . Incorporating the relation for fi, assumed in equation ( 4 9 , into the differential equation (43), we derive the relations
3
For a single particle moving in a field of force, such as an electron in the hydrogen atom, T = '/& = E U
at
=
*.Td(.-2tivt)
-
and therefore and while, since A = h/(pv), Substituting the last two relations into (43), the result is the ordinary differential equation Smce in general U is a function of the coordinates of all the electrons with respect to the nucleus, A must vary from point to point in the field of force. Thus we may regard the motion of the electrons as governed by that of the associated de Broglie waves. Each electron in an atomic system follows the direction along which these waves are "refracted" in the field of force which is due to the simultaneous presence of all the particles. The Schroedinger (S.) equation is a partial differential equation, of the same nature as the differential equation for the vibration of a string, which indicates the manner in which the wave pattern, produced by the motion of an electron in a given field of force, varies with the space coordinates. We, therefore, introduce a function, +, analogous to the amplitude in a physically vibrating system, which defines the "amplitude" of the wave motion and is known as the S. function or space function. Its actual physical interpretation will be considered in a subsequent section. The simplest type of problem is that in which we are dealing with the motion of a single particle of mass, p, in a field of force for which the potential energy, U, is a function of only one co*I ordinate, x. Hence, $ is a function of x and t only, and we denote this by writing = +(x, t). Referring now to equation (29) which is valid for any vibrating system in which the motion occurs along only one axis of coordinate, we write the eqnation in the form,
+
In consequence of equation (42) this equation assumes the form
This is the form of Schroedinger's eqnation for one coordinate, and it will be observed that since U is a function of x, the equation cannot be solved as readily as the differential equations considered in the previous sections. The mathematical technic of solution thus depends upon the nature of the function U = U(x).
:.
MOTION OF ELECTRONS IN ABSENCE OF FIELD OF FORCE
In absence of a field, that is, for'the case U = 0, the S. eqnation assumes the form
Since the coefficientof 4 is a constant, this equation has the same form as equation (9a) or (32). Thus, the solution is
*
= A&%
+ Bc-imz
(47)
where orz = 8aapE/h2,and A and B are arbitrary constants. As emphasized in a previous section, this equation has physical significance only if a = 2s/A, that is, if
where vX takes the place of u, the velocity. As in the solution of equation (29) we assume that *(*,t ) = *(+-2a"' (44) where the function corresponding to g(t) in equation (30) is assumed to be of the exponential form with a negative sign in the exponent. Evidently this satisfies which is de Broglie's relation. What is the physical interpretation of the function the partial differential equation (43), and the choice of In a previous section it was shown that the ex? the single exponential function instead of the combinapression tion (2*'"' a-2"'"') as in equation (35) is merely con&z-*I) &d.I ventional. As a matter of fact, since we are interested
+
-*
represents a simple harmonic wave for which the direction of propagation is from left to right, wbiie the expression .-ikw+.r) = r-zniu~ . . S-~W
represents a simple harmonic wave motion from right to . ". left. Thus, if in equation (47), we wish to designate a wave motion propagated from left to right, we can indicate this by putting B = 0. To obtain an interpretation of + we proceed as follows: Multiplying by its complex conjugate, the result is
+
-
-
++, = AA&x
.
.-2&1
. .-
