ELEMENTS of the QUANTUM THEORY* IV. THE LINEAR HARHONIC OSCILLATOR
SAUL DUSHMAN Research Laboratorv, General Electric Co.. Schenectady, New York
T
HE motion of a particle of mass, & acted upon by a restoring force proportional to the displacement is defined by the diierential equation, considercd in Chapter 11, of the form,
and (1Wb)
b = 4r2pvo/h
Hence, we can replace (108a) by the equation where k is a constant. The general solution of this equation, as shown previously, has the form p = 27rv0, and vo denotes the frequency where w = q of vibration, while 6 is the phase angle. In order to describe the behavior of such a Smear harmonic oscillator from the point of view of wave mechanics, it is necessary, as a first step, to express the total energy, E, in the Hamiltonian form, that is, as a function H@,q) of the coordinates and corresponding momenta. Now, in this case, E, the total energy, is the sum of the kinetic energy, T = & ~ / d t ) ~ / 2and the potential energy
+ (a - ba&
d2+/dqE
=
0
(low
Let us introduce the new variable Since phvo has the same dimensions as f i in the expression A = h/for the de Broglie wave-length of a particle having kinetic energy, E, i t follows that d b has the dimensions of a reciprocal length, andtherefore z must be a dimensionless quapfify, that is, a pure number. Also from (110) it is seen that and d4/dq* = bd'/dxP.
..
Hence, the S. equation (10%) becom'es That is, E
=
Pa H(P,d = 2s
+-kp9 2
(1074
Now in order that q5 shall have a physical significance, where p = p(dx/dt) = momentum, and g is used in- it is necessary to obtain such solutions of equation (111) stead of z to designate the coijrdinate of position. In- as will permit qb to be finite and continuous for all values troducing the frequency of vibration, vo = */ ZT, of z ranging from - m to m . While in the case of a linear harmonic oscillator, in ordinary mechanics, we derive the relation the particle may vibrate only between the limits z = E - U = E - kp4/2 E - 2r2~oXpp' (1076) * d a / b (since a t these limits the value of U,the po-
+
-
and therefore the corresponding Schroedinger equation assumes the form * This is the fourth of a series of articles presenting a more de-
tailed 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 he found on this month's Mathematical Problem Page (p. 392).
tential energy, becomes equal to the total energy, E), there are no such limitations to the range of values of x in which q5 may be finite. For, according to the principle of indeterminism, there is no correlation between simultaneous values of position and velocity; the particle may be anywhere along the coordinate axis x. But it is obvious that $5, the probability of occurrence of the uarticle, as a function of x, must decrease continuously to zero as the values z = * m are approached. It is the introduction of these "boundary conditions"
that is essential for the solution of equation ( 1 1 1 ) . It will be observed that for values of z>-, equation ( 1 1 1 ) assumes the form d'+/dza
- xa+ = 0.
Writing this in the "operator" form, it is seen that this equation is equivalent to the two first-order equations, &/+ = +xdz. that is,
3 = n(n - 1 ) a . x -.+ dx'
(n - l ) ( n- 2 ) r
- V . ... . .
If we substitute these equations in ( 1 1 3 ) it follows, since this equation is valid for any value of z,that the coefficient of each power of x must vanish identically. This leads to the "recursion formula,"
The series will end with xu, if the coefficients a, + %, a. + &, etc., are each equal to zero, that is, if
d(ln+) = + d ( x P / 2 ) .
where in denotes logarithm to base e. Hence the complete integral is
+
= A,PVZ
+ A$+-%V2
where A, and A2 are two arbitrary constants. + cannot increase without limit for very large absolute values of x, it is evident that A1 = 0 . We thus assume as a solution of ( 1 1 1 ) the relation
F Since
+
=
0-1'/2.*(1)
(112)
where $ ( x ) is afunction of x, the nature of which must be determined from further considerations. From equation (112) it follows that
while dl4 -= drP
-+
-
Since e-'V2 is not equal to zero (e~ceptfor x = * m ) , we obtain, by substitution of the value for dZ+/dx2indicated by ( 1 1 1 ) and ( 1 1 2 ) , the equatiRn to be solved in the form
Let us express $ ( x ) in the form of a polynomial,
where n = 0 , 1, 2 , etc. It is evident that this deduction is analogous to that which was previously derived for the case of an electron in a "box." The fact that (a/b) may assume only those values corresponding to the series of odd integers 1, 3, 5, etc., indicates that the S. equation ( 1 1 1 ) can have solutions which are physically significant only for a discrete series of values of the energy, E. These values constitute the characteristic energy walues, corresponding to a series of energy levels of the linear harmonic oscillator, and the corresponding solutions for +, constitute a series of characteristic functions. Substituting for (a/b) in equation ( 1 0 9 ) it is found that the energy of the oscillator may assume any one of the series of discrete values defined by the relation where E. is therefore the ezgenwalue correspondmg to the eigenfunction 6.. :, Thus, while in the older quantum theory, the energy states of the linear harmonic oscillator were found to be integral values of hv, the S. solution leads to an addiFrom (117) it tional term in the energy of value hv& also follows that the frequency in the nth state is
This result is in better agreement with spectroscopic observations than that derived from the older theory. Furthermore, equation ( 1 1 7 ) shows that the energy of a linear oscillator does not vanish a t the absolute zero, but becomes equal to huo/2, thus solving definitely the problem of the existence of a "Nullpunktsenergie." We.shal1 now proceed with the determination of the corresponding eigenfunctions. From equations ( 1 1 5 ) and ( 1 1 6 ) it follows that
If + ( z ) , as defined by equation ( 1 1 2 ) , is to vanish for m , it follows that zne-2'/2 must also approach n(n - I)& = - [ 2 n - 2 ( n - 2)la, - r zero for large values of x. As x becomes infinite, x", as tends to become infinitely large, so well as @ , = 2 . 2 ~. 2 that the value of the product is indeterminate. and that If, however, we make n $finite, that is, let the series in ( 1 1 4 ) have no powers greater than xn, then it can be shown that x ' ' - ' * / ~ will always decrease to zero as z approaches * = . whence we obtain the expression for $-, From equation ( 1 1 4 ) we obtain the relations x =
where a, is an arbitrary constant. Now in investigating various types of functions, Hermite, a noted mathematician of the 19th century, discovered a set of functions known as "Hermitian polynomials," which are defined by the relation,* H&)
dq-xr) = (-1)Vt. d~"
rence of the particle in the region zz > z > X I and , since the oscillating particle must certainly be located in the range, x = + m, it is necessary that we introduce a constant N, such that,
(119)
where n = 0, 1, 2, etc. The expression defined by this relation is known as the nth Hermitian polynomial and is identical with (118) if we put a. = 2". The first five members of the series are readily determined. He(%) = (-l)0+=..-=' = 1. d(e-z2) HI(%) = ( - I ) @ ' . dr = 22.
where N. is known as the normalizingfactor for the nth eigenfunction +", and +JN, is designated the normalized eigenfunction. In order to illustrate the method used in determining the value of N, and also for the purpose of demonstrating the orthogonal properties of the eigenfunctions which satisfy the S. equation (111) it is necessary to introduce the values of certain definite integrals. In any standard table of integrals? it is shown that
It will be observed that in the first of these integrals, + changes sign with change of sign in z. The function e-"'.z2" + ' is therefore of the type designated as "odd," and resembles the trigonometric function sin z. Thus the area under the plot of the function for the range z = 0 to z = m is positive, while the area for the range z = 0 to z = - m is negative, and consequently the total area for the range - m to m is zero. That is
z"
From equation (119) the following relations may be derived:
The last relation makes it possible to derive the higher members of the series from the lower members. Combining equations (120) and (121), i t follows that and differentiating with respect to z, Hf,,+,(x) - 2H,(x) - 2zH',(x)
+ Hn,(x) = 0.
Hence, 2(n
+ l)H.(r)
- 2H,(x) - ZxH',(x)
t HV,(r) =
0
+
On the other hand, in the case of the second integral, the function s-"'.z" is always positive, whether z is positive or negative. The function is thus of the type designated as "even," and resembles in this respect the function cos z. Consequently
.
Q
That is, H",(z) - 2zH',,(r)
+ 2nH,(r)
=
0.
Evidently the last equation is identical with (113), if we put (alb) - 1 = 2n. It is therefore seen that the functions H . ( x ) satisfy the differential equation (113). Thus the eigenfunctions which satisfy the S. equation (111) are of the form We now have to consider the physical interpretation of these functions. As mentioned previously, +.6& or $P,dz (since in the present case is a real function) defines the probability of occurrence of the particle in the element of distance dz a t the point z. It follows
By means of this equation and the expressions for H,,(z) following we derive the values:
+"
that
-
is a measure of the probability of occur
* These functions and their properties are described more fully in the mathematical treatises of Couraut-Hilbert and Frank.-v. MISES.
where No, N,, Nz denote the corresponding normalizing factors. More generally, if we express in the form 9. = e-*'/2(a,z" fa , _ ~ x " - + ~ . . . . . .) -
+,
t For example, L. Silberstein's "Mathematical Tables," G. Bell and Sons, Ltd., London.
where a,, a,, - 2, etc, are the coefficients in the power, series of the nth Hermitian polynomial, it is evident that
where 2 denotes that the sum of the serLes of terms is to be taken for the range of values r f s = 2% 2n - 2, and so forth. Hence, r s is always even, and it follows that I, must always have a definite value,
+
The value of N, may be derived as follows. By definition, N,Z =
J--- e - ~ 1 . ~ . 2 d X ,
where H, is written instead of H,(x). (119) it follows that Nms= (-1)"
['-
From equation
.d x H". d r
a"-
= u, and consider the dx" result of differentiating the product H.u. Since
Now let us put
d (Ha") = H,u'
-
H . ~ ]- m
J:-
~.u'd.x
+S : - H ' , ~ ~ ~ .
That the first expression on the left vanishes is due to the fact that the presence of the term eCX' in all the derivatives of this function makes u (and consequently uH,,) vanish a t x = * m . Combining the last relation with equation (120) it follows that
LS-"
H,u'dx = -2nJ--~H,_,udx.
That is,
*
E m d"(C"3 .-. d~
=-2nJ:= (-1)s2sn(n - 1)
d"-'(e-q ~ " - 1 . dr-,
J--- H " - * . d n - = ( e - X = )
dx,
and by continuing the same procedure, we finally deduce the relation ( - I ) - N X P = (-1)" 2 W )
- $---
r-~aH3mdx
where m is not equal to n. Assume n = m p. Then we can proceed to calculate I,, as in the case m = n, and we shall obtain the result
+
I,,
=
J---+n+n+mdz
=K
1:-
..
J--- Ho.-~'dz.
That is,
Figure 20-shows plots of the normalized eigenfunctions, that is, of @,/N,, for the values n = 0, 1,2, 3,
=J . - ~ ~ H , + ~ H ~ ~ ~
H,~-='dx.
where K is a factor involving 2 b n d the product n(n 1). . . . (n - m). Now from equation (119) it follows that K-~..-z'd% =
J-".,
)
and 4. It will be observed that for n = 0, 2, and 4, the curves are symmetrical with respect to a change from positive to negative values of x (even functions), while for n = 1 and 3, the curves are antisymmetric with respect to such a change in sign of x. Excluding the nodes at x = * m , the number of these is equal to n in all cases. This arises from the fact that H.(x) is a polynomial in x", and may therefore be expressed in the form (x - a ) ( % - a%). . . . .(x - a,). Consequently H,(x) must have n roots, that is, n points along the axis of x at which H.(x) = 0. We shall now consider the integral I., = J---+.+,dr
+ H',u,
we obtain by integration the relation =0 =
FIGURE 20.-ErOENRUNCTIONS FOR LOWER ENERGY STATES oa LINEAROSCILLATOR
----
J-
d ~ ( e - ~d'X) dzn
and it is evident that the differential coefficient d*(e-x')/dx9 will consist of a series of terms of the form a,z'e-x' where r has the maximum value p, and the signs of the coefficients 611 be alternately plus and minus. If p is odd, it is evident from equation (122b) that the integral I,, will be equal to zero. If p is even, it is not self-evident that the same result will be valid, and a more elaborate proof is required (see Appendix I). However, the result may be demonstrated by an actual calculation for two of the simpler cases. Thus, let us consider the product W a . From the values of the corresponding Hermitian polynomials, it is seen that
= -1 6 . 3 6
2
2
- 0,
Similarly,
From equation (126) it follows that
Adt =
Adx 2r"" V'2n 1 -
+
'
22
That is, the value of +,,& alternateshetween positive Introducing equation (127), according to which and negative values, as may he seen by inspecting the curves shown in Figure 20, with the result that I,, = 0. This deduction may be stated in the generalized form that the solutwns of the Schroedinger equation form an orthogonal set of eigenfunctions. As mentioned in a previous section this conclusion is valid for the solutions obtained for the S. equation in all cases, and it is found that A = 2u. may be demonstrated to be a logical consequence of a very fundamental mathematical theorem. In a sub- and that therefore sequent section it will he shown that this property of dx the solutions of the S. equation is of extremely great Pdx = (128) r d 2 n + 1 - r2 imoortance in dealinrc with a number of oroblems which arise in quantum mechanics. This result shows thatpincreases from l / ( x d - ) We shall now consider further the interpretation, on at = O to at x = zo = ' d G o n the other the basis of wave mechanics, of.thebehavior of a particle which, according to classical mechanics, executes hand, S.'S solution leads to finite values of 6" even for values of 1x1 > Izoj. That is, qhm2 has a definite value harmonic vibrations with frequency vo. for d l r e s of X (or ¶) which are forbidda by chsical From equation (110) it is observed that in terms of . . mechanzcs. q, the actial displacement, the dimensionless variable In the region outside the classical limits, the potenis defined by the relation, tial energy, U,is greater than the total energy, E, and x = nd = i ~uqhence the de Brodie wave-length is imaeinarv. From In classical mechanics, the maximum amplitude of the S. equation ( i l l ) oscillation is given by
-
as is evident from equation (107b), since for this value of q. E - U = 0. Substituting for E from equation (119) and using equation (1l o ) , the corresponding maximum value of z is found to be Hence, the motion of the oscillating particle, from the classical mechanics point of view, would he given by the relation x =
z / n + l n + l
sin(Zuv.$).
showing that z oscillates between
-v'%Zx
(126)
+ &2fl and
From this it is possible to calculate the probability Pdz that the particle will be found between z and z dz.* Since P d z is proportional to the element of time, dt, required for the particle to pass from z to z dz,
+
+
P d x = Adt
where A is a constant which satisfies the condition that
*The following remarks constitute an amplification of the calculation given by Condon and Morse, p. 51.
- .
Hence for z = zo, l* f n2d d2$/dz2 = 0, which shows that this constitutes a point of inflection. For values of 1x1 > 1x01, 4 decreases continuously without exhibiting any nodes and, as mentioned previously, for very large values of z, varies as' e - ~ ' / ~that , is. decreases rapidly without increase in z. Plotting the values of @,/N2. against z, a series of curves such as those shown in Figure 21 is obtained, and designated by A . (The significance of 1/Nn2consists in the fact that it makes the area under each of these curves equal to 1/2. Sincesimilar curves, symmetrical with respect to the @axis, may be plotted for negative values of z, the tofal area under the curve giving P / N 2 as a function of all values of z is equal to 1.) The curves designated by B in Figure 21 give the probability distribution function as calculated from equation (128), that is, according to classical mechanics. The difference in results derived from the two points of view is due, as emphasized in the remarks on the "tunnelling effect," to the complete lack of association in quantum mechanics between position and velocity. That is, we have here another illustration of the application of the Principle of Indeterminism. That the probability for the occurrence of the oscillating particle in the region 1x1 > 1 d G fis con-
+
rod, then
Similarly, in the case of two coordinates of position, the coordinates zu,yo of the center of gravity are given by the relations:
F I G ~~~.-PROBABIL~TY E OP OCCURRENCE OF OSCILLATPARTICLE ACCORDINGTO QUANTUM MECHANICS (CURVESA) AND ACCORDING TO CLASSICALTHEORY ING
(CURVESB)
siderable, is readily shown by calculating the value of (2/N3 [-@dx in the case of the first characteristic function, 40 = r-"/2, corresponding to E = huo/2. Here we find (shce xu = I),
