Frank 0. Ellison a n d C. A. Hollingsworth University of Pinsburgh Pittsburgh, Pennsylvania 15260
The Probability Equals Zero Problem in Quantum Mechanics Or, How does an electron get from first to second to third without touching second?
In the study or teaching of elementary quantum chemistry, there is a particularly vexing question which often arises, and which manifests itself in several related ways. One primary example concerns the problem of a single particle of mass m .. . restricted to a one-dimensional hox of lrngth I, (i.e., U i.x < L ) hut otherwise suhiect to no iorres. Suppose that the panicle isin its first exciteditate (n = 2 ) , for which the eigenf&ction is $!x) = (2/L)'J2 sin(2rrxlL), and the probability density is p(x) = J.*(x)J.(x) = (21L) sin2(2rxlL) (1) This density function is plotted in Figure 1.According to the Born Postulate (I) as written in all textbook and literature sources of which we are aware, the probability for observing the particle between x and x dx (or, in the element dx located a t x ) is given by
+
dP = p(r)dx = J.*(x)J.(x)dx which, for eqn. (I), gives
(2)
dP = (21L) sin2(2~x/L)dr
(3)
We say then that if many measurements of the x-coordinate of the particle in this n = 2 state are performed, the incidence of various positions will conform to the prohability density given by eqn. (1)and displayed in the figure. Now the question: How does the particle pass from the left-half region to the righthalf region if it is never ohsewed between x = (112)L a n d x = (11211 d l ? Other versions of the same puzzle
+
11
How does a 2on electron eet from one lobe to the other if it is
in dx located at x = O? 3) How does an electron in the 2s orbital of the hydrogen atom pass from a region r > 2a (a = 0.5292 A) to a region r < 20 if it is never observedbetween r = 2a and r = 2a + dr? 41 How does an electron in an unperade molecular orbital, e.&, . o,, = I s A - Ise, pass from one end of the molecule to the other if it is never observed in an infinitesimalplane bisecting and perpendicular to the internuelear axis? These are questions which frequently bother students, and teachers, in the study of quantum mechanics. There are probably three popular ways used to dismiss the problem. The first wav. vou might sav. " . is s i m- ~-l vto "wave" the " problem away. The question, "How does a particle pass from this particular element of space to that particular element of space?" implies that we can precisely follow the motion, or orbit, of an individual particle just as astronomers follow the motion of individual planets in the solar system. But this is not so in quantum mechanics. T h e statistical description is the description of matter. Questions about the particle path as it goes from one region to another are outside of the domain It can also be said that the probability that a particle be at any point of zero volume or on any surface of zero thickness is zero, whether or not the point or surface is at a nude of the probability density. In other words, there is always zero probability in zero volume. When considering continuous distributions,we must consider only volume elements that may be arbitrarily small, but nonzero.
Robability density as function of xfor particle in on+dimensionalbox of length L in its first exited state ( n = 2).
of science in which wave-particle duality is an absolute reouirement. The second and a more satisfactory way has been presented very nicely by Powell (2) and by Szabo (3) in this Journal. They show that in the relativistic extension of quantum mechanics, there are no nodes in the probability densities. The third way is suggested by a footnote in the elementary text entitled "Chemical Bonding" .. bv. Audrev L. Companion ( 4 ) .Shr emphasiles, torexamplr, that the nr;dalplnnr separatme the 'Lo0 lobes, alluded in uurstinn number t l ) above, has gathema&d hut not physical&nificance; i.e., it has zero thickness. Thus, "it makes no sense to speak of finding a particle in that plane. In a slice of finite thickness. . .including that plane, there is a non-zero probability of finding the electron." In this article, we shall show specifically how to calculate that non-zero probability. Thus, we will show that one need not wave the problem away, nor seek recourse to relativistic quantum mechanics. We will need to carefullv reexamine the Born Prohabilitv Postulate and, in fact, suggest an extension which conforms more closelv with the mathematicallv established definition ofpr&ahilky. When this is accompiished, we will find acceptable answers to all of the above questions even in the simple non-relativistic quantum theory. We shall show that the probability for observing a particle in a given element of volume need not vanish even though the wave function J. vanishes somewhere in that element!
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Some Elementary Calculus Concerning Differentials Consider the functional relationship between the dependent variable y and the independent variable x Consider further an arbitrary increment in r which we shall call A*. I n the calculus. this increment of the independent variable is defined as the differential o f r and denoted by either the symbols dx or A x dx = A%
(5)
Volume 53, Number 12, December 1976 / 767
The actual change in the dependent variable y which results when x changes by Ax = dx is written as Ay
the Born Probability Postulate, eqn. (2). However, our remarks in the previous section should make i t clear that the actual probability A P that the random variable x will fall between x andx +dx, orx Ax, is only given approximately by eqn. (14) (cf., eqn. (9)).
+
On the other hand, the differential of the dependent variable y is defined by the formula dy = f'(x)dx = f(r)Ax
(7)
fYx) = 'lim (Aylk)
(8)
where &-0
Therefore, although dx = Ax by definition, we see that the differential dy defined by eqn. (7) is quite distinct from the "actual" increment Ay caused by an arbitrary increment Ax = dx. The distinction between dv and Av and the nondistinction between dx and Ax are often overlooked, forgotten, and seldom of importance in actual applications of the calculus to scientific problems. In practice, eqn. (7) is frequently integrated to find v = f(x). Also. in ~ractice.we often use the aDproximation to find the change in y caused hy a "very, very small change" in x. There are circumstances in whicb the error in this approximation can be significant. Consider the simple example
An Extended Born Probability Postulate
In order to deal more precisely with the quantum mechanical probability problem in elements of space a t whicb the probahility density is zero, we here propose an Extended Born Prohahility Postulate. We stipulate, as is done in al! basic treatices on probahility theory (5), that probahility be defined as the integral of the probability density function as is done ineqn. (11). I t follows from eqn. (11) that the exact increment A P that t fall in the interval x < t < x + Ax is given by
Assuming that p(t) is continuous and has continuous derivatives, consider the Taylor series expansion about the point t =x
"
A?'=
y=f(x)=~2-x+2
1
p(t) = Z 7p'k)(l)(t -x)k +.. . (16) a=, k. where p("(x) denotes the kth derivative of p(x). Eqns. (15) and (16) combine to give
" -p'"(x) 1
a=o k!
Jl"+A'(t - z)kdt + . . .
for which
=?P ( ~ ) ( x ) A x ~++. ~. (k + I)! k=o
and
-
Ay = f(x + Ax) - f(x)
+ Axz
(10) Suppose that we were asked to find Ay if x = 2 and Ax = 0.01. The approximate eqn. (9)gives Ay dy = 0.03; the exact eqn. (10) gives Ay = 00.301. Suppase that we werevaskedto find Ay if x = 112 and Ax = 0.01. The approximate eqn. (9) gives Ay dy = 0; the exact eqn. (10) gives Ay = 0.0001. = (Zx - 1)Ax
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Some Elementary Definitions in Probability Theory
In probability theory (5), the probability P ( t < x) that a random variable t be observed such that t < x is expressed hy the cumulative distribution function F(x)
r-
P(t b r ) = F ( x ) =
p(t)dt
(11)
Here, p(t) is called a probability density function, and since F(m) = 1, we must have
JI p(t)dt
=1
(12)
The lower limits on the integrals in eqns. (11) and (12) and the upper limit in eqn. (12) may be modified if the total range of t is more limited; e.g., in spherical polar coordinates, 0 $ r < -,O
