Robert A. Harris and Herbert L. Strauss University of California Berkeley, California 94720
I
Paradoxes from the Uncertainty Principle
The uncertainty principle is a fundamental part of the foundations of quantum mechanics and provides vital information on what can or cannot be measured ( I ) . At first glance, the statement of the uncertainty relations seems simple enough. Let us consider measuring two quantities A and B, which are represented by the operators A and B. The uncertainty in the measurement of the average value of A is given by c~~ = ( A A Z )= (A2- ( ( A ) ) 2 ) = Jdu+*[A2- ((A))2]$ (1) where J. is the wave function of the system, the integration is over the volume of the system and the brackets, (),denote an average value (or a matrix element). An equation identical to eqn. (1) also holds for B, of course. The uncertainty principle applies to measurements of both A and B simultaneously and states that when [A,B] = (AB - BA) = ih
(2)
then ~ r ~ =~(AA2) c e (aR2) ~ 2 1/4h2 (3) T h e proof of eqn. (3) may be found in most texts (2) and rests on the usual boundary conditions that J. is integrable and normalized, that
(A) = Jdu+'A$ is well-defined and that
(4)
( A ) = Sdu (A*V)+ (5) An onerator for which eons. (4) and (5) are true simultaneouky is called ''~ermitik." The boundary conditions seem reasonable and straightforward in their application, hut their careless application to eqns. (2) and (3) leads to a host of well-known paradoxes. We shall consider two of these, one dealing with the position, x, and its conjugate momentum, p, and the second with an angle variahle, +, and its conjugate momentum, L.
-ih
a] ax
= ih
(6)
( I x d ) = Sdu $ p * ( x ~ P X ) + ~dx = I S ~ U + ~ ~ X+ ~h ,I [AS,d ~ $ ~ * ~= +0 ~ l(7)
and
~ J i p=
(8)
(9)
the eigenvalue equation and the Hermitian property of eqns. (4) and (5). Combining eqns. (6), ( 3 , and (8) we have 0 = ih
(10)
an impossible result. Many different ways of stating that the arguments leading up to eqn. (10) are faulty have been given in the literature (3). If the off-diagonalmatrix elements of eqn. ( 6 ) are evaluated, no difficulties arise even for eigenfunctions of p or x. I
374 1 Journal of Chemical Education
the "free" particle wave function where N is a normalizing constant. This eigenfunction says that the probability of finding the particle between x and x dx is
+
probability at x
=
1
J.,*(x)$,(x)dx = 7dx
IN1
(13)
a uniform probability everywhere. Such a situation is physically impossible, and it is little wonder that the integrals of eqns. (7) and (8) are badly defined (the reader may wish to write them out in detail!).' We could have also "derived" eqn. (10) by trying to apply eqn. ( 6 )to an eigenfunction of position. In this case it is the momentum that is undefined, and the wave function implies the likely occurrence of an arbitrary large momentum and thus of the energy. This is again impossible. If we restrict application of the uncertainty principle to physically realizable situations and reasonable wave functions then the steps of eqn. (7) do not follow and the paradox disappears. Angle-Angular Momentum For an angular variable eqn. (2) reads
-1
[$,I,] = [Q,-ih J
am
= ih
(14)
Here difficulties appear even when we do not try to apply eqn. (14) to an eigenfunction of @ or L. We have (15)
4
to an eigenfunction of p.
(ih) = Jdu$,*ih$, = i h S d ~ + , * +=~ih In these equations we have used
The eigenfunction of momentum is the solution of eqn. (9) which is
om%,,2 t 1h2
Posiilon, Momentum The x,p paradox comes from applying the relation [X,PI = [ x ,
These involve more sophisticated discussions of boundary conditions or alternate proofs or interpretations of eqn. (3). However, we can obtain the essence of the argument, and a correct answer, by arguing on physical grounds. Equation (3) for our case says
Many perfectly admissible wave functions have a small value of n ~ 2and , then eqn. (15) implies a correspondingly large am2. However, the entire range of is only from 0 to Zs, so that the uncertainty in + is surely no more than Z~T,and eqn. (15) presents a contradiction! The difficulty is intimately connected to the consequences of periodic boundary conditions, and its existence was recognized from the earliest days of quantum mechanics ( 4 ) . Equation (15) can be applied to t,wo different physical situations. 1)If @ describes position on an infinite helix, then the to +- and the uncertainty in values of @ can range from @ can he arbitrary large. In this case, eqn. (15) is just like eqn. (11) for a Cartesian coordinate and momentum, and the additional contradiction disappears. 2) If describes position on acircle, then @ and @ ZsN, where N is any integer, specify the same physical point, and functions of must have the same value a t + that they have a t + 2 s N . Once again many suggestions for resolving the contradiction have been made. (3).We consider one simple resolution. Applying eqn. (15) to an eigenfunction of L, we have
+
--
+
+
+
d $L = m h + ~ dm
L+L = --ih-
Equation (16) has the solution
+
(16)
+L
1 = =e'"t
..
rn = 0, *I, * 2 , .
(17)
an eigenfunction which is perfectly admissible just because of the conditions on Then
Lzr
am2 '
m.
e-"'?P
dd
-
(d)?
1 -earn'
4%
(18)
Now we take the operator as 4=$t2rN
and actual calculation gives
(a4p =
&
-
+
(19)
]I2
Literature Cited
1 1%
t 2 r r ~ ( 2 ~ ) 2( 2 - ~ ) %2r . -
1
w2(k+ NZ- NQ + N - N , ) We have chosen to define + with N in evaluating the m2 term X -+2rrN'.2s [(2;)2
=
we can take N = N' and take eqn. (15) to refer to the different values of o measured the same number of times around the circle. There is one more situation in which the uncertainty principle can lead to anomalous results. Consider resolving the x, p difficulties by applying periodic boundary conditions to the free particle wave function of eqn. (12). The anomaly that arises can he dealt with in the same manner as for the +, L case. We leave the details to the reader.
(20)
and to define @withN' in evaluating the (6).Equation (20) allows Am to be arbitrarily large with the appropriate choice of N and N'. The indeterminacy of N and N' arises because on any observation of 4 we don't know how many times around the circle the system has gone. We have no contradiction with a small q 2provided we remember this indeterminacy. On the other hand, when UL is large, q2is small, and
(I) neisenherg, W.. 2.Pkysik 43.17211927). 121 Strsurr, H.L., "Quantum Mechanw." P~entiee.Hall, Englewood Cliffs. N.J.. I%% Levina, 1. N.,"Quantum Chemistry." 2nd Ed.. Allyn & Bemn. Bostun, 1914;Afkins. P.W.. "Molex~lsrQuantum Mechanics: Ciarendon Press, Oxford, 1970 this bmk mentions the problem of angular varisblen i n a Footnoto. Dawdov, A. 8.."Quantum Mechanics." tec H a m D, (tronslatarl Addinon.Wealey, Reeding, Mass. 1965: D. R. Bates. D. R.. "Quantum Theory: Vol I, Academic. N.Y.. 1961; Davydov and Bates write eqn. I151 without further comment. 131 Carruthers P.. and Neb. M. M. Rru. Mod. Phys., 10.411 11968) this referencegives a comprehensive review of many alternate ways of considering the difficulties we arr discussing; Dsvidson, E. R., J. Cham. Phys, 42,1461 (1965):Yaria,R., J.Chom Phys., 44.425 (1966). It is a pleanun to acknowledge Davidrun'r paper which started eonsiderable discussion of there paradoxes among many chemists. (41 Jordan, P.. Z. Physik, 44, l 119271. In disc-ingangular variables, Jordanae*nowldges conversationswith Direr We msysgeulatofhat t h e e paradore caused mncemeven the"!
Volume 55, Number 6, June 1978 / 375
