William Laurita
Kutztown Stote College Kutztown, Pennsylvania 19530
Heisenberg's uncertainty principle1 says that the positiou and momentum of a particle may not be precisely determined simultaneously and, in an equivalent statement, that the energy and time of passage of a particle past a particular point may not be precisely determined simultaneously. These restrictions, although surprising, are nevertheless consistent with the "perverse" nature of a particle in a box. Such a particle is confined t o one dimensional motion, and is unable t o penetrate confining walls. The classical description reveals only a particle bouncing from wall to wall with any momentum and any corresponding energy permitted, but the quantum mechanical picture is more restrictive. Quantum mechanically allowed momentum and energy values may be obtained in the usual manner or in the following more pictorial f a ~ h i o n . ~Solutions of the Schrodinger equation describing possible energy states of the system are found to be standing waves confined t o the box. The wavelengths of these are E., "Fnndamentals of Qusntom See far example: PERSICO, Mechanics." Prentiee-Hall. New York. 1950. See far example: CASTELLN, G. w., "Physical Chemistry," Addison-Wesley, New York, 1964,p. 408.
Demonstration of the Uncertainty Principle identical to the wave1engt.h~of a vibrating string, anchored t o and stretched between the walls. The occurrence of a node a t each wall requires that an integral number of '/* wave lengt,hs (X/2) will fit exactly between the walls; thus: where n is an integer and a is the distance between the walls. Eliminating X between eqn. (1) and the de Broglie relationship ( p = *h/X where p is momentum, nzu, and h is Plank's constant) leads t o the quantum mechanically correct allowed values of momentum: p, = +nh/2a
(2)
Since the energy of t,he particle in the box is entirely ldnetic, the allowed energy values consistent with eqn. ( 2 ) are: Momentum of the particle is restricted t o a series of allowed values separated'in magnitude by the minimum allowed momentum, h/2a. At the same time, the energy is restricted to the series of values given by eqn. (3). ~, The uncertainty principle is not a part of the classical description of the particle in a box, but it cannot b e
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avoided if the quantum mechanical description is talcen into account. The position of the particle is evidently given by the coordinate of the center of the box + a / 2 ; thus the uncertainty in position (Ax) is "a." It is apparent that Ax may be made as small as desired, but a natural perversity becomes apparent when an attempt is made t o measure the momentum of the same particle. Carrying with him the most sensitive velocity meter conceivable, an observer enters the box with the intention of measuring the momentum of the particle. During the necessary measurements of position and time the particle, exhibiting its particle nature, makes its presence detectable by exchanging momentum with the measuring device. Classically the exchange may be as slight as desired, but quantum mechanically the most gentle interaction will be accompanied by a t least the exchange of the smallest allowed change in momentum, namely h/2a. Since we cannot decide whether the particle, during the interaction, lost or gained momentum, the measurement will be in doubt by a t least +h/2a. That is, the momentum will be nh/2a + h/2a. I n an attempt t o find out whether the particle gained or lost momentum, we might try to measure the momentum of the velocity meter before and after the interaction, since this difference represents the momentum transferred. T o this end, we introduce anot,her velocit,y meter hut, by the same reasoning, this second velocity meter is able to measure t,he initial and firial moment,um of t,he first velocity meter only to the same degree of accuracy as is possible in the measurement of the particle, namely: p, =th/2a and p, + h/2a. The change in moment,um of the first velocity meter after the interaction is (pi - p,) + h/a. This large uncertainty in the difference precludes its use in any attempt to calculate a more precise value of t,he momentum of the particle. Evidently the best momentum measurement yields
tice
PEHSICO, E., "Fondamentala of Quantum llechm>ie," Prew Hall, New York, 1950, p. 184.
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Journal of Chemical Education
nh/2a + h/2a; and the uncertainty in momentum (Ap) is a t least: ~p
=
(n + l ) h / 2 a - ( n - l ) h / 2 a = h/a
(4)
When a is taken t o be Ax,
If more vigorous interactions are allowed, Ap must be larger than the value given by eqn. (4) and a statement of the uncertainty principle results: ApAs 2 h
(6)
I n agreement with the fact that momentum and energy of a free particle are unquantitized, Ap + 0 when Ax+ m.
An equivalent statement of the uncertainty principle is: AEAT 2 h. Where, for the particle in the box, AE is the uncertainty in energy of the particle and AT is the uncertainty in its time of passage past a particular point.3 From the uncertainty in momentum, the uncertainty in energy (eqn. (3)) is seen to be: AE = ( n
+ l)sh'/8maP- (n - 1 ) 2 h ~ / 8 m d= nh2/2maa (7)
AT is evidently the time required for the particle to travel the distance a, since in this time the particle is certain to pass every point in its travel from wall t o wall. From eqn. (2): v
=
nhl2am
(8)
where v is the velocity of the particle. By definition: v = a/AT
(9)
Equating eqns. (8) and (9): Multiplying eqns. (7) and (10) yields: AEAT = h
(11)
Again AE may be greater than the value given by eqn. (7) hence: AEAT 2 h
(12)
