2-D Wave Packets and the Heisenberg Uncertainty Principle Giles Henderson Eastern Illinois University, Charleston, IL 61920 Undereraduate students t v ~ i c a l l vencounter t h e ~ e i s e n b &uncertainty ~ principl"Lin thek second or third vear. This principle places a nonclassical restriction on the precision i f coniuga'te variables: momentum and position or energy and time. Many authors ( I d ) present the student with a measurement scenario based on the Heisenberg microscope (6) in which the experimental determination of one conjugate variable alters the value of the other such that the precision of the respective variables are inversely proportional to each other:
where Ap and k are the uncertainties or precision in momentum and position respectively, ti is Planck's constant divided by 2n, and AE and At are the uncertainties or precision in energy and time, respectively. For example, refinements i n the experimental measurement of the instantaneous position of a moving particle results in ever ereater ~erturbationson its momentum. Althoueh these restrictiks are inconsequential and too small touobserve for macroscopic objects that are successfully described by Newtonian mechanics, they are of great importance for small atomic particles in which classical mechanics fail and quantum mechanics are employed. This discussion may leave the student with the impression that the uncertainty principle is merely an experimental limitation and that we may subscribe to the Einstein philosophv that even thoug6 the conjugate variables may not be-simultaneously measured with unlimited precision, a particle does in facthave a precise momentum-and position at a specified instant (that perhaps is only known by God). Other authors (7-13) emphasize the statistical nature of wave mechanics and illustrate the need to construct a waue packet comprised of a superposition of energy eigenfunctins to obtain a localized description of free particles. The time-dependent position probability for some specified velocity or momentum distribution is then characterized by a Fourier analysis of the wave packet. This approach underscores the intrinsic nature of the uncertainty principle and reveals that it is a direct consequence of the particle-wave duality inherent to quantum descriptions of moving particles. Exposure to both viewpoints may provide the student with a satisfying conceptual understanding of the Heisenberg uncertainty principle. However, the second or third year undergraduate may not be suficientlv " emerienced with Fourier transforms to carry out or even appreciate the analytical formalisms described above. In this studv we wish to ex~lore the outcome of directly superimposing matter waves without the inter-
vention of any calculus-based Fourier analysis. Since this approach involves replacing the Fourier transform inteerals with direct summations of manv free-particle wave Functions, it is an ideal task for the d c r o ~ o & ~ u t eMorer. over. the readv availabilitv of sophisticated .. era~hics . utilines enables i l i u s t r a t ~ nthe ~ pn,l;lrm for part~clesw ~ t htwo spatial demees offrwdwn, rather than the olcen c~tedbut physicall~unusualparticles in one dimension. Of course the ultimate graphical representation of position probability for particles with three spatial degrees of freedom would require something like a three-dimensional hologram in which the intensity of the image was scaled to the quantum probability (or some other independent variable). Computational Methods Let us consider a pulsed beam of n particles that has been collimated in an r v plane and restricted to a ~articular beam width by a n ipLrture or slit. We wish have a sufticientlv large sample of articles to realize a meaningful statistical ikerpretation of our results (n t 100) an2 yet not so large that the computational task exceeds practical time limits for a typical laboratory microcomputer. Using n = 251, we find that an IBM PS Model 30 286 equipped with an Intel 80287 math coprocessor requires approximately two hours to compute the 2911 points used to plot the probability surface in Figure 5a. The BoxMuller (14) method may be employed to generate a Gaussian random distribution of momenta to the n particles. Each moving particle is described by a deBroglie traveling wave yi, which is a solution to the time-dependent, two-dimensional Schrodinger equation for a free particle:
td
where m is the mass of the particle; r and $ are circular polar coordinates, and t is the time. The eigenfunctions of eq 1may be written for the ith particle in the form:
Pulsed Particle B e a m
A
III
Editor's Note: Aprevious paper by this author, "Howa Photon is Created or Absorbed [J. Chem. Educ. 1979, 56, 6316351 is being reissued in an interactive version by JCE: Software. The fioures have been animated so the reader can see the progreeif the phenomena that are described. See the Abstract on page 978 fordetails.
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Figure 1. A pulsed beam of 250 particles,which has been collimated in the x,y plane and restricted to a particular beam width by a slit.
X Figure 2.The real component of eq 3 evaluated at t= 0.
Figure 4.The real and imaginary components of they = 0 cross section of Figure (3a).
Figure 3. The time evolution of a two-dimensional wave packet is compared to the classical position of a panicle pulse: (a)t = V/4A and later at (b) t = VAOh. where J,(kir) a n d Yn(kir)are the well-known nth-order Bessel functions of the first and second kind, respectively (15);ki = 2a/h, is the wavenumber of the ith particle; h is the deBroglie wavelength; r (= [x2+ Cy - yl)211iz) and $ are circular polar coordinates; x and y are Cartesian spatial coordinates for which the origin is located a t the center of the slit (see Fig. 1);y is the y-cwrdinate of the ith particle and is restricted to f'/z the magnitude of the aperture or slit width. These time-dependent wave functions are traveling waves that exhibit phase velocities given by
k3
"'G
Figure 5. The spatial distribution of the wave packet reveals an increase in the ycomponent of the momenta as a result of our attempt to decrease thcuncertainty in the y-position by narrowing the slit width from (a)21 to (b) I h. We may set n = 0 for a collimated beam of particles undergoing linear motion (16).By separating the real and imaginary components of eq 2 we may write t h e n = 0 wavefunctions in the form: Volume 70 Number 12 December 1993
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Figure 7. The momenta spectra, a and b, depict the Gaussian distributions of particle momenta employed in the wave packets illustrated in Figures 8a and 8b, respectively.
Figure 6. Contour plots of the Figure 5 position probability amplitudes.
v; = [ ~ ~ ( k cos j r )(
~ t+)Yo(kir)sin (qt)]
+ i [-J0(kir)sin (qt) + Yo(kir)eos (qt)] = Rei + Ini
(3)
The zero-order Bessel functions may be evaluated to high precision using polynomial approximations (17). It is evident from Figure 2 that these waves propagate from their origin with circular symmetry and that the probability amplitude is not localized for a single particle moving with a precise momentum. Aposition probability distribution of the particle pulse a t a specified time after passing our slit is then calculated from the complex square of the superposition wave packet:
These auantum orobabilitv surfaces are then lotted as a three-dimensioLa1 projection a t a specified kalue o f t usina Axum maohicsutilities (18).In some instances these qu&tum de&ptions are directly compared with classical description by overlaying a dot pattern in which each dot locates the x,y coordinate of one of the n particles a t the specified time as calculated classically from the particles initial position and momentum. This method is initiated by generating a data file that contains a set of n (kj ,yJ values. The momenta of the n particles are represented by 974
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a Gaussian distribution of ki with a specified mean and standard deviation. The initial y; coordinates of the particles are assigned random values ranging over the slit < v j < a12. Both uniform and normal (Gaussaoerture: ian) random n u h e r s may be generated using standard functions available in the Axum soRware utilities. A verv short and simple computer code may be written in B&, Fortran, Pascal, or any other convenient language to evaluate a P(x,y) surface over a n appropriate x,y domain in accord with eqs 3 and 4. An outline of our algorithm follows: Dimension the k and y arrays for n values. Open data files for the k and y data. Open an output file for the gy,P surface. Read and store then (k,yi) values from the data fde. Defme a value oft. Defme a value of the aperture. Set up nestled Imps to sweep the g y grid. Initialize the Z Re; = Z hi= 0. Set up internal summation loop for then vi functions. Evaluate the J,,and Y, Bessel functions. Evaluate Rei and hicomponents of yrj using eq 3. Evaluate X Ret and Z Inj. Evaluate P(x,y)= using eq 4. Write +y,P ta the output file. The output generated by this routine is then imported as a data file for the Axum graphics utility, which carries out the three-dimensional projection using a hidden-line algorithm and direction cosine transformations to rotate the probability surface to a user's specified viewing angle. Results and Discussion Figure 3a compares the classical positions with the quantum position probability function for a collimated
(a)
+*+AkCONTOURS . I 5ko =
(b)
+*$ CONTOURS Ak = .075ko
Figure 8. The quantum probability surfaces reveal an increase in the x component of the particles' spatial distribution as a result of the
Figure 9. Contour plots of the Figure 8 position probability amplitudes.
beam of 250 particles moving in the +x direction with a Gaussian distribution of velocities and momenta. The corresponding momentum spertrum for this distr~butionis presented as a histomam in Fimre 7a. The slit aperture is identified by dashed lines in the x,y-quantum plane. The x-axis is annotated in units of h = hlmu and the c&ulations were-camed out at a t corresponding to F t = 4h, where ij and h are the mean velocity and deBmglie wavelength of the 250 particles. The time evolutions of the classical equations of motion and the time-dependent quantum probability fnnction are revealed in_Figure 3b in which the time has been advanced to t = 10 h fi. As we examine the classical x,y planes in Figure 3a and 3b we observe a spreading of the pulse and a decay in the particle number density as ex~ected.The auantum ~robabilitvsurface also exhibits a corresponding delocalization us evident from the decay in ocak orobubilitv umolitude and the increase in the width bf the~probabili~y sukace in the x-direction. This temporal behavior is clearly a natural consequence of the dephasing of the individual wavefunctions as time advances. It is pleasing to note that the group velocity of the quantum wave packet coincides exactly with the classical mean velocity of the particle pulse. A y = 0 cross section of Figure 3a reveals the real and imaginary components of the position probability as shown in Figure 4. Although the individual components exhibit wavelike fringe structure, it is evident that they are mutually complimented to give a smooth probability profile.
Not all of the classical features of this system correlate with their quantum counterparts. If, for example, we use a smaller slit aperture, the classical beam width is narrowed. However the deBroglie matter waves may experience an angular diffraction resulting in an increase in the width of the y position probability distribution in direct contrast to the decrease in the classical distribution. These effects are clearly evident in Figures 5 and 6. This behavior is perhaps familiar to the student since it is precisely the same phenomenon that helshe hadwill encounter(d) with single slit optics. In the optics experiment, we observe the square of superimposed Maxwell (electromagnetic)waves undergoing diffraction as the slit width approaches the magnitude of the wavelength of the radiation. In Figures 5 and 6 we observe the corresponding effect on the square of superimposed deBroglie (matter)waves. In both instances the Heisenberg uncertainty principle applies: in our attempt to restrict the uncertainty in t h e y position of the particles (or photons) by narrowing the slit width, we evidently cause an increase in the velocitv comDonent u.. in this direction. We can also investigate the effect of changing the width of the momentum distribution (APJ on the width of the position distribution (Ax) a t a specifiedtime. Figure 7 compares the momentum spectrum of the original velocity distribution (a) with a modified distribution (b) in which the standard deviation AP has been halved. Figures 8 and 9 compare the corresponding position probability surfaces.
decrease in the width of the momenta distribution (Fig. 7).
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It is immediately evident that the decrease in the uncertainty (standard deviation) of the x momentum as depicted in the momentum spectra results in a n increase in the uncertainty in the position as described by the quantum probability surface. This outcome is precisely what we would exnect from the Heisenbere uncertaintv nrincinle. The simple computer algorithm and graphics provide students the opportunity to verifytdiswver these relationships by a naive, direct summation of wave functions in place of a formal and more elegant Fourier transform. In our view. this provides a direct, hands-on experience with a convincing and understandable outcome.
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Acknowledgment The author is grateful to Professor Doug Brandt for his manv comments and sueeestions durine the course of this project.
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Literature Cited 1. ahiff, L. I. Quantum Morhonics;MGrsr Hill: New York, 1955; pp 7-12.
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2. Kamrnann, W. Quantum Chpmislry:Aeademic: New York, 1957; pp23E-240. 3. M m h . J. S.Am J. Phys 1975,4% 9 7 4 8 . IPhvs. 1979.47 4. Corben. H. C A m . . . 10361037. 5. NO& J. H. Physiml dhemisfv, 2nd ed.; Scott, Fmeaman: Boston, 1989: pp 68& 633. 6. Heisenberg, W Z Phys 1927.43, 1972-1977, Publishing: Amsterdam, 1958; 7. Messiah, A. T. Quantum M&nics;North-Holland Vol 1, pp 12S133. 8. s h e w i n , C. w ~ntmduetionm ~ u a n t u m~ x h o n i c s ;H&: N ~ W Y ~1959: ~ L ,pp 12% 137. 9. Merzbaeher. E. QuonNmMeehonics, 2nd ed.; Wiley: New York, 1970; pp 17-27. lo. Fench A. P.; Taylor, E. F. An Intmduelion to Quantum Physics; Norton: New York. 1978: w 327351. 11. Ref I. oo 8&84. 12. Flygare, W H.MolPeuhrStruefunandDynomlcs;Renti-Hall: EnglewmdCliff~, N. J., 1978; pp 73-75. 13. Sakursl, J. J. M d m Quantum Mechanics: Addison-Wesley: Redwood, CA. 1985;
...
nn 5 7 6~~. 0 77
~
14. Press,W. H.: Flannery. B. P; Teukolaky, S. A.; Vetterling, W T. Numerziol Reeips: ThP&~fS~&nflfrcCompuling;CarnbridgeUniversity: Cambridge. 1986; ~ ~ 2 0 % q ""a. "
15, Abramodtl M.; Segun,I. A. Editors, Handbod ofMothemntidFundlom; &ver: New York, 1965: p 358. 16. Ref. 2, pp 181-183. 17. Ref. 15 oo 369-370. 18. Arvrn
