The Particle Inside a Ring: A Two-Dimensional Quantum Problem

Research: Science and Education

The Particle inside a Ring: A Two-Dimensional Quantum Problem Visualized by Scanning Tunneling Microscopy Mark D. Ellison Department of Chemistry, Ursinus College, Collegeville, PA 19426; [email protected]

Quantum mechanics is central to an understanding of modern chemistry. However, it is a difficult subject to teach and one that students have great difficulty grasping (1). It has long been recognized that incorporating aspects of modern research is one way to increase student interest in a subject (2), yet such examples can be difficult to find in quantum chemistry. A balance must be struck between mathematical straightforwardness and physical relevance. Often, examples lean toward mathematical tractability so as not to overwhelm undergraduate students. This approach has the advantage of allowing students to become familiar with solutions to the Schrödinger equation but often leaves students grasping for links to physical systems with which they are familiar.

B

Figure 1. (A) STM image of electron density in a ring of Fe atoms on a Cu(111) surface and (B) cross section of the image. From Crommie, M. F.; Lutz, C. P.; Eigler, D. M. Science 1993, 262, 218–220. Reprinted with permission from AAAS. Reprint Courtesy of International Business Machines Corporation, copyright © International Business Machines Corporation.

1282

One such physical system that is presented in nearly every undergraduate physical chemistry textbook is the one-dimensional particle-in-a-box model (3–20). This model is well-suited for the students’ first exposure to the Schrödinger equation. The fact that the potential inside the box is zero makes the solution of the resulting ordinary differential equation rather clear-cut. Additionally, the imposition of boundary conditions clearly leads to quantized energy levels in the system. Thus, the particlein-a-box model does indeed serve as a valuable introduction to the Schrödinger equation and its important implications for bound systems. However, the particle-in-a-box model has, at best, a somewhat tenuous connection to real physical systems. There are, of course, the well-known laboratory experiments using cyanine dye molecules (21–23). A slight modification to a particle-on-a-ring system allows for a crude model of the electronic structure of benzene (4, 7, 13, 14, 16). However, the fact that these models make only qualitative connections to actual physical systems is a serious drawback. This need not be the case, for there are a number of examples from modern science that can be used to illustrate quantum mechanics. One such area of active research that should be known to physical chemistry students is scanning probe microscopy (SPM), which generally encompasses two complementary techniques, atomic force microscopy (AFM) and scanning tunneling microscopy (STM). These experimental techniques have, for example, enabled scientists to measure forces involved in stretching DNA molecules (24), to visualize elementary steps of chemical reactions (25), to probe the electronic structure of surfaces (26), and even to manipulate single atoms (27). The high-resolution imaging made possible by AFM and STM can stir students’ imaginations and motivate them to learn more about the underlying science. STM in particular can be used to illustrate several topics in quantum mechanics. Many physical chemistry textbooks discuss tunneling in some detail (3–5, 7–13, 15, 18, 19). Although the details might be beyond the level of an undergraduate physical chemistry course, students can grasp the qualitative features of tunneling, such as the exponential dependence on tip–sample distance (28). Recently there has been some debate in this Journal about the exact nature of the image (29–32), but it is generally recognized that an STM image reflects the overlap of orbitals in the tip and sample (33). This allows STM to observe the wavelike nature of matter directly. In 1990, researchers at IBM published their work on using STM to manipulate Xe atoms on a Ni surface in ultrahigh vacuum and at very low temperatures (27). By adjusting the tip–sample voltage, they were able to pick up individual atoms and deposit them at specified locations on the surface. Shortly thereafter, the researchers arranged Fe atoms in a circular pattern of average diameter of 142.6 Å on a Cu(111) surface and observed standing electron waves within this “quantum corral”, as shown in Figure 1A (34). Another study imaged the standing waves at steps on a Cu(111) surface (35). The visual images

Journal of Chemical Education • Vol. 85 No. 9 September 2008 • www.JCE.DivCHED.org • © Division of Chemical Education


of the electron waves are striking and directly demonstrate the wavelike nature of matter (36). Additionally, they can be quantitatively understood using the two-dimensional Schrödinger equation. Therefore, a particle confined to a circular well, or a “particle in a quantum corral”, is a suitable example for students that shows an application of the quantum mechanics to a physically real and interesting system. Introducing the particle-in-a-quantum-corral model to students after the particle-in-a-box model has additional advantages. First, it extends the solution of the Schrödinger equation to a system in two dimensions. This is a more natural progression than the typical jump from a particle in a onedimensional box to a particle in a three-dimensional box. Second, the solution involves separation of variables, preparing the students for use of this technique to find the solutions to the Schrödinger equation for the hydrogen atom. Third, the solution to the Schrödinger equation for the particle in a quantum corral involves an established differential equation with known solutions, the Bessel equation and Bessel functions, respectively. When presented with the solution to the Schrödinger equation for the hydrogen atom, students are, in short order, introduced to two known differential equations with known solutions (the Laguerre and Legendre polynomials). The fact that known differential equations with known solutions result from solving the Schrödinger equation is obvious to teachers but not necessarily to students. Solving the Schrödinger equation for the particle in a quantum corral introduces this idea simply, making students familiar with it once the Schrödinger equation for the hydrogen atom is solved. These advantages plus the connection to a realworld example that is visually captivating make the particle in a quantum corral a useful system to present in an undergraduate quantum chemistry course. Of the many physical chemistry and quantum chemistry textbooks that I inspected (3–20), only the one by Berry, Rice, and Ross (7) presents and solves this system. Introduction to the Model A schematic of the system is shown in Figure 2. A particle of mass μ is confined within a circular region of radius r0. The spherical coordinate system is used, with r being the distance between the particle and the origin, ϕ being the angle between r and the x axis, and θ having a constant value of 90° or π/2. We assume that the potential inside the corral is zero and that the particle cannot escape the corral. That is, V = 0 for 0 ≤ r < r0 and V = ∞ for r ≥ r0. With θ constant, the Schrödinger equation for this system simplifies to

2

1 v 2 v h r 2 N r 2 vr vr

2 1 v : E : r 2 vG 2

(1)

z

y G

Because r and ϕ are independent variables, a natural approach is to posit that the solution will be a product of functions of the individual variables: Ψ(r,ϕ) = R(r)Φ(ϕ). That is, separate the solution into the product of a radial part and an angular part, the same approach that is used for the hydrogen atom. Substitute this expression into eq 1, noting that Φ is a constant

r

N

x Figure 2. Coordinate system and variables for a particle in a quantum corral. The corral is a circle of radius r0 in the xy plane.

with respect to derivatives of r and R is a constant with respect to derivatives of ϕ:

2

h 1 v 2 vR 1 v2 ' ' 2 r R 2 2N vr r vr r vG 2

ER ' (2)

Next, divide both sides by R, Φ, and the constants outside the brackets:

1 1 v 2 vR r R r 2 vr vr

1 1 v 2 ' 2 NE 2 ' r 2 v G2 h

(3)

Now multiply both sides by r 2 and collect terms dependent on r on the left side and terms dependent on ϕ on the right side:

1 v 2 vR r vr R vr

1 v2' 2 N E 2 r 2 ' v G2 h

(4)

The left side of eq 4 has terms that depend only on r, and the right side has terms that depend only on ϕ. Because r and ϕ are independent variables, the only way the two sides of eq 4 can be equal is if they are equal to the same constant. Following the convention of the solution of the Schrödinger equation for the hydrogen atom, we let each side of the equation equal the constant ‒m 2: 1 v2' m 2 ' v G2

1 v 2 vR r R vr vr

2 NE 2 r m 2 2 h

(5)

(6)

Equation 5 is exactly the same as what will be obtained for the Φ part of the hydrogen-atom solution. It rearranges to

Solution of the Model

R = 90p

r0

v2' vG 2

m2 ' 0

(7)

which has the solutions (8) ' A e q i mG 1/2 where i = (‒1) and A is a normalization constant. Because the wavefunction must be single-valued, the circular symmetry of the system means that Φ(ϕ) = Φ(ϕ + 2π). This condition restricts

© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 9 September 2008 • Journal of Chemical Education

1283

Research: Science and Education v= 0 v= 1 v = 1

1.0 0.8

Table 1. Values of x for Which the Bessel Functions J0, J1, and J2 Are Equal to Zero J0

J1

J2

2.405

3.832

5.136

5.520

7.016

8.417

8.654

10.173

11.620

11.792

13.324

14.796

14.931

16.471

17.960

Jv(x)

0.6 0.4 0.2 0.0

0.2 0.4 0.6 5

0

10

x

15

20

25

Figure 3. Bessel functions J0, J1, and J−1.

the values of the quantum number m in eq 8 to 0, ±1, ±2, ±3, and so forth. This is the solution to the angular part of the problem, and we now turn to the solution to the radial part. Starting with eq 6, bring m2 to the left side and multiply both sides by ‒R to obtain v 2 vR r vr vr

2 N E h

2

R r 2 m2R 0

Energy Levels of the System (9)

Taking the derivative of the first term and factoring R from the second and third terms yields

vR v2R 2r r2 vr vr 2

2 N E 2 r m2 R 0 2 h

(10)

Then rearranging, we finally have

r2

v2 R vr 2

2r

vR vr

2N E 2 r m2 R 0 2 h

(11)

This is a form of Bessel’s equation that appears frequently in science and engineering, particularly in situations with cylindrical symmetry (37–39), such as the particle in a quantum corral. Specifically, eq 11 is analogous to

x2

d 2y dx

2

x

dy B2x2 v 2 y 0 dx

(12)

Comparing eqs 11 and 12 shows that α = (2μE/h2)1/2. Because m is an integer, the solutions to eq 11 are Bessel functions of the first kind, Jv(αr) and J‒v(αr) (38, 39). Bessel functions are a convergent infinite polynomial series, and Bessel functions of the first kind are given by the following formulas (38, 39)

Jv x x v

e

¥

p 0

1 p x 2 p

2 2 p v p ! v p !

(13)

For integer v, the following condition holds

J v x 1 Jv x

v

The total wavefunction, then, is : N Jm

(14)

2 NE h

1

2

2

(15)

r ei m G

where N is a normalization constant. To satisfy the boundary condition Ψ = 0 for r ≥ r0, the Bessel function must equal zero at r = r0. As in the particle in a 1-D box, the application of the boundary condition results in quantization of the allowed energy levels. However, unlike the particle in a 1-D box, there are no analytical expressions for the x values at which the wavefunction for the particle in a quantum corral is zero. However, applied mathematics textbooks have tables of Bessel function values that allow the determination of x values at which the Bessel function is zero (38). Alternatively, these x values can be determined graphically or by using a symbolic mathematics engine such as Mathcad (40), Maple (41), or Mathematica (42). Because J‒1 = ‒J1, these Bessel functions both equal zero at the same x values. Table 1 lists x values at which the first few Bessel functions equal zero. Also, for |m| > 0, Jm = 0 at x = 0, but this is a trivial solution, because it would imply a corral of zero radius. Consequently, for m = 0, the allowed energy values are

E

2

2

h 2. 405 , r0 2N

2

h 5.5520 2N r0

2

,

2

h 8. 654 2N r0

2

, | (16)

This result can be extended to a more general formula for any value of m

Because the Bessel functions are series functions, their graphical form is not readily apparent. Figure 3 shows J0, J1, and J‒1.

1284

These functions oscillate sinusoidally, and the amplitude steadily decreases as x increases. Unlike the sine and cosine functions, however, the Bessel functions do not equal zero at regularly periodic intervals.

E

2

h

2 N r02

z n,m2

(17)

where zn,m is the nth zero of the Bessel function Jm (Table 2) and n and m can be thought of as analogous to quantum numbers for this system. In particular, n is analogous to the quantum number



Energy/J n

|m| = 0

|m| = 1

10–22

1.763 ×

|m| = 2

10–21

3.167 × 10–21

1

6.945 ×

2

3.659 × 10–21

5.911 × 10–21

8.507 × 10–21

3

8.993 × 10–21

1.243 × 10–20

1.621 × 10–20

4

1.670 × 10–20

2.132 × 10–20

2.629 × 10–20

5

2.677 × 10–20

3.258 × 10–20

3.873 × 10–20

for a particle in a 1-D box, whereas m is an angular momentum quantum number. As an illustration, for an electron in a corral of radius 71.3 Å, corresponding to the STM-constructed quantum corral (34), the energies of the first five levels are given in Table 2. The relative energies for the energy levels in Table 2 are shown in Figure 4. Note that, for |m| > 0, the levels are doubly degenerate because the energy depends on |m|. The energy levels for a particle in a quantum corral are qualitatively similar to those of a particle in a 1-D box in several ways. First, the spacing increases as energy increases. Second, the energy is inversely proportional to the square of the radius of the corral, so a small corral will have large energy spacings whereas a large corral will have small energy spacings. Finally, the energy is inversely proportional to the mass of the particle, so the lighter the particle, the more pronounced the wave nature of the system. Wavefunctions and Probability Distributions The form of the wavefunctions for the particle in a quantum corral is given in eq 15. Because the Bessel functions are an infinite series, it is not possible to write the complete wavefunction explicitly without truncating some of the terms. The wavefunctions are of particular interest in terms of the probability of finding the particle, which is proportional to Ψ*Ψ. Because the Bessel functions are real, the probability can be expressed as

:* : u

Jm

2NE h

2

1

Relative Energy

Table 2. Energy Values for Low-Lying 71.3 Å Radius Quantum Corral Energy Levels

|m| = 0

|m| = 1

|m| = 2

Figure 4. Energy Levels of the quantum corral system.

respectively, of a 71.3 Å radius quantum corral. Figures 5–8 were generated by plotting eq 18 in MathCad (40). White and light gray represent a high probability of finding the particle, and dark gray and black represent a low probability. It is apparent that as n increases, so does the number of nodes. For lower n, there is a larger probability of finding the particle near the center of the ring, whereas as n increases, so does the probability of finding the particle farther away from the center of the ring. Figures 6 and 7 display the relative probability distribution for |m| = 1 and |m| = 2, respectively. Similarly to Figure 5A–C, these figures display the probability distributions for n = 1, 3, and 5, respectively. The probability distributions for |m| > 0 are distinctly different for those for m = 0. First, for |m| > 0, the probability of finding the particle near the center of the ring is low, whereas it is high for m = 0. Second, although it is difficult to see, these distributions are broader than those for m = 0. The main feature, however, of these probability distributions is the low probability of finding the particle near the center of the ring. The probability distributions for |m| = 1 and |m| = 2 have the same number of radial nodes and are similar in appearance. Indeed, the differences between them are subtle. For |m| = 1, the central ring is taller, but the successively outward rings are shorter than those for |m| = 2. Also, the rings for |m| = 2 are broader than those for |m| = 1. This pattern continues for increasing m.

2 2

r

Jm

Modeling Electron Waves

e i m G e i m G 2 NE h

2

1

2 2

(18)

r

The integral of the square of a Bessel function does not have a closed analytical form, so finding the normalization constant in eq 15 is not straightforward. It is possible, however, to work with non-normalized wavefunctions and make comparisons of relative probabilities. STM measures electron density (33), which is related to Ψ*Ψ. Therefore, graphs of the square of the Bessel function can be used to make comparisons to the STM images. Figure 5A–C shows the top and perspective views of J02, for n = 1, 3, and 5,

Because the ultimate test of a model is comparison to experimental data, the prediction of the probability of finding the electron made by the particle-in-a-quantum-corral model should be compared to the STM data. The STM researchers used scanning tunneling spectroscopy to determine which energy levels were populated during the experiment. Their results indicated that the electrons were predominantly in the states (n,m) = (5,0), (4,2), and (2,7) (34). The authors fit their STM data to a linear combination of the probability distributions of these states and found good agreement between their data and the fit, as shown in Figure 1B. This experimental validation indicates that the particle-in-a-quantum-corral model works well to describe the electron waves observed when electrons are confined to a circular region of the surface of a metal. Figure 8 shows a linear


1285


A

B

C

Figure 5. (top) Top view, (bottom) perspective views of the relative probability of finding a particle in a quantum corral for m = 0 and (A) n = 1, (B) n = 3, and (C) n = 5.

A

B

C

Figure 6. (top) Top view, (bottom) perspective views of relative probability of finding a particle in a quantum corral for |m| = 1 and (A) n = 1, (B) n = 3, and (C) n = 5.

A

B

C

Figure 7. (top) Top view, (bottom) perspective views of relative probability of finding a particle in a quantum corral for |m| = 2 and (A) n = 1, (B) n = 3, and (C) n = 5.

Figure 8. (top) Top view, (bottom) perspective view of relative probability of finding a particle in a quantum corral for a linear combination of three quantum states.

1286



combination of the square of the Bessel functions corresponding to the above three states. It is in good qualitative agreement with the STM data (Figure 1), correctly showing five “ripples” in the pattern, the largest peak in the center, and larger gaps between peaks 2 and 3 and peaks 4 and 5. These results provide further confirmation of the wave nature of matter and demonstrate the high accuracy with which this simple model can describe an actual physical system. Conclusions The particle in a quantum corral is a model of an actual physical system that offers several pedagogical advantages. It extends the idea of a particle in a one-dimensional box to two dimensions, providing a bridge between one-dimensional and three-dimensional systems. Solution of the Schrödinger equation for this system involves separation of variables and reveals a well-known equation with established solutions. Application of boundary conditions results in quantization of energy, and some of the energy levels are degenerate. The wavefunctions do not have an easily expressible normalized form, but it is possible to make qualitative comparisons of relative probabilities. These probabilities agree well with STM data, indicating the accuracy of the model. By learning about this model, students can appreciate the application of quantum mechanics to understand exciting, cutting-edge science. Acknowledgment I acknowledge Erica Ellison for a critical proofreading of this article. Literature Cited 1. Nicoll, G.; Francisco, J. S. J. Chem. Educ. 2001, 78, 99–102. 2. Sands, M. Physics Today 2005, 58, 49–55. 3. Alberty, R. A.; Silbey, R. J. Physical Chemistry, 1st ed.; John Wiley and Sons, Inc.: New York, 1992. 4. Atkins, P. W.; de Paula, J. Physical Chemistry, 7th ed.; W. H. Freeman and Company: New York, 2002. 5. Ball, D. W. Physical Chemistry; Thomson Brooks/Cole: Pacific Grove, CA, 2003. 6. Barrow, G. M. Physical Chemistry, 5th ed.; McGraw-Hill: New York, 1988. 7. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: New York, 2000. 8. Engel, T.; Reid, P. Physical Chemistry; Pearson Benjamin Cummings: San Francisco, 2006. 9. Fayer, M. D. Elements of Quantum Mechanics; Oxford University Press: New York, 2001. 10. Laidler, K. J.; Meiser, J. H.; Sanctuary, B. C. Physical Chemistry, 4th ed.; Houghton Mifflin Company, Boston, 2003. 11. Levine, I. N. Quantum Chemistry, 4th ed.; Prentice-Hall: Englewood Cliffs, NJ, 1991. 12. Levine, I. N. Physical Chemistry, 5th ed.; McGraw-Hill: Boston, 2002. 13. McQuarrie, D. A. Quantum Chemistry; University Science Books: Mill Valley, CA, 1983. 14. McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, 1997.

15. Mortimer, R. G. Physical Chemistry, 2nd ed.; Harcourt Academic Press: San Diego, 2000. 16. Noggle, J. N. Physical Chemistry, 2nd ed.; Scott, Foresman and Company: Glenview, IL, 1989. 17. Pauling, L.; Wilson, E. B., Jr. Introduction to Quantum Mechanics with Applications to Chemistry; Dover Publications, Inc.: New York, 1963. 18. Raff, L. Principles of Physical Chemistry; Prentice-Hall: Upper Saddle River, NJ, 2001. 19. Ratner, M. A.; Schatz, G. C. Introduction to Quantum Mechanics in Chemistry; Prentice-Hall: Upper Saddle River, NJ, 2001. 20. Woodbury, G. Physical Chemistry; Thomson Brooks/Cole: Pacific Grove, CA, 1997. 21. Garland, C. W.; Nibler, J. W.; Shoemaker, D. P. Experiments in Physical Chemistry, 7th ed.; McGraw-Hill: Boston, 2003. 22. Horng , M.-L.; Quitevis, E. L. J. Chem. Educ. 2000, 77, 637–639. 23. Moog, R. S. J. Chem. Educ. 1991, 68, 506–508. 24. Williams, M. C.; Rouzina, I.; Bloomfield, V. A. Acc. Chem. Res. 2002, 35, 159–166. 25. Dobrin, S.; Harikumar, K. R.; Polanyi, J. C. Surf. Sci. 2004, 561, 11–24. 26. Hamers, R. J.; Kohler, U. K. J. Vac. Sci. Technol. A, 1989, 7, 2854–2859. 27. Eigler, D. M.; Schweizer, E. K. Nature 1990, 344, 524–526. 28. Ellison, M. D. J. Chem. Educ. 2004, 81, 608. 29. Scerri, E. J. Chem. Educ. 2000, 77, 1492–1494. 30. Scerri, E. J. Chem. Educ. 2002, 79, 310. 31. Spence, J. C. H.; O’Keeffe, M.; Zuo, J. M. J. Chem. Educ. 2001, 78, 877. 32. Zuo, J. M.; Kim, M.; O’Keeffe, M.; Spence, J. C. H. Nature 1999, 401, 49–52. 33. Chen, C. J. Introduction to Scanning Tunneling Microscopy; Oxford University Press: New York, 1993. 34. Crommie, M. F.; Lutz, C. P.; Eigler, D. M. Science 1993, 262, 218–220. 35. Crommie, M. F.; Lutz, C. P.; Eigler, D. M. Nature 1993, 363, 524–527. 36. STM Vision Gallery Home Page. http://www.almaden.ibm.com/ vis/stm/gallery.html (accessed May 2008). 37. Cullen, C. G. Linear Algebra and Differential Equations, 1st ed.; Prindle, Weber and Schmidt: Boston, 1979. 38. Kreyszig, E. Advanced Engineering Mathematics, 8th ed.; John Wiley and Sons, Inc.: New York, 1999. 39. Zill, D. G. A First Course in Differential Equations with Modeling Applications, 8th ed.; Thomson Brooks/Cole: Pacific Grove, CA, 2005. 40. Mathcad, version 11; Mathsoft, Inc.: Needham, MA, 2003. 41. Maple, version 10; Maplesoft, Inc.: Waterloo, Ontario, Canada, 2005. 42. Mathematica, version 6; Wolfram Research: Champaign, IL, 2007.

Supporting JCE Online Material

http://www.jce.divched.org/Journal/Issues/2008/Sep/abs1282.html Abstract and keywords Full text (PDF)

Links to cited URLs and JCE articles


1287

The Particle Inside a Ring: A Two-Dimensional Quantum Problem

Recommend Documents