Research: Science and Education
Two Particles in a Box Igor Novak Department of Chemistry, National University of Singapore, Singapore 117543, Singapore;
[email protected] The particle in a box is the simplest bound quantum mechanical system for which the Schrödinger equation can be solved exactly. The particle in a box thus appears regularly in most quantum chemistry courses and textbooks and in many articles in education journals. Numerous examples of the model have been explored in one, two, and three dimensions. Some of the more interesting versions are the “particle in a champagne bottle” (1) and the “particle in an equilateral triangle box” (2). In this article I present another variation of the model: “two particles in a box”. The significance of this model is that it can be used at a very early stage in a quantum chemistry course to introduce the behavior of bosons and fermions. In particular, it demonstrates that identical fermions try to avoid each other and that this effect arises not from Coulombic repulsion, but from symmetry constraints imposed on the wave function. The model presented here is suitable for advanced undergraduate students taking courses in quantum chemistry, and for statistical thermodynamics courses where Bose condensation may be studied. Discussion Consider a system that consists of two indistinguishable neutral particles (for example neutrons) located in an infinitely deep, 1-D potential well of unit width. The interaction between particles is assumed to be negligible. The spatial parts of their total wave functions can be constructed readily from the known wave functions for the one-dimensional particle in a box (3). However, an acceptable wave function must recognize that the particles are indistinguishable; therefore it must be either symmetric or antisymmetric with respect to particle exchange. This constraint gives only two possible normalized wave functions for any state described by quantum numbers n1 and n2. The wave functions are: Ψs = 21/2[sin(n1π x1) sin(n2πx2) + sin(n1πx 2) sin(n 2 π x1)] Ψa = 21/2[sin(n1π x1) sin(n2πx2) – sin(n1πx 2) sin(n 2 π x1)]
where x1 and x2 are positions of particles in a state described by quantum numbers n1 and n2. (Ψs and Ψa can describe a pair of either bosons or fermions, depending on the spins of the particles; for n1 = n2 the normalization constant of Ψs is unity.) The energy of such a state equals E(n1,n2) = (n12 + n22)h 2/8m. If the two quantum particles were distinguishable, their normalized wave function would be nonsymmetric (i.e., neither symmetric nor antisymmetric): Ψn = 2 sin(n1π x1) sin(n2πx2) Ψs and Ψa wave functions exhibit several properties of
pedagogical interest in trying to explain the properties of bosons and fermions. The strong Pauli principle, which imposes indistinguishability on the wave function, requires that the total wave function ψtot = ψspat ψspin be either symmetric or antisymmetric with respect to exchange of bosons
Figure 1. Spatial distributions of two particles in a box for E(1,2) state ψs2 ψa2 ψn2.
or fermions, respectively. Since Ψa = 0 if n1 = n2 , two identical particles in the same energy level must be described by a symmetric spatial (ψspat) wave function—if the particles are bosons, Ψspin must be symmetric; if the particles are fermions, Ψspin must be antisymmetric. When two identical particles occupy different energy levels (n1 ≠ n2) no such restriction applies. The point is that certain spatial distributions for particles in the same energy level are simply not allowed, even if there is no interaction between them. At this point one can show how these rules lead to weak Pauli principle, namely, two electrons in the same orbital must have opposite spins. To examine this effect on spatial distributions, we wish to monitor more closely the positions of particles when they are described by different wave functions, Ψs , Ψa, or Ψn . This can be achieved either graphically, by plotting joint spatial distributions for the two particles, or numerically, by calculating their average separation 具d 冔. With the numerical approach, quantitative trends in spatial distribution for different states can be readily observed. I shall present both approaches, so that teachers or students can select the method most suitable to them. Figure 1 describes particle distribution for a single state (n1 = 1, n2 = 2) (Appendix 1). Particles described by the Ψs wave function are able to occupy the same positions in space (as indicated by the peaks along the x1 = x2 diagonal line on the plot) and show “attraction”. On the other hand, Ψa particles have a negligible probability of being found near each other (as indicated by the trough along the diagonal x1 = x2 line) and thus can be said to “repel” each other. The distinguishable particles (Ψn ) are an intermediate case in the sense that (as indicated by the peaks and troughs along x1 = x2 line) they can occupy the same space, but with lower probability than Ψs particles. The case Ψn demonstrates “independence”. One can use the numerical integration routine in Mathematica software to calculate average particle separation (Appendix 2), which can then be taken as a numerical expression of particle distribution in a particular state (Table 1). Analytical integration with Mathematica does not work. The average distance between particles was calculated according to the formula (3) 具d 冔 = ∫01 ∫01|x1 – x2|ψ2(n1,n2)dx1 dx2 where Ψ is a normalized wave function describing a pair of non-interacting particles, in the state designated by quantum numbers n1 and n2. The results indicate that the relationship between average distances is always 具d s 冔 < 具d n 冔 < 具d a 冔. The indistinguishable particles in a particular state (n1 ≠ n2), which
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are described by ψs , are always closer on avTable 1. Average Distance between Two Particles in a Unit Box erage than those described by ψa. If particles n2 n1 are considered distinguishable, their average 1 2 3 4 5 6 7 8 9 distance is intermediate. 1 0.206 0.157 0.245 0.264 0.271 0.275 0.277 0.278 0.279 If n1 = n2, 具d n 冔 = 具d s 冔. The differences 0.206 0.270 0.277 0.279 0.280 0.281 0.281 0.281 0.282 — 0.382 0.308 0.294 0.289 0.287 0.286 0.285 0.284 between average distances decrease at high 2 0 . 3 0 1 0 . 2 0 9 0 . 2 8 9 0 . 3 0 5 0 . 3 1 1 0 . 3 1 4 0 . 3 1 6 0 .317 quantum numbers/energies. Let us compare 0.301 0.315 0.317 0.318 0.319 0.319 0.320 0.320 values in Table 1 with a pair of classical — 0.420 0.345 0.331 0.327 0.324 0.323 0.322 (nonquantized) particles in a unit box. The 0.319 0.322 0.323 3 0.319 0.221 0.298 0.313 separation of classical particles would always 0.327 0.319 0.324 0.325 0.326 0.326 0.327 be 1⁄3 . This value is equal to the average dis— 0.427 0.352 0.338 0.334 0.331 0.330 tance between two points selected randomly 4 0.325 0.225 0.302 0.317 0.322 0.324 0.325 0.328 0.328 0.329 0.329 0.329 from a unit interval [0,1] and assuming that — 0.430 0.355 0.341 0.336 0.334 the points are uniformly distributed inside 1 5 0 . 3 2 8 0 . 2 2 7 0 . 3 0 4 0 . 3 1 8 0 .323 the box (4 ). Any deviation in Ψn from ⁄3 is 0.328 0.330 0.330 0.330 0.330 due to quantum mechanical effects. — 0.432 0.356 0.342 0.337 The fact that 具d n 冔 converges to the 6 0 . 3 3 0 0 . 2 2 9 0 . 3 0 5 0.319 value of 1⁄ 3 is an example of the correspon0.330 0.330 0.331 0.331 — 0.432 0.356 0.343 dence principle. The preceding discussion can 7 0.330 0.230 0.305 be a starting point for teaching bosons and 0.330 0.331 0.331 fermions (especially Bose and Fermi gases) — 0.433 0.357 in statistical mechanics. Here the advantage 8 0.331 0.230 of this model is pedagogical. Since electrons 0.331 0.332 are fermions, which also happen to be nega— 0.433 tively charged, a misconception may arise in 9 0.331 0.331 students’ minds about fermionic repulsion: — they may consider the repulsion to be due to a Coulombic force. In this model there is no NOTE: The numbers in each cell indicate (top to bottom) the average distances for such difficulty. Because the two unspecified eigenstates described by Ψs (symmetric), Ψn (nonsymmetric), and Ψa (antisymmetric), particles are neutral, any attractive/repulsive respectively. Only the upper triangle is shown because ψ(n 1, n 2) = ψ(n 2,n 1). effects are due solely to symmetry constraints, which stem from the indistinguishability of Conclusions particles. The discussion of bosonic “attraction” via the model can be related to the phenomenon of Bose–Einstein condenI believe that this version of the particle-in-a-box model sation (BEC). BEC has only recently been observed; it occurs has several useful pedagogical features: when one cools an extremely dilute gas composed of neutral atoms to temperatures as low as 10᎑6 K. The diluteness of gas is 1. The notion of fermionic repulsion and bosonic attracnecessary to prevent collisions, which would otherwise lead to tion follows naturally from the model and is shown to the formation of molecules or clusters and hence to familiar be unrelated to Coulombic or any other (e.g. van der phase transitions giving liquids or solids (5, 6 ). Waals) forces. The transition temperature at which BEC occurs is given 2. The model provides a logical introduction to a disas λdB/r ≈ 1.377. The equation can be derived from ref 6; r cussion of advanced and interesting topics such as is the average interparticle separation and λdB is de Broglie Bose–Einstein condensation. This may stimulate wavelength that represents position uncertainty associated students’ interest in the study of an otherwise rather 2 1/2 with thermal motion. The expression λdB = (h /2πmkBT ) abstract topic of statistical thermodynamics. shows that at low temperatures λdB increases and allows 3. The model encourages lecturers and students to use bosons to undergo quantum mechanical phase transition so information technology methods (computer algebra that their wave functions overlap and form a condensate, a software) to explore quantum mechanics. coherent cloud of atoms (5, 6 ). How does this tie in with the 4. The model can help clarify the reasons for Hund’s rule— previous discussion? The average particle separation r will for example, why the triplet state has lower energy than affect the BEC transition temperature. Bosons have smaller the singlet for the electron configuration with two interparticle separation than fermions (they “attract” each unpaired electrons. In the triplet state the spatial part other), which leads to a smaller λdB. A smaller λdB implies of the total wave function is antisymmetric, whereas that BEC occurs at higher T. The conclusion is then obvious in the singlet state it is symmetric. Our model shows why bosons form such coherent phases and fermions do not. that particles described by antisymmetric spatial wave A final pedagogical caveat must be reiterated: the neutral functions are further apart; this implies that their bosons do not condense because there are “attractive” forces Coulombic repulsion will be smaller and the state as a operating among them, nor can the lack of “Fermi–Dirac” whole more stable. condensates be attributed to “repulsive” forces between fermions.
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Acknowledgment
Appendix 1
I wish to thank the anonymous referee for very useful, in-depth comments.
Mathematica routine for plots of ψs2, ψa2 ψn2 in n1 = 1, n2 = 2 state:
Literature Cited 1. Miller, G. R. J. Chem. Educ. 1979, 56, 709–710. 2. Li, W.-K.; Blinder, S.M. J. Chem. Educ. 1987, 64, 130–132. 3. Eisberg, R.; Resnick, R. Quantum Physics of Atoms, Molecules, Solids and Nuclei; Wiley: New York, 1974; p 332. 4. Burrows, B. L.; Talbot, R. F. Int. J. Math. Educ. Sci. Technol. 1988, 19, 109–117. 5. Ketterle, W. Phys. Today 1999, 52, 30–35. 6. Burnett, K.; Edwards, M.; Clark, C. W. Phys. Today 1999, 52, 37–42.
Plot3D[2*(Sin[Pi*x1]*Sin[2*Pi*x2]+Sin[Pi*x2]*Sin[2*Pi*x1])^2, {x1,0,1},{x2,0,1}] Plot3D[2*(Sin[Pi*x1]*Sin[2*Pi*x2]Sin[Pi*x2]*Sin[2*Pi*x1])^2, {x1,0,1},{x2,0,1}] Plot3D[4*(Sin[Pi*x1]*Sin[2*Pi*x2])^2,{x1,0,1},{x2,0,1}]
Appendix 2 Mathematica routine for numerical integration of 具da 冔 in n1 = 1, n2 = 2 state: 2*NIntegrate[Abs[x1-x2]*(Sin[Pi*x1]*Sin[2*Pi*x2]Sin[2*Pi*x1]*Sin[Pi*x2])^2, {x1,0,1},{x2,0,1}]
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