Research: Science & Education
Variational Treatment of Attractive Central Power-Law Potentials B. Cameron Reed Department of Physics, Alma College, Alma, MI 48801-1599 The variational principle (VP) is a powerful method for establishing upper limits on the ground-state (and excitedstate) energies of potentials for which the Schrödinger equation is difficult or impossible to solve analytically (1). From a pedagogical point of view, the beauty of the VP lies in its easy interpretation and the fact that it demands analytic techniques no more complex than sophomore-level calculus (2). This paper addresses application of the VP to three-dimensional attractive central potentials (ACPs) of the form V(r) = { κ/rn (κ, n > 0; 0 ≤ r ≤ ∞). Such potentials can be expected to possess negative-energy bound states; when n = 1, one recovers the familiar Coulomb potential. Originally developed as a classroom example, this potential proves to contain some unexpected behavior. The question of whether negative-energy states can be bound by ACPs has appeared occasionally in the pedagogical and research literature, but with conflicting results. Swenson (3) utilized the virial theorem to argue that (i) negative-energy bound states will exist for 0 < n < 2, (ii) that a bound state of exactly E = 0 will exist for n = 2, and (iii) that no negative-energy states exist for n > 2. Liebman and Yorke (4) subsequently took issue with Swenson’s conclusion ii, arguing that application of the virial theorem when n = 2 is invalid because the kinetic and potential energies evaluate to infinity in that case. They further pointed out that Landau and Lifshitz (5) show that when n = 2, there exists a critical value of the strength parameter κ such that if κ > h 2 / 8m, then the only solution is one of E = { ∞; whereas if κ < h 2 / 8m, then the lowest allowed state is one of E = 0. Case (6) solved the Schrödinger equation explicitly for n = 2, finding that complex wave functions arise for κ > h 2 / 8m. While bound states clearly exist for the case of n = 1, the classic “Coulomb” attractive central potential, one is inclined to wonder if Swenson’s conclusion (i) in fact holds up for n ≠ 1, and what causes n = 2 to be a special case. These questions can be addressed at a level understandable to undergraduate students by applying the VP to this potential, an approach evidently not previously considered. While the resulting calculations lead to a constraint that restricts consideration to 0 < n < 2, an apparently new conclusion emerges: that, depending on the strength parameter κ and the mass of the particle involved, there exists a value of n such that the ground-state energy is a maximum; that is, the ground-state is minimally bound. Further, n → 2 can be treated as a simple limiting case, from which the existence of a critical value of κ arises quite naturally. This critical value proves to be directly related to the types of ACPs that will exhibit such minimally bound ground-states.
orbit and a0 is the Bohr radius:
32π ε20h 2
(2)
2
and
a0 =
4πε0h
2
(3)
m ee 2
Any energy and distance could be used in eq 1; E 1 and a0 are convenient in view of their familiarity and the fact that they lead to an easy check of the results in the case of the Coulomb potential, which is described by m = me, K = 2, and n = 1. The essence of the VP is that one chooses a trial wave function φ (in this case a real function of radial distance r) containing a free parameter. The true ground state energy E0 can be shown to satisfy E0 ≤ E
(4)
where E is given by ∞
E = 4π 0
φ *φ r 2dr
(5)
and where * is the Hamiltonian operator
*={
2
h ∇2 + V(r) 2m
(6)
(Since we seek limits on the ground state, there is no angular momentum contribution to consider.) The variational parameter is then adjusted to minimize E, yielding an upper limit on E0 . As a trial wave function, consider the normalized hydrogenic ground-state wave function
φ (r) =
β3 / 8πexp ({βr / 2)
(0 ≤ r ≤ ∞)
E=
n n εβ2 KE 1a 0 β Γ 3 – n – 4 2
n 0; 0 ≤ r ≤ ∞)
(1)
where E 1 is the absolute value of the energy of the first Bohr
(8)
where
ε = h 2 / 2m
To expedite later computations, it is useful to cast the ACP into a form where the strength parameter is dimensionless. One convenient choice is to rewrite the potential as
(7)
where β is the variational parameter. Choosing the trial function to be of this form provides a built-in check of the results: it should yield the exact result for E in the case of the Coulomb potential. After some algebra, one finds
Variational Calculation
a V(r) = {KE 1 r0
m ee 4
E1 =
(9)
where m is the mass of the particle trapped in the potential. The n < 3 restriction on eq 8 arises from integrals involved in the computation of E. However, we will see below that a further constraint arises on E, restricting us to 0 < n < 2. Minimizing E with respect to β requires
β=
ε nKE 1a n0 Γ (3 – n)
1 n– 2
Vol. 74 No. 8 August 1997 • Journal of Chemical Education
(10)
935
Research: Science & Education Back-substituting this result into eq 8 yields
E = E1 / µ
1 2 / n– 2
2 1– n
0
(11)
4 Kµ nΓ 3 – n
1.0
W 1.5
-1
where µ is the ratio of the mass of the particle to the mass of the electron,
µ= m me
2.0
(12)
The variational method is predicated on minimizing E; on computing the second derivative of E from eq 8, one finds that it yields a minimum only for n < 2. While one might anticipate this constraint to be a manifestation of the choice of trial wave function, this may in fact not be the case: on trying a more general trial wave function of the form φ (r) ~ rp/2 exp ({βr/ 2), the resulting expression for E has the same dependence on β as eq 8, and the p-dependences fold into multiplicative constants in such a way that the second derivative of E still yields a minimum only for n < 2. In other words, the n < 2 constraint may be physical as opposed to a lack of suitable cleverness in choosing φ (r); perhaps n ≥ 2 lies in the realm of relativistic quantum mechanics. Comments Equation 11 is the central result of this paper. Clearly, the ground-state energy of an ACP depends on n and the combination (Kµ) of the dimensionless parameters K and µ. Figure 1 illustrates ground-state upper limits for ACPs of various (Kµ) as a function of n; the quantity plotted on the vertical axis is the dimensionless energy W = E/(E1 / µ). A number of conclusions follow immediately: 1. Equation 11 and Figure 1 confirm Swenson’s conclusion i: negative-energy bound states are possible for ACPs with n < 2. 2. The prediction of eq 11 for the Coulomb potential, (K, n, µ) = (2, 1, 1), is W = {1, the exact value. This is to be expected on the basis that the trial wave function corresponds to the exact solution of the Schrödinger equation in this case. 3. The most striking feature of Figure 1 is that for (Kµ) > 0.5 (see below) there exists a least-bound ground state of the ACP; as (Kµ) increases, the value of n at which the maximum energy occurs and the energy of the ground-state both decrease (i.e., the ground-state becomes more tightly bound). While the calculations leading to this conclusion are sufficiently straightforward that one must suspect this feature to have been previously discovered, it does not, to my knowledge, appear in any commonly used textbooks, nor have I come across it in research papers. Curiously, the ground-state of the Coulomb potential is about as unbound as it could possibly be.
Numerical solutions of the radial Schrödinger equation verify the presence of the maxima seen in Figure 1. Equation 11 actually proves to yield fairly stringent upper limits. For (Kµ) = 1, for example, it predicts {W = (0.435, 0.250, 0.260) for n = (0.5, 1.0, 1.5), whereas the true solutions are { W ~ (0.438, 0.250, 0.292); the variational calculation overestimates the true energies by about (0.7, 0, 11) percent. The physical interpretation of these least-bound ground-states is left as an open question. To some extent, the behavior of Figure 1 can be justified semiintuitively: as n increases from zero, the force corresponding to eq 1 falls off more rapidly with distance; hence one might expect the ground-states to become less bound. However, there seems
936
0.50 0.75
Kµ = 2.5
-2
-3 0
0.5
1
1.5
2
n
Figure. 1: Upper limits on ground-state energies for attractive central potentials of the form of eq 1 for various values of (Kµ); W = E/(E1/µ). See eq 11.
no simple way to understand on the basis of fundamental physics why the curves should exhibit maxima. Finally, it is instructive to study the behavior of E as n → 2. In this limit, the gamma function Γ (3 – n) approaches unity from above. Equation 11 reduces to
E E1 / µ
n →2
⇒ 1 2Kµ 4
∞
1–2/n
(13)
Two fates are possible for E: if (Kµ) ≤ 0.5, then E → 0, whereas if (Kµ) > 0.5, E → {∞. Swenson’s conclusion ii therefore represents only half of the story. Inverse-square ACPs with (Kµ) > 0.5 must exhibit minimally bound ground states for 0 < n < 2, since E approaches { ∞ as n approaches both 0 and 2; the existence of a critical value of the strength parameter (actually, of Kµ) for the inverse-square ACP thus emerges naturally from the variational approximation. In terms of the present notation, the Landau–Lifshitz (5) threshold value of the strength parameter that arises from the exact solution of the Schrödinger equation in this case is (Kµ) = 1/4 [E = { ∞ for (Kµ) > 1/4; E = 0 for (Kµ) < 1/4]. That the present result for the critical value of (Kµ) differs from this is not surprising in view of the fact that we have utilized an approximate trial wave function. For non-inverse-square ACPs with (Kµ) < 0.5, the minimum binding energy evidently approaches zero as n → 2. We cannot address Swenson’s conclusion iii in view of the n < 2 restriction on eq 11. Acknowledgment I am grateful to Gene Deci for a number of useful comments. Literature Cited 1. Pauling, L.; Wilson, E. B. Introduction to Quantum Mechanics with Applications to Chemistry; Dover: New York, 1985; Chapter vii. 2. Bendazzoli, G. L. J. Chem. Educ. 1993, 70, 912–913. 3. Swenson, R. J. Am. J. Phys. 1981, 49, 694. 4. Liebman, J. F.; Yorke, E. D. Am. J. Phys. 1983, 51, 274. 5. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics (Nonrelativistic Theory), 3rd ed.; Pergamon: Oxford, 1977; Section 35. 6. Case, K. M. Phys. Rev. 1950, 80, 797–806.
Journal of Chemical Education • Vol. 74 No. 8 August 1997
