Some results for the wave function at the origin for S-wave levels
aa r X i v : . [ h e p - ph ] A p r Some results for the wave function at the origin for S -wave levels V. Gupta ∗ and G. S´anchez-Col´on † Departamento de F´ısica Aplicada.Centro de Investigaci´on y de Estudios Avanzados del IPN.Unidad M´erida.A.P. 73, Cordemex.M´erida, Yucat´an, 97310. MEXICO. (Dated: October 28, 2018)
Abstract
Starting with the S -wave radial equation for an attractive central potential V ( r ), we give resultsfor the n (principal quantum number) and the µ (reduced mass) dependence of R n (0), the S -waveradial wavefunction at the origin, for potentials with definite curvature. PACS numbers: 03.65.-w, 03.65.GeKeywords: radial wavefunction, central potential ∗ [email protected] † [email protected] . INTRODUCTION Discovery of quark-antiquark atoms like charmonium in 1974 led to general investigationsof the Schr¨odinger equation with a central potential V ( r ) representing the q ¯ q -potential [1].The motivation was to obtain results based on general properties of V ( r ) like its shape, sinceits precise form was then (and still is) unknown.Some results were obtained for the S -wave ( ℓ = 0) bound state radial wave function R n ( r ), n the principal quantum number [2]. Specifically, it was shown that R (0) is larger(smaller) than R (0) provided V ( r ) was everywhere convex (concave), V ′′ ( r ) > V ′′ ( r ) < R (0) with the reduced mass µ was also related to the curvature ofthe potential. This was directly proved from the radial equation for n = 1 [3]. Both typesof results mentioned above were proved for large n using the WKB approximation [4].In this paper we show that both types of results follow directly from the S -wave radialSchr¨odinger equation for all n . For notational simplicity, define: S n (0) = [ R n (0)] . (1)We will prove that for an attractive central potential V ( r ) and n = 1 , , , . . . : Case (a). If V ′ ( r ) > V ′′ ( r ) = 0 for all r , then: S n (0) − S n +1 (0) = 0 and ∂∂µ (cid:20) µ S n (0) (cid:21) = 0 . (2) Case (b). If V ′ ( r ) > V ′′ ( r ) < r and V ′ ( ∞ ) is finite, then: S n (0) − S n +1 (0) > ∂∂µ (cid:20) µ S n (0) (cid:21) > . (3) Case (c). If V ′ ( r ) > V ′′ ( r ) > r and V ′ (0) is finite, then:2 n (0) − S n +1 (0) < ∂∂µ (cid:20) µ S n (0) (cid:21) < . (4)Case (a) corresponds to an attractive linear potential. This case is exactly solvable.Indeed, there are well known exactly solvable examples for the concave (Coulomb potential)and convex (harmonic oscillator) cases which satisfy the above inequalities. Explicit solutionsof convex power law potentials r k with k > r ) potential satisfy theabove inequalities [5]. With all this evidence at hand we believe that the above inequalitiesare really theorems. In the next section we establish the notation and preliminaries, inSec. III we present a result for non-zero ℓ , followed by our arguments for the S -wave resultsin Sec. IV. A confirmation of the results on the µ dependence of R n (0) via a dimensionalanalysis is presented in Sec. V. The concluding section contains some discussion. II. NOTATION AND PRELIMINARIES
The radial equation, for a two-body system with reduced mass µ in an attractive centralpotential V ( r ) for u nℓ ( r ) = rR nℓ ( r ) is: − C ( µ ) u ′′ nℓ ( r ) + [ W ℓ ( r ) − E n ] u nℓ ( r ) = 0 , (5)where C ( µ ) = ¯ h µ (6)and W ℓ ( r ) = V ( r ) + C ( µ ) ℓ ( ℓ + 1) r . (7)The radial wavefunction R nℓ ( r ) for energy E n is real so its modulus square is the sameas its square. Consequently, the inequalities in the introduction are usually stated for themodulus square. The energy of the bound state increases with the principal quantum num-ber n , thus E < E < E . . . . The potential obeys the standard restrictions, namely,lim r → [ r V ( r )] = 0. Also, recall that R nℓ ( r ) behaves as r ℓ as r tends to zero. For an attrac-tive force, the asymptotic behaviour ( r → ∞ ) of the radial wavefunction u nℓ ( r ) will be likeexp( − ar ), a >
0. 3ultiply the radial equation by u ′ nℓ and integrate from zero to infinity. The term with E n gives zero. One integration by parts gives: C ( µ ) [ u ′ nℓ (0)] δ ℓ = Z ∞ W ′ ℓ ( r ) u nℓ ( r ) dr. (8)The term W ℓ ( r ) u nℓ ( r ) from the partial integration does not contribute. This is obvious forthe upper limit r = ∞ . One has to be careful at the lower limit r = 0. However, since V ( r ) is less singular than r − and u nℓ ( r ) ∼ r ℓ +1) as r →
0, the lower limit also does notcontribute. All this is well known. Before specializing to S -wave it is interesting to considerthe above equation for non-zero ℓ . III. RESULT FOR NON-ZERO ℓ In this case, since the left hand side of Eq. (8) is zero, the equation simply says thatthe expectation value of the effective force W ′ ℓ ( r ) is zero. Alternatively, it implies that theexpectation value of V ′ ( r ) for a general potential is related to that of r − . Explicitly: h V ′ ( r ) i nℓ = 2 C ( µ ) ℓ ( ℓ + 1) (cid:28) r (cid:29) nℓ . (9)This general result (probably known personally to many [6]) deserves to be better known.It is is very useful. For example, for a Coulomb potential it immediately gives the correctrelation between the expectation values of r − and r − . IV. S -WAVE RELATIONS For ℓ = 0, Eq. (8) reduces to: C ( µ ) S n (0) = Z ∞ V ′ ( r ) u n ( r ) dr = h V ′ ( r ) i n . (10)This is a well-known result and provides the basis for the arguments leading to the proof ofthe results given in Sec. I. 4 ase (a). V ′′ ( r ) = 0 for all r . This is the case of the attractive linear potential V ( r ) = λr . So, V ′ ( r ) = λ is a positiveconstant for all r . In this case, Eq. (10) reduces to simply C ( µ ) S n (0) = λ, (11)since u n ( r ) is normalized, that is, Z ∞ u n ( r ) dr = 1 . (12)Thus, in this case C ( µ ) S n (0) is a constant (the potential strength), independent of µ or n as required. It is well known that the linear potential is exactly solvable in terms ofAiry functions. The above relation for C ( µ ) S n (0) has been noted earlier using the explicitsolutions [7]. Case (b). V ′′ ( r ) < V ′ ( r ) > r , with V ′ ( ∞ ) finite. A well-known exactly solvable example of this case is the Coulomb potential. Performan integration by parts in Eq. (10) to obtain: C ( µ ) S n (0) = V ′ ( ∞ ) − Z ∞ V ′′ ( r ) f n ( r ) dr, (13)where f n ( r ) = Z r u n ( r ′ ) dr ′ . (14)Note that f n ( ∞ ) = 1 because the radial wavefunction is normalized. The term V ′ ( r ) f n ( r ),from the integration by parts at r = ∞ gives V ′ ( ∞ ) while that at r = 0 vanishes. This isbecause V ( r ) is less singular than r − as r tends to 0 while one expects f n ( r ) ∼ r as r tends to 0 because u n ( r ) ∼ r . Physically, f n ( r ) represents the probability of finding theparticle (two-body system) between 0 and r .To prove that S n (0) − S m (0) > n < m , we appeal to the virial theorem. For S -wavelevels it states: 5 n = h U ( r ) i n = Z ∞ U ( r ) u n ( r ) dr, (15)where U ( r ) = V ( r ) + 12 rV ′ ( r ) . (16)So, if U ( ∞ ) is finite, then an integration by parts gives: − [ E n − E m ] = Z ∞ U ′ ( r )[ f n ( r ) − f m ( r )] dr > . (17)For n < m , the left hand side is always positive, so the integral is positive. Now, f n ( r ) isthe probability of finding the bound particle between 0 and r in the eigenstate with energy E n . Physically, we expect that for all r : f n ( r ) ≥ f m ( r ) , for n < m. (18)To check this out for the Coulomb potential, V ( r ) = − ( e/r ), the f n ( r ) ( n = 1 , , ,
4) areplotted in Fig. 1. Thus, it is clear that if U ′ ( r ) > r , then the Virial theorem [9] issatisfied because f n > f m for n < m . For example, the power law potentials, V ( r ) = −| λ | r α with − < α < U ′ ( r ) > V ′′ ( r ) < r .Given the above, from Eq. (13) it follows that: C ( µ ) [ S n (0) − S m (0)] = − Z ∞ V ′′ ( r )[ f n ( r ) − f m ( r )] dr > , (19)for n < m since V ′′ ( r ) < r . This gives the first inequality in Eqs. (3).For the variation with respect to the reduced mass, we note that with increasing µ , thebounded system will shrink in size. So, physically one expects that f n ( r ) will increase, thatis, ∂ [ f n ( r )] /∂µ >
0. Thus, taking the derivative with respect to µ of Eq. (13), since V ′ ( ∞ )is a constant, gives the second inequality in Eqs. (3). Case (c). V ′′ ( r ) > V ′ ( r ) > r , with V ′ (0) finite. In this case we perform a slightly different integration by parts in Eq. (10) to obtain: C ( µ ) S n (0) = V ′ (0) + Z ∞ V ′′ ( r ) g n ( r ) dr, (20)6here g n ( r ) = Z ∞ r u n ( r ) dr = 1 − f n ( r ) . (21)The term V ′ ( r ) g n ( r ), from the integration by parts, gives V ′ (0) for r = 0. For the upperlimit r → ∞ it vanishes because u n ( r ) represents a bound state.From the above two equations we obtain: C ( µ ) [ S n (0) − S m (0)] = Z ∞ V ′′ ( r )[ g n ( r ) − g m ( r )] dr = − Z ∞ V ′′ ( r )[ f n ( r ) − f m ( r )] dr. (22)Power law potentials V ( r ) = | λ | r α with α > α = 2 gives the isotropic harmonic oscillator which is exactly soluble. Figure 2gives plots of f n ( r ) for n = 0 , , ,
3. The differences f ( r ) − f ( r ), f ( r ) − f ( r ), and f ( r ) − f ( r ), are plotted in Figs. 3–5, respectively. Unlike the Coulomb case, in this casethe differences are slightly negative for small r in a small region near the origin and afterthat they are positive for all r . Even so, the integrals R ∞ r [ f n ( r ) − f m ( r )] dr (in the Virialtheorem, Eq. (17)) and R ∞ [ f n ( r ) − f m ( r )] dr (in Eq. (22)) are positive. Consequently,physically one expects that the integrals R ∞ r α − [ f n ( r ) − f m ( r )] dr and R ∞ r α − [ f n ( r ) − f m ( r )] dr will be positive for power law potentials V ( r ) = | λ | r α with α ≥
2. This issupported by explicit solutions for convex power law potentials with α > V ′′ ( r ) >
0, this analysis leads to S n (0) − S m (0) <
0, the first inequality in Eqs. (4).For the variation with respect to the reduced mass µ , we obtain from Eq. (20): ∂∂µ (cid:20) µ S n (0) (cid:21) < , (23)since the variation with µ of g n ( r ) is opposite to that of f n ( r ). This concludes the proofof the S -wave results given in Sec. I. V. DIMENSIONAL ANALYSIS CONFIRMATION OF THE VARIATION OF µ − S n (0) WITH REDUCED MASS µ . Consider the power law potential 7 ( r ) = λr α , (24)sign of λ is chosen depending on the range of α so that V ′ ( r ) is positive and V ( r ) has boundstates. For example, for the Coulomb potential α = − λ < a , for bound states will depend on λ , ¯ h , and the reduced mass µ ,the parameters in the Schr¨odinger equation. Dimensional analysis gives a ∼ (cid:18) ¯ h | λ | µ (cid:19) α , (25)For Coulomb case λ = − e , α = −
1, so a is just the Bohr radius.Since the wavefunction is normalized S n (0) has dimensions of (length) − so, dimensionally,1 µ S n (0) ∼ µ (cid:18) | λ | µ ¯ h (cid:19) α , (26)this formula gives the required dependence on µ for the various power law potentials. Weapply it to the cases treated earlier. Case (a).
For the linear potential λ > α = 1, there is no µ dependence, so ∂∂µ (cid:20) µ S n (0) (cid:21) = 0 , (27)in agreement with Eqs. (2) and (11). Case (b).
For potentials where V ( r ) = −| λ | r α with − < α <
0, the power of µ in Eq. (26) ispositive, in accord with Eq. (3). Case (c).
For potentials where V ( r ) = | λ | r α with 1 < α , the power of µ in Eq. (26), (1 − α ) / (2 + α ),is negative, in accord with Eq. (4). 8 I. CONCLUDING REMARKS
The proofs of the S -wave results presented above have appealed to the physical meaningof the quantities involved and how they are expected to change physically with the energyof the bound state (or n ) and the reduced mass µ . All known soluble examples of attractivepotentials with curvature of the same sign for all r support the results given in the intro-duction. Such potentials imply that the bound states lie in a single potential well. This isimportant for the physical arguments presented here. A potential with more than a singlewell cannot possibly have curvature of the same sign everywhere. There are lot of solvablepotentials for S -waves [8] without definite curvature for which the results given in the Intro-duction may or may not hold. It would be an interesting challenge to find a counter exampleto the inequalities presented in this work. Acknowledgments
It is a pleasure to thank Antonio Bouzas and Andr´es G. Saravia for discussions and help.The authors would like to thank CONACyT (M´exico) for partial support. [1] C. Quigg and J. L. Rosner, “Quantum Mechanics with Applications to Quarkonium”, Phys.Rep. , 167 (1979). This is a compendium of many interesting and instructive results.[2] A. Martin, Phys. Lett. B , 192 (1977). For related work see A. Martin, Phys. Lett. B ,330 (1977) and H. Grosse, Phys. Lett. B , 343 (1977) and references therein.[3] J. L. Rosner, C. Quigg, and H. B. Thacker, Phys. Lett. B , 350 (1978).[4] V. Gupta and R. Rajaraman, Phys. Rev. D , 697 (1978).[5] H. J. W. M¨uller-Kirsten and S. K. Bose, J. Math. Phys. , 2471 (1979) and references therein.[6] V. Gupta and A. Khare, unpublished (1977). It is mentioned for ℓ = 1 in: A. Khare, Nucl.Phys. B , 533 (1979).[7] S. K. Bose, A. Jabs, and H. J. W. M¨uller-Kirsten, Phys. Rev. D , 1489 (1976) and referencestherein.[8] A. Galindo and P. Pascual, Quantum Mechanics I, p. 254 (Springer-Verlag, Berlin, 1990).
9] Virial theorem holds for more complicated potentials, here we are concerned with a limitedclass of power law potentials for which V ′ ( r ) and V ′′ ( r ) have the same sign for all r .
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FIG. 1: First four lowest S -wave levels probabilities, Eq. (14), for the Coulomb potential, showingthat f n ( r ) − f m ( r ) > n < m and all r . FIG. 2: First four lowest S -wave levels probabilities, Eq. (14), for the Harmonic Oscillator potential,showing that in this case f n ( r ) − f m ( r ) ≥ n < m and all r except for a small region nearthe origin. r FIG. 3: The difference f ( r ) − f ( r ) for the Harmonic Oscillator potential, showing that in thiscase f n ( r ) − f m ( r ) is slightly negative for small r in a small region near the origin and after thatit is positive for all r for n < m . r FIG. 4: The difference f ( r ) − f ( r ) for the Harmonic Oscillator potential, showing that in thiscase f n ( r ) − f m ( r ) is slightly negative for small r in a small region near the origin and after thatit is positive for all r for n < m . The negative region is smaller as n increases. r FIG. 5: The difference f ( r ) − f ( r ) for the Harmonic Oscillator potential, showing that in thiscase f n ( r ) − f m ( r ) is slightly negative for small r in a small region near the origin and after thatit is positive for all r for n < m . The negative region is smaller as n increases.increases.