Abstract
In contrast to the nondegeneracy theorem, we present various scenarios in one-dimensional quantum mechanics that demonstrate how the Wronskian of two bound-state eigenfunctions with the same energy eigenvalue can be zero without implying that the eigenfunctions are linearly dependent. It is shown that the nondegeneracy theorem fails only when the potential makes different bound-state solutions corresponding to the same energy vanish at the singular point or region of singularity.
Keywords:
Nondegeneracy theorem; Two-fold degeneracy; Linearly independent eigenfunctions
The one-dimensional time-independent Schrödinger equation
requires continuous eigenfunctions. As a second-order differential equation, it has two linearly independent solutions that correspond to the same energy eigenvalue. However, the occurrence of the two-fold degeneracy is rare in describing one-dimensional bound states. Indeed, the absence of degenerate one-dimensional bound states is ensured by the nondegeneracy theorem (see, e.g. §21 in [1[1] L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1958).], Theorem 15 in [2[2] R. Shankar, Principles of Quantum Mechanics (Plenum Press, New York, 1994).], and also Problem 2.42 in [3[3] D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Upper Saddle River, 1995).] with the proviso that “the potential does not consist of isolated pieces separated by regions where ”). Motivated primarily by justifying the presence of degenerate bound states for the one-dimensional hydrogen atom [4[4] S. Flügge and H. Marschall, Rechenmethoden de Quantentheorie (Spring-Verlag, Berlin, 1952) p. 69.], Loudon revisited the nondegeneracy theorem and properly concluded that it is not necessarily valid for a potential with singular points [5[5] R. Loudon, Am. J. Phys. 27, 649 (1959).]. Since then a lot of controversy surrounds that problem [6[6] M. Andrews, Am. J. Phys. 34, 1194 (1966)., 7[7] L.K. Haines and D.H. Roberts, Am. J. Phys. 37, 1145 (1969)., 8[8] M. Andrews, Am. J. Phys. 44, 1064 (1976)., 9[9] J.F. Gomes and A.H. Zimerman, Am. J. Phys. 48, 579 (1980)., 10[10] M. Andrews, Am. J. Phys. 49, 1074 (1981)., 11[11] J.F. Gomes and A.H. Zimerman, Am. J. Phys. 49, 579 (1981)., 12[12] L.S. Davtyan, G.S. Pogosyan, A.N. Sissakian and V.M. Ter-Antonyan, J. Phys. A 20, 2765 (1987)., 13[13] H.N. Nuñez-Yepez, C.A. Vargas and A.L. Salas-Brito, Eur. J. Phys. 8, 189 (1987)., 14[14] M. Andrews, Am. J. Phys. 56, 776 (1988)., 15[15] U. Oseguera, Eur. J. Phys. 11, 35 (1990)., 16[16] J.M. Cohen and B. Kuharetz, J. Math. Phys. 34, 12 (1993)., 17[17] U. Oseguera and M. de Llano, J. Mat. Phys. 34, 4575 (1993)., 18[18] K. Bhattacharyya and R.K. Pathak, Int. J. Quantum Chem. 59, 219 (1996)., 19[19] A.N. Gordeyev and S.C. Chhajlany, J. Phys. A 30, 6893 (1997)., 20[20] S. Kar and R.R. Parwani, Eur. Phys. Lett. 80, 30004 (2007).]. Recently, the possibility of double degeneracy for a particle in a box has also been explored [21[21] S. De Vincenzo, Braz. J. Phys. 38, 355 (2008).].
The usual proof of the nondegeneracy theorem considers the Wronskian of two eigenfunctions and corresponding to the same energy eigenvalue:
For bound states, because and vanish for large , so
Some authors simply divide (3) by , disregarding the zeros of and , to obtain
Therefore,
It follows that
where is an arbitrary constant.
Loudon has cast doubt on the validity of (4) by drawing attention to possible troubles in the regions or at the points where the eigenfunctions have zeros [5[5] R. Loudon, Am. J. Phys. 27, 649 (1959).]. To further elaborate on this problem we examine the behaviour of the eigenfunctions in the vicinity of a possible singular point , where we assume linear dependence on each side of the point:
We segregate the problem into two classes of eigenfunctions based on the behaviour of .
-
Class I: .
For this class, is also not equal to zero. As a result, we have
which implies that and are linearly dependent functions. To better understand what happens with the logarithmic derivatives, we substitute (7) into (3) and find that is indeterminate. Here, indicates the limit of as approaches from . By setting into (7), we obtain
which states that the logarithmic derivatives are continuous functions at . This always occurs when the potential is regular at . However, if then
Note that the logarithmic derivative in this last case can be integrated across as in (5), despite its jump discontinuity. This scenario typically occurs when the potential gives a dominant contribution proportional to at (see, e.g. [3[3] D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Upper Saddle River, 1995)., 16[16] J.M. Cohen and B. Kuharetz, J. Math. Phys. 34, 12 (1993)., 22[22] C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977), v. 1.]).
-
Class II: .
In this class, and there is no logical connection between and as in Class I. Furthermore, Eq. (3) does not establish a connection between the first derivatives of and at the right and at the left of . The logarithmic derivative is also meaningless for this class. This situation, with for independent of for , arises from an infinite potential at the right or at the left of . Examples of this include the problem of an infinite double well and also in the problem of a finite double well as the barrier width tends to infinity (see, e.g. Problem 2.44 in [3[3] D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Upper Saddle River, 1995).], Complement G in 22[22] C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977), v. 1., Sec. 6.6 in [23[23] J.M. Lévy-Leblond and F. Balibar, Quantics: Rudiments of Quantum Physics (North-Holland, Amsterdam, 1990).], and also Sec. 8.3.9 in [24[24] K.K. Wan, From Micro to Macro Quantum Systems (Imperial College Press, London, 2006).]). Two-fold degenerate spectra also appear for the potentials proportional to (see, e.g. [25[25] D.R.M. Pimentel and A.S. de Castro, Rev. Bras. Ens. Fis. 35, 3303 (2013).]) and (see, e.g. [26[26] D.R.M. Pimentel and A.S. de Castro, Rev. Bras. Ens. Fis. 36, 1307 (2014).]). For example, if we have
then the two-fold degeneracy can appear if the potential dominates at (with for , and for ) [26[26] D.R.M. Pimentel and A.S. de Castro, Rev. Bras. Ens. Fis. 36, 1307 (2014).], a conclusion that differs from that one found in Ref. [5[5] R. Loudon, Am. J. Phys. 27, 649 (1959).]. A two-fold degenerate spectrum is also seen for a Dirac delta potential embedded in a box in the strong coupling limit [16[16] J.M. Cohen and B. Kuharetz, J. Math. Phys. 34, 12 (1993).].
A necessary condition for linear independence of two functions and is that the Wronskian does not vanish. With one obtains but the converse is not necessarily true. If and is infinitely differentiable at with , it is easy to conclude that is proportional to . Differentiating (3) repeatedly yields
Thus,
Even if and , we find , according to the theory of distributions. However, if , then the process of repeated differentiation can not establish a connection between the th order derivatives of and . This implies that we can not connect for and for . In other words, we can not establish whether and are linearly dependent. Therefore, we can not ruled out any possibility of a two-fold degeneracy.
In conclusion, we presented in a straightforward way that the vanishing of the Wronskian of two bound-state eigenfunctions does not guarantee their linear dependence and that the zeros of the eigenfunctions may cause the nondegeneracy theorem to fail, as noted by Loudon [5[5] R. Loudon, Am. J. Phys. 27, 649 (1959).]. Additionally, we presented fair scenarios that illustrate two-fold degeneracies for bound states in one-dimensional quantum mechanics. The nondegeneracy theorem fails only when the potential has the pathological feature that makes different bound-state solutions corresponding to the same energy vanish at the singular point or region. In this case, the lack of connection between the derivatives of those eigenfunctions on each side of the singular point leads us to agree with Andrews [8[8] M. Andrews, Am. J. Phys. 44, 1064 (1976)., 10[10] M. Andrews, Am. J. Phys. 49, 1074 (1981)., 14[14] M. Andrews, Am. J. Phys. 56, 776 (1988).]: the singular point or region acts as an impenetrable barrier.
Acknowledgements
Grants 149369/2021-2 and 09126/2019-3, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil. The authors express their gratitude to the referee for providing valuable comments and suggestions to improve the previous version of the manuscript.
References
-
[1]L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1958).
-
[2]R. Shankar, Principles of Quantum Mechanics (Plenum Press, New York, 1994).
-
[3]D.J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, Upper Saddle River, 1995).
-
[4]S. Flügge and H. Marschall, Rechenmethoden de Quantentheorie (Spring-Verlag, Berlin, 1952) p. 69.
-
[5]R. Loudon, Am. J. Phys. 27, 649 (1959).
-
[6]M. Andrews, Am. J. Phys. 34, 1194 (1966).
-
[7]L.K. Haines and D.H. Roberts, Am. J. Phys. 37, 1145 (1969).
-
[8]M. Andrews, Am. J. Phys. 44, 1064 (1976).
-
[9]J.F. Gomes and A.H. Zimerman, Am. J. Phys. 48, 579 (1980).
-
[10]M. Andrews, Am. J. Phys. 49, 1074 (1981).
-
[11]J.F. Gomes and A.H. Zimerman, Am. J. Phys. 49, 579 (1981).
-
[12]L.S. Davtyan, G.S. Pogosyan, A.N. Sissakian and V.M. Ter-Antonyan, J. Phys. A 20, 2765 (1987).
-
[13]H.N. Nuñez-Yepez, C.A. Vargas and A.L. Salas-Brito, Eur. J. Phys. 8, 189 (1987).
-
[14]M. Andrews, Am. J. Phys. 56, 776 (1988).
-
[15]U. Oseguera, Eur. J. Phys. 11, 35 (1990).
-
[16]J.M. Cohen and B. Kuharetz, J. Math. Phys. 34, 12 (1993).
-
[17]U. Oseguera and M. de Llano, J. Mat. Phys. 34, 4575 (1993).
-
[18]K. Bhattacharyya and R.K. Pathak, Int. J. Quantum Chem. 59, 219 (1996).
-
[19]A.N. Gordeyev and S.C. Chhajlany, J. Phys. A 30, 6893 (1997).
-
[20]S. Kar and R.R. Parwani, Eur. Phys. Lett. 80, 30004 (2007).
-
[21]S. De Vincenzo, Braz. J. Phys. 38, 355 (2008).
-
[22]C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977), v. 1.
-
[23]J.M. Lévy-Leblond and F. Balibar, Quantics: Rudiments of Quantum Physics (North-Holland, Amsterdam, 1990).
-
[24]K.K. Wan, From Micro to Macro Quantum Systems (Imperial College Press, London, 2006).
-
[25]D.R.M. Pimentel and A.S. de Castro, Rev. Bras. Ens. Fis. 35, 3303 (2013).
-
[26]D.R.M. Pimentel and A.S. de Castro, Rev. Bras. Ens. Fis. 36, 1307 (2014).
Publication Dates
-
Publication in this collection
29 May 2023 -
Date of issue
2023
History
-
Received
18 Jan 2023 -
Reviewed
20 Mar 2023 -
Accepted
24 Apr 2023