Abstract
We consider the effect of space dimension N, on the results predicted by two approximation techniques applied on physical quantum systems. In the first, we apply degenerate perturbation theory to perturbed N-dimensional infinite cubical well. It is found that the energy difference for splitting decreases as N increases and it vanishes in the infinite dimensional space. In the second, we apply the sudden approximation to the electronic structure change implied by beta-decay of the tritium nucleus. It is found that as N increases the ionization probability increases.
Keywords:
1 Introduction
Approximation methods are of great importance and widely used in quantum mechanics and have been enormously successful in obtaining approximate solutions to quantum systems. These methods, for both stationary problems and time-dependent problems, are usually applied to quantum systems in space dimension N
⩽ 3. Over the past two decades, the generalization of the three-dimensional problems to higher dimensions received a considerable development mathematical and theoretical physics. For example, the N-dimensional analogy of the hydrogen atom has been investigated extensively over the years (Avery and Herschbach, Citation1992; Kirchberg et al., Citation2003; Saelen et al., Citation2007"). In addition, the generalization to higher dimensions is useful in random walks (Mackay et al., Citation2002), in Casimir effect (Bender et al., Citation1992), in harmonic oscillator (Oyewumi et al., Citation2008; Al-Jaber, Citation2008; Rothos et al., Citation2009"), and in mathematical physics (Bredies, Citation2009; Szmytkowski, Citation2007; Bouda and Ghabri, Citation2008; Golovnev, Citation2006"). It is interesting to expose undergraduate students to simple, but illustrative, quantum systems in higher space dimensions. Therefore, it is tempting to apply specific approximation methods to certain quantum systems in N-dimensional space. It must be emphasized that our intention herein is not to find approximate solutions but rather to expose the reader to higher dimensional problems and to examine the effect of the space dimension on the results of two well-known approximation methods in quantum mechanics. We choose two approximation methods, one for stationary quantum system and another for time-dependent one. For the first method, we consider the time-independent degenerate perturbation theory applied to N-dimensional infinite cubical well that is perturbed with some perturbation, . The reason behind the selection of degenerate perturbation theory lies on its wide recent applications in various settings: In studying degeneracy breaking due to short-ranged impurities in atomically doped carbon nanotubes (Bondarev and Lambin, Citation2005; McCann and Fal’ko, Citation2005"), in investigating the effects of magnetic anisotropy in magnetic molecules (Kostantinidis and Coffey, Citation2002; Meng and Wessel, Citation2008; Bialynicki and Sowinski, Citation2007; Bergman et al., Citation2007"), in discussing quantum systems coupled to oscillators (Muthukumar and Mitra, Citation2002; Hagelstein et al., Citation2008"), in dealing with many-body interacting systems (Stauber et al., Citation2000; Morawetz, Citation2002"), in studying degenerate atomic systems (Vilain et al., Citation2001), and in calculating relativistic corrections for low-lying excited states when a hydrogen atom is placed in a strong magnetic field (Poszwa and Rutkowski, Citation2004). For the second method we choose the sudden approximation and apply it to β-decay of tritium atom. The sudden approximation has been widely used for describing abrupt change in the Hamiltonian of the system. For example, some workers used the sudden approximation in calculating photo-ionization cross-section of atoms induced by strong fields (Kazansky and Kabachnik, Citation2007; Yudin et al., Citation2007"). Others applied this method in the study of the multiple electron loss of heavy projectiles in fast ion-atom collisions (Matveev et al., Citation2009; Voitkiv et al., Citation1999; Salop and Eichler, Citation1979; Richards, Citation1981; Kapusta and Mocsy, Citation1999"). In addition, there has been an investigation, using sudden approximation, on how hydrogen atoms respond to perturbation by intense ultrafast laser pulses of duration shorter than the inverse of the initial-state energy (Lugovskoy and Bray, Citation2006; Lugovskoy and Bray, Citation2005"). More interestingly, the sudden approximation has been used in the calculations of transition probabilities of electronic excitations resulting from β-decay in atomic processes (Frolov and Talman, Citation2005; Saenz and Froelich, Citation1997; Froelich and Saenz, Citation1996; Wauters and Vaeck, Citation1996; Claxton et al., Citation1992"). The organization of the present paper goes as follows: In this section, we give an introduction. Section 2 deals with degenerate perturbation theory applied to a perturbed N-dimensional infinite cubical well. In Section 3, the sudden approximation method is applied to the process of β-decay of tritium atom in N dimensions. Section 4 is devoted for results and conclusions.
2 Perturbed N-dimensional infinite cubical well
We consider an infinite cubical well of side a in N space dimensions whose potential is given by(1) The stationary states solutions for Schrödinger equation are
(2) with
a positive integer for all values of i. The corresponding allowed energies are
(3) Notice that the ground state
is non-degenerate with energy
(4) Since we are interested in the effect of space dimension on splitting degenerate states, we need to consider the first excited state which is N degenerate, namely
(5) all have the same (unperturbed) energy, which is
(6) We introduce the perturbation
(7) We now construct the matrix W, whose elements are the matrix elements
of
with respect to the unperturbed wave functions
, with
(8)
(9) The diagonal elements are
(10)
(11) While the off-diagonal elements are calculated as follows:
Since the
integral in the above equation vanishes, we have
(12) The other off-diagonal elements are
(13) Eqs. Equation(11)
(11) and Equation(13)
(13) reduce to the three-dimensional values (N
= 3), namely
and
, respectively (Griffiths, Citation1995). Defining
and with the help of Eqs. (11)–(13) we are able to construct the matrix W as
(14) The eigen-values,
are easier to find for the matrix
rather than W. This is achieved by setting
(15) The characteristic equation of Eq. Equation(15)
(15) can be found by using (Hohn, Citation1973)
(16) The identification
and
, yields the characteristic equation for Eq. Equation(15)
(15) , with the result
(17) The roots of the above equation are
, and
, with
being
degenerate.
Therefore, the energies, to first order, of the three distinct energy levels are(18) where
is the (common) unperturbed energy given by Eq. Equation(6)
(6) . We note that the perturbation
lifts partially the degeneracy N to
, and splits
into three distinct energy levels. It is interesting to note that the amount of splitting decreases as the space dimension N increases and it vanishes in the infinite dimensional space
. This can be easily checked by taking the limit of the second terms in Eq. Equation(18
(18) ) as
. This shows that
becomes weaker and hence the amount of splitting becomes smaller as the space dimension gets higher.
3 Sudden approximation for β-decay in N dimensions
In order to investigate the effect of space dimension on the results of the sudden approximation method, we consider an N-dimensional tritium atom that consists of a nucleus (containing one proton and two neutrons) and an electron. It is an unstable nucleus, since by β emission it decays into the nucleus
(containing two protons and one neutron), an electron
and an antineutrino
via the reaction
. The purpose herein is to investigate the influence of β decay on the atomic electron in N-dimensional space. We assume that initially the tritium atom is in the ground state. In β decay process, the emitted electron from the tritium nucleus has an energy which, in most cases, is of the order of several keV. Therefore, the velocity v of the emitted electron is usually much larger than the velocity
of the atomic electron in the ground state of tritium. Thus, if the
denotes the first Bohr radius, the β-electron will traverse the atom in time
, which is much less than the periodic time
of the circulating atomic electron. Therefore, when tritium nucleus
decays by beta emission into
, the nuclear charge ‘seen’ by the atomic changes instantaneously from ze to
, with
and
. For time
, the initial
atom has the Hamiltonian
(19) and for
, the final
atom has the Hamiltonian
(20) The eigen functions of
and
are the hydrogenic wave functions corresponding to
and
, respectively. The tritium atom is initially in the ground state
and thus the probability amplitude
of finding the atomic electron in a discrete eigenstate
of the
hydrogenic Hamiltonian
at
is
(21) The hydrogenic wave function is
(22) where
are the normalized radial hydrogenic wave function in N dimensions and
are the hyperspherical harmonics that form orthonormal set, namely
(23) The substitution of Eq. Equation(22
(22) ) into Eq. Equation(21
(21) ) and the use of Eq. Equation(23
(23) ) show that the only non-vanishing probability amplitudes
are those corresponding to s-states
, therefore
(24) The radial solutions are given by (Al-Jaber, Citation1998)
(25) where
, with
are the associated Laguerre polynomials and the normalization constant
is given by
(26) For
and
, Eqs. Equation(25)
(25) and Equation(26
(26) ) become
(27)
(28) with
. Using
and substituting Eqs. Equation(27)
(27) and Equation(28
(28) ) into Eq. Equation(24
(24) ), we get
(29) Upon substituting
and
in the constants
and
we get
(30) and with the help of the formula
(31) the integral in Eq. Equation(29
(29) ) yield
, where
is the gamma function. Therefore, Eq. Equation(29
(29) ) becomes
which simplifies to
(32) The above equation clearly shows the dependence of the probability amplitude on the space dimension N, and it reduces to its well-known value for the three-dimensional case (Bransden and Joachain, Citation2000), namely
for N
= 3.
The probability, that the
ion be found in its ground state is
(33) which gives the expected result [42],
for N
= 3. The total probability for excitation and ionization of the
ion is therefore
(34) Eqs. Equation(33)
(33) and Equation(34
(34) ) clearly show that as the space dimension N increases
decreases (increases), and if
, then
. This means that when tritium atom undergoes a β-decay, the higher the space dimension the higher the probability of excitation and ionization of
ion and the lower to be in the ground state.
4 Conclusions and results
In the present paper, the role of the space dimension N on the results obtained by the approximation methods has been investigated. This gives an opportunity for involving students in an in-depth study of quantum systems in higher space dimensions, To that end, two approximation methods were applied to quantum systems in N-dimensional space. In the first method, degenerate perturbation theory was employed for a perturbed N-dimensional infinite cubical well. It was found that the perturbation lifts partially the degeneracy N, of the unperturbed system, to (N
− 2). It was also observed that the perturbation splits the unperturbed energy
, into three distinct energy levels with one of them being (N
− 2) degenerate. It is remarkable to notice that the amount of energy difference for splitting decreases as the space dimension increases and it vanishes in the infinite dimensional space. The decrease of the matrix elements with N seems to be due to the multiplicative effect of factors smaller than 1, the number of which increases with N. In the second method, the sudden approximation method was applied to the process of β-decay of tritium atom in N dimensions. Assuming that the tritium atom is initially in the ground state, it was found that the probability that the
ion to be in the ground state depends on the space dimension N. In addition, this probability decreases as N increases and it vanishes as
. This means that the total probability of excitation and ionization increases as N increases and approaches unity in the infinite dimensional space. One may argue that the results come from the fact that as N increases, the eigenvalues of the attractive 1/r potential get closer to zero. Thus, the ionization becomes easier, and it becomes even a spontaneous transition in the limit of infinite N.
Therefore, our results clearly show the effect of the space dimension on the results predicted by approximation methods when applied to quantum systems.
References
- S.M. Al-Jaber Hydrogen atom in N dimensions Int. J. Theor. Phys. 37 1998 1289 1298
- S.M. Al-Jaber A confined N-dimensional harmonic oscillator Int. J. Theor. Phys. 47 2008 1853 1864
- J. Avery D.R. Herschbach Hyperspherical strumian basis functions Int. J. Quantum Chem. 41 1992 673 686
- C.M. Bender S. Boethtcher L. Lipatov Almost zero-dimensional quantum field theories Phys. Rev. D 46 1992 5557 5573
- D.L. Bergman R. Shindou G.A. Fiete L. Balents Degenerate perturbation theory of quantum fluctuations in a pyrochlore antiferromagnet Phys. Rev. B 75 2007 094403 094433
- B.I. Bialynicki T. Sowinski Quantum electrodynamics of qubits Phys. Rev. A 76 2007 062106 062128
- I.V. Bondarev Ph. Lambin Van der Waals coupling in atomically doped carbon nanotubes Phys. Rev. B 72 2005 035451 035465
- A. Bouda A. Ghabri Quantum law of motion: analysis and extention to higher dimensions Int. J. Theor. Phys. 47 2008 1068 1078
- B.H. Bransden C.J. Joachain Quantum Mechanics 2000 Prentice-Hall Publisher London, England
- K. Bredies A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in banach Inverse Problem 25 2009 015005 015024
- T.A. Claxton S. Schafroth P.F. Meier Molecular final states after β decay of tritium-substituted molecules Phys. Rev. A 45 1992 6209 6216
- P. Froelich A. Saenz Calculation of the β decay spectrum of the T2 molecule beyond the sudden impulse approximation Phys. Rev. Lett. 77 1996 4724 4727
- A.M. Frolov J.D. Talman Final-state probabilities for -decaying light atoms Phys. Rev. A 72 2005 022511 022516
- V. Golovnev Thin layer quantization in higher dimensions and co- dimensions J. Math. Phys. 47 2006 082105 082114
- D.J. Griffiths Introduction to Quantum Mechanics 1995 Prentice-Hall Publisher New Jersey, United States
- P.L. Hagelstein I.U. Chaudhary Excitation transfer in two two-level systems coupled to an oscillator J. Phys. B: At. Mol. Opt. Phys. 41 2008 135501 135510
- F.E. Hohn Elementary Matrix Algebra 1973 Collier–Macmillan Publisher London, England
- J. Kapusta A. Mocsy Hydrogen like atoms from ultrarelativistic nuclear collisions Phys. Rev. C 59 1999 2937 2940
- A.K. Kazansky N.M. Kabachnik An attosecond time-resolved study of strong field atomic photoionization J. Phys. B: At. Mol. Phys. 40 2007 F299 F305
- A. Kirchberg J.D. Lange A. Wipf Algebraic solution of the super-symmetric hydrogen atom in d dimensions Ann. Phys. 303 2003 359 388
- N.P. Kostantinidis D. Coffey Magnetic anisotropy in the molecular complex V15 Phys. Rev. B 66 2002 174426 174434
- A.V. Lugovskoy I. Bray Sudden perturbation of hydrogen atoms by intense ultrashort laser pulses Phys. Rev. A 72 2005 063402 063410
- A.V. Lugovskoy I. Bray Interaction of Rydberg atoms with two contrapropagating ultrashort laser pulses Phys. Rev. A 73 2006 063401 063408
- T.D. Mackay S.D. Bartlett L.T. Stephensen B.C. Sanders Quantum walks in higher dimensions J. Phys. A: Math. Gen. 35 2002 2745 2754
- V.I. Matveev S.V. Ryabchenko D.U. Matrasulov K.Y. Rakhimov S. Fritzsche T. Stöhlker Electron loss of fast heavy projectiles in collision with neutral targets Phys. Rev. A 79 2009 042710 042717
- E. McCann I.V. Fal’ko Degeneracy breaking and intervally scattering due to short-ranged impurities in finite single-wall carbon nanotubes Phys. Rev. B 71 2005 085415 085428
- Z.I. Meng S. Wessel Phases and magnetization process of an anisotropic Shastry–Sutherland model Phys. Rev. B 78 2008 224416 224427
- K. Morawetz Relation between classical and quantum particle systems Phys. Rev. E 66 2002 022103 022106
- B. Muthukumar P. Mitra Noncommutative oscillators and the commutative limit Phys. Rev. D 66 2002 027701 027703
- K.J. Oyewumi F.O. Akinpelu A.D. Ayboola Exactly complete solutions of the pseudoharmonic potential in N-dimensions Int. J. Theor. Phys. 47 2008 1039 1057
- A. Poszwa A. Rutkowski Hydrogen atom in a strong magnetic field. II. Relativistic correction for low-lying excited states Phys. Rev. A 69 2004 023403 023408
- D. Richards A sudden approximation for heavy-particle collisions J. Phys. B: At. Mol. Phys. 14 1981 1465 1476
- V.M. Rothos H.E. Nistazakis P.G. Kevyebidis D.J. Frantzeskakis Stability of localized structures in generalized DNLS equations near the anti-continuum limit J. Phys. A: Math. Theor. 42 2009 025207 025219
- L. Saelen R. Nepstad J.P. Hansen L.B. Madsen The N-dimensional Coulomb problem: stark effect in hyperspherical coordinates J. Phys. A: Math. Theor. 40 2007 1097 1104
- A. Saenz P. Froelich Effect of final-state interactions in allowed β decays. II. Reliability of the β decay spectrum for T2 Phys. Rev. C 56 1997 2162 2184
- A. Salop J. Eichler Sudden approximation cross-sections for ionization of H atoms by energetic C6+ and He2+ impact J. Phys. B: At. Mol. Phys. 12 1979 257 264
- T. Stauber R. Zimmermann H. Castella Electron phonon interaction in quantum dots: a solvable model Phys. Rev. B 62 2000 7336 7343
- R. Szmytkowski Closed forms of the Green’s function and the generalized Green’s function for the Helmholtz operator in the N-dimensional unit sphere J. Phys. A: Math. Theor. 40 2007 995 1010
- P. Vilain P. Öhberg L. Santos A. Sanpera M. Lewenstein Four-wave mixing in degenerate atomic gases Phys. Rev. A 64 2001 023606 023610
- A.B. Voitkiv G.M. Sigaud E.C. Montenegro Nonperturbative theory of projectile-electron loss in fast collisions with heavy atomic targets Phys. Rev. A 59 1999 2794 2803
- L. Wauters N. Vaeck Study of the electronic rearrangement induced by nuclear transmutations: a B-spline approach applied to the β decay of 6He Phys. Rev. C 53 1996 497 500
- G.L. Yudin S. Patchkovskii P.B. Corkum A.D. Bandrauk Attosecond photoelectron interference in the separable Coulomb–Volkov continuum J. Phys. B: At. Mol. Opt. Phys. 40 2007 F93 F103