Journal of Physics: Conference Series Related content An approach to quantum anharmonic oscillators via Lie algebra To cite this article: M Jafarpour and D Afshar 2008 J. Phys.: Conf. Ser. 128 012055 View the article online for updates and enhancements. - Comment Francisco M Fernández - Anharmonic oscillators F J Gómez and J Sesma - On the growth of almost soluble Lie algebras S G Klement'ev and V M Petrogradskii Recent citations - A Closed Form Solution for Quantum Oscillator Perturbations Using Lie Algebras Clark Alexander This content was downloaded from IP address 207.241.231.80 on 23/07/2018 at 05:38 V International Symposium on Quantum Theory and Symmetries Journal of Physics: Conference Series 128 (2008) 012055 IOP Publishing doi:10.1088/1742-6596/128/1/012055 An approach to quantum anharmonic oscillators via Lie algebra M Jafarpour and D Afshar Physics Department, Shahid Chamran University, Ahvaz, Iran E-mail: mojtaba jafarpour@hotmail.com and da afshar@yahoo.com Abstract. We present a new Lie algebraic method to study the eigenvalues and eigenfunctions of quantum anharmonic oscillators. We consider the Hamiltonians for the simple harmonic and anharmonic oscillator as the two generators of a Lie algebra, whose other generators may be found exactly or up to any order of the parameter involved. Specifically, the closed commutator algebra for the quartic anharmonic oscillator is established in a perturbation sense. An element of this Lie group, turning out to be the four-photon operator, transforms the quartic anharmonic oscillator Hamiltonian to the harmonic one in a perturbation sense; thus, facilitating the calculation of the eigenvalues and eigenfunctions of the former. 1. Introduction Simple harmonic oscillator is an idealized model to describe many phenomena in physics and chemistry. Anharmonic oscillator is a deviation from this idealized model to a realistic one. Rayleigh-Schrödinger perturbation theory has been widely used, providing the energy of an anharmonic oscillator, as a formal power series of the perturbation parameter involved [1, 2, 3, 4]. However, the power series diverges even for small coupling constants; thus, appropriate techniques must be applied to alleviate this problem [5, 6, 7, 8, 9, 10, 11, 12]. Several alternative approaches have also been developed; among them, a Lie algebraic method using canonical transformations [13], multiple scale method [14, 15, 16], quasilinearization method [17, 18], a variational method based on the squeezed states [19, 20] and Naundorf method [21] are mentioned. In this work, we use a Lie algebra method to relate the Hamiltonian of an anharmonic oscillator to that of a harmonic one, in a perturbation sense. This is achieved via a canonical transformation, which is unique to the structure of the specific Hamiltonian under study. We introduce our method in section 2, where we apply it to two prototype trivial examples: a harmonic oscillator with a linear perturbation term and one with a quadratic perturbation term that just brings about a frequency shift. In section 3, a unitary transformation is found that relates the Hamiltonian of a quartic anharmonic oscillator to that of a harmonic one. It is observed that the matrix representation of the transformed Hamiltonian is diagonal, up to the first order in the perturbation parameter involved. The eigenvalues of the anharmonic oscillator are evaluated and the results are compared with those obtained from Rayleigh-Schrödinger perturbation theory. Finally, we deal with the discussion and conclusions in section 4. c 2008 IOP Publishing Ltd  1 V International Symposium on Quantum Theory and Symmetries Journal of Physics: Conference Series 128 (2008) 012055 IOP Publishing doi:10.1088/1742-6596/128/1/012055 2. Our Lie algebra method The anharmonic oscillator, described by the Hamiltonian p2 x2 + + λxn , 2 2 Hn = H0 + λxn = is a deviation from the harmonic one, described by the Hamiltonian H0 . generators Li , including L1 = H0 and L2 = Hn , satisfying [Li , Lj ] = X (1) If N hermitian cijk Lk , (2) k form a closed commutator algebra, the operators Li specify a Lie group, whose generators they are. The elements of this group are given by U = exp(−i N X αi Li ), (3) i=1 where αi ’s are real parameters. Now, an element of the group may be found that transforms H0 to Hn . This coming about, the eigenvalues and the eigenfunctions of the anharmonic oscillator can be obtained in terms of those of a harmonic one in a simple manner. Now, let’s assume n = 1 in equation (1); we have H1 = 1 a + a† p2 x2 + + λx = a† a + + λ( √ ), 2 2 2 2 (4) which may be considered as a harmonic oscillator in an external electric field. a and a† are the ordinary the commutator [H0 , H1 ] = √ annihilation and creation operators. Considering † † λ(a − a)/ 2, we realize that the operators H0 , H1 , i(a − a) and the identity operator I form a closed commutator algebra and are the generators of a Lie group. The Glauber operator U1 = exp(αa† − α∗ a), with α assumed real in this section, is an element of this group. The transformation of H0 to H1 is carried out in the following manner H1 = U1† H0 U1 − λ2 , 2 (5) √ where, α has been replaced by λ/ 2 in U1 . Using (5) we find λ2 1 H1 [U1† |nia ] = [(n + ) − ][U1† |nia ] , 2 2 where {|nia } are a-mode number states; meaning that the eigenvalues of H1 are given by λ2 1 , En = (n + ) − 2 2 (6) and the eigenfunctions, called the displaced number states [22] are expressed by U1† |nia = e λ √ (a−a† ) 2 |nia . (7) Now, we consider our second prototype trivial example. The Hamiltonian of a shifted-frequency harmonic oscillator, the case n = 2 in equation (1), is given by H2 = 1 a + a† p2 x2 + + λx2 = a† a + + λ( √ )2 . 2 2 2 2 2 (8) V International Symposium on Quantum Theory and Symmetries Journal of Physics: Conference Series 128 (2008) 012055 IOP Publishing doi:10.1088/1742-6596/128/1/012055 In this case, we have [H0 , H2 ] = λ(a†2 − a2 ); therefore, the Hermitian operators H0 , H2 , i(a†2 − a2 ) and I, form a closed commutator algebra and are the generators of a Lie group. The squeezing operator U2 = exp(βa†2 − β ∗ a2 ), with β assumed real, is an element of this group. The transformation of the fundamental mode operators a and a† into the new ones, b and b† under U2 , is carried out by the following Bogoliubov transformations b = U2† aU2 = µa + νa† , b† = U2† a† U2 = µa† + νa, where, µ = cosh(2β) and ν = sinh(2β). As µ2 −ν 2 = 1, we have also [b, b† ] = 1. The transformed Hamiltonian under U2 is given by H2 = √ √ 1 1 + 2λU2† H0 U2 = 1 + 2λ(b† b + ), 2 (9) where, λ = 2µν/(µ − ν)2 has been assumed. Using equation (9) we find H2 [U2† |nia ] = or √ 1 1 + 2λ(n + )[U2† |nia ] , 2 √ 1 1 + 2λ(n + )|nib , 2 implying that eigenfunctions of the shifted-frequency harmonic oscillator, the b-mode number states, number states [22], are given by |nib = U2† |nia , and the corresponding energies √ or squeezed 1 by 1 + 2λ(n + 2 ). H2 |nib = 3. Quartic anharmonic oscillator Now we consider the quartic anharmonic oscillator. Its Hamiltonian in terms of the fundamental mode operators a and a† , may be given by H4 = a† a + a + a† 1 + λ( √ )4 . 2 2 (10) We can show that [H0 , H4 ] = −iL3 − 3iL4 − 2iL5 , where, (11) L3 = iλ(a†4 − a4 ), L4 = iλ(a†2 − a2 ), L5 = iλ(a†3 a − a† a3 ). Inspecting the mutual commutators between the operators H0 , H4 , L3 , L4 and L5 , we are finally led to prove that H0 , H4 , iλ(a†4 − a4 ), iλ(a†2 − a2 ), iλ(a†3 a − a† a3 ), λ(a†4 + a4 ), λ(a†2 + a2 ), λ(a†3 a + a† a3 ) and I form a closed commutator algebra up to the first order, if the perturbation parameter λ is small, and therefore they are the generators of a Lie group, up to that order. We now focus on the following element of this group, known as the four photon operator [23] U4 = exp[Aλ(a†4 − a4 ) + Bλ(a†2 − a2 ) + Cλ(a†3 a − a† a3 )], 3 (12) V International Symposium on Quantum Theory and Symmetries Journal of Physics: Conference Series 128 (2008) 012055 IOP Publishing doi:10.1088/1742-6596/128/1/012055 where A, B and C are real parameters. This operator transforms the fundamental mode operators, a and a† to the new ones c and c† . Using Baker-Campbell-Hausdorff relation up to the first order, they may be expressed by the following expressions c = U4† aU4 = a + λ(4Aa†3 + 2Ba† + 3Ca†2 a − Ca3 ), † c = U4† a† U4 † 3 † 2 †3 = a + λ(4Aa + 2Ba + 3Ca a − Ca ). (13) (14) It should be emphasized that c and c† also obey the canonical commutation relation [c, c† ] = 1; therefore, we have introduced a new class of Bogoliubov transformations by (13) and (14). Assuming A = 1/16, B = 3/4 and C = 1/2, the four photon operator transforms H0 to H4 , up to the first order in the parameter λ, as follows 3λ 2 3λ 1 = H4 − (Na + Na ) − , 2 2 4 3λ 3λ Na (Na + 1) − , H0 = U4 H4 U4† − 2 4 where, Na = a† a is the normal number operator. Using equation (15), we find U4† H0 U4 = c† c + (15) 1 3λ 3λ + + n(n + 1)][U4† |nia ] . 2 4 2 Thus, the eigenstates of the quartic anharmonic oscillator, our c-mode number states, are given by 3 1 1 †4 4 †2 2 †3 † 3 (16) |nic ≡ U4† |nia = e−λ[ 16 (a −a )+ 4 (a −a )+ 2 (a a−a a )] |nia , H4 [U4† |nia ] = [n + and the first order perturbed energies are expressed by 1 3λ 3λ + + n(n + 1) . (17) 2 4 2 The above results are in complete agreement with those obtained, using first order RayleighSchrödinger perturbation theory [24]. Furthermore, although (16) and (17)are formally derived up to first order in the parameter λ, it is worthwhile to investigate the contribution of higher order terms to the energy using (16). For example, we calculate the expectation value of H4 in the c-mode number states basis up to the fourth order; we find En = n + E0 = 0.5 + 0.75λ − 2.625λ2 + 20.8125λ3 − 104.098λ4 + ... , (18) Bender and Wu also give the following perturbation series, for the ground state energy E0,B , of the quartic anharmonic oscillator which we write down up to the fourth order also [1] E0,B = 0.5 + 0.75λ − 2.625λ2 + 20.8125λ3 − 241.289λ4 + ... . (19) We note that the result (18) agrees with (19), up to the third order in the parameter λ. We have also ascertained that our result (18) is in agreement, up to the third order in the parameter λ, with those obtained from multiple scale method [16], and also those from Rayleigh-Schrödinger perturbation theory [24]. The agreement between (18) and (19) should not surprise the reader. We should remember that this situation is similar to the one we encounter in Rayleigh - Schrödinger perturbation theory. Moreover, both series are divergent even for small values of λ, and one has to truncate the series, to get any physical result. Obviously, a judicious choice of the truncation order is essential to the finding of good results [12]. For example, using (18) we calculate the ground state energy up to the fourth order; for λ = 0.1 we find E0 = 0.5591527. Comparing this result with the exact value E0 = 0.559146327183519576 obtained by Vinette and Čı́žek [25], the error is about 0.001 percent. 4 V International Symposium on Quantum Theory and Symmetries Journal of Physics: Conference Series 128 (2008) 012055 IOP Publishing doi:10.1088/1742-6596/128/1/012055 4. Discussion We have introduced a concise method to study anharmonic oscillators. It is based on the construction of a Lie group, whose generators include H0 and Hn . The generator of the group H0 may be transformed to the generator Hn by a unitary transformation, endowed by an element of the group. If a closed commutator algebra is formed exactly, the method can be used to obtain exact eigenvalues and eigenfunctions of Hn . However, the commutator algebra is generally obeyed up to some order of the parameter λ, thus the eigenvalues and eigenfunctions of Hn are found approximately. We applied the method to obtain the energy eigenvalues of a quartic anharmonic oscillator; the results are in agreement with the other methods, up to third order in the parameter λ. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] Bender C M and Wu T T 1969 Phys. Rev. 184 1231-60 Bender C M and Wu T T 1971 Phys. Rev. Lett. 27 461-5 Bender C M and Wu T T 1973 Phys. Rev. D 7 1620-36 Banks T, Bender C M and Wu T T 1973 Phys. Rev. D 8 3346-66 Seznec R and Zinn-Justin J 1979 J. Math. Phys. 20 1398-408 Fernández F M 1995 J. Math. Phys. 36 3922-30 Janke W and Kleinert H 1995 Phys. Rev. Lett. 75 2787-91 Weniger E J 1996 Phys. Rev. Lett. 77 2859-62 Ivanov I A 1996 Phys. Rev. A 54 81-6 Ivanov I A 1998 J. Phys. A:Math. Gen. 31 6995-7003 Skála L, Čı́žek J, Weniger E J and Zamastil J 1999 Phys. Rev. A 59 102-6 Weniger E J. 1997 Phys. Rev. A 56 5165-68 Fernández F M 1992 Phys. Rev. A 45 1333-8 Bender C M and Bettencourt L M A 1996 Phys. Rev. Lett. 77 4114-7 Bender C M and Bettencourt L M A 1996 Phys. Rev. D 54 7710-23 Auberson G and CapdequiPeyranere M 2002 Phys. Rev. A 65 032120 Mandelzweig V B 1999 J. Math. Phys. 40 6266-91 Liverts E Z, Mandelzweig V B and Tabakin F 2006 J. Math. Phys. 47 062109 Jafarpour M and Afshar D 2002 J. Phys. A:Math. Gen. 35 87-92 Jafarpour M and Afshar D 2005 Nuovo Cimento B 120 335-344 Gómez F J and Sesma J 2005 J. Phys. A:Math. Gen. 38 3193-202 Nieto M M 1997 Phys. Lett. A 229 135-43 Fisher R A, Nieto M M and Sandberg V D 1984 Phys. Rev. D 29 1107-10 McRae S M and Vrscay E R 1997 J. Math. Phys. 38 2899-921 Vinette F and Čı́žek J 1991 J. Math. Phys. 32 3392-404 5