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Discrete & Continuous Models & Applied Computational Science 2022, 30 (1) 39-51
ISSN 2658-7149 (online), 2658-4670 (print) http://journals-rudn-ru/miph
Research article
UDC 519.711.3
DOI: 10.22363/2658-4670-2022-30-1-39-51
The quantization of the classical two-dimensional Hamiltonian systems
Irina N. Belyaeva
Belgorod State National Research University 85, Pobedy St., Belgorod, 308015, Russian Federation
(received: December 26, 2021; revised: January 18, 2022; accepted: February 18, 2022)
Abstract. The paper considers the class of Hamiltonian systems with two degrees of freedom. Based on the classical normal form, according to the rules of Born-Jordan and Weyl-MacCoy, its quantum analogs are constructed for which the eigenvalue problem is solved and approximate formulas for the energy spectrum are found. For particular values of the parameters of quantum normal forms using these formulas, numerical calculations of the lower energy levels were performed, and the obtained results were compared with the known data of other authors. It was found that the best and good agreement with the known results is obtained using the Weyl-MacCoy quantization rule. The procedure for normalizing the classical Hamilton function is an extremely time-consuming task, since it involves hundreds and even thousands of polynomials for the necessary transformations. Therefore, in the work, normalization is performed using the REDUCE computer algebra system. It is shown that the use of the Weyl-MacCoy and Born-Jordan correspondence rules leads to almost the same values for the energy spectrum, while their proximity increases for large quantities of quantum numbers, that is, for highly excited states. The canonical transformation is used in the work, the quantum analog of which allows us to construct eigenfunctions for the quantum normal form and thus obtain analytical formulas for the energy spectra of different Hamiltonian systems. So, it is shown that quantization of classical Hamiltonian systems, including those admitting the classical mode of motion, using the method of normal forms gives a very accurate prediction of energy levels.
Key words and phrases: Hamilton function, normal form, Weyl-MacCoy rules, Born-Jordan rule, quantum normal form, computer modeling, energy spectra
Introduction
Representation of the original classical Hamilton function in normal form as the sum of homogeneous polynomials in canonically conjugate coordinates and momenta [1] allows us to carry out its quantum-mechanical description.
The main provisions of the new quantum mechanics were discovered by W. Heisenberg in 1925 [2]. In the same year, the paper was published by
© Belyaeva I.N., 2022
This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/
M. Born and P. Jordan [3], in which the commutation relation for the quantum-mechanical operators of coordinate q and momentum p was obtained for the first time in the form
pq — qp = h/2ni, (1)
as well as the rule of matching to the classical monom of the form qmpn, (m, n = 1,2,3,...) of its quantum analog, which we present as
1 n
BJ{qmpn = pnqm} = q^(2)
The results obtained by P.A.M. Dirac [4] should be added to this. In this paper P.A.M. Dirac wrote: "In a recent work [2], Heisenberg put forward a new theory, which suggests that not the equations of classical mechanics are erroneous in any way, but that mathematical operations, by which physical results are derived from these equations, need modification. Thus, all the information provided by the classical theory can be used in the new theory... We make the basic assumption that the difference of the Heisenberg products of two quantum quantities is equal to the Poisson bracket of these quantities multiplied by ih/2n":
here {q;p} is the Poisson bracket, [q,p] is the commutator for the operators, q,p, h is Planck's constant.
In 1927, G. Weyl published a paper [5], (see also [6]), in which the author, on the basis of group-theoretic ideas, proposed the following rule of correspondence between classical quantities and their quantum analogs in integral form. Let the classical function f(q,p) be determined by the following Fourier integral
fi^,P) = ff exp(гaP + trq)((a,r)dadr, then the corresponding function F(q,p) in quantum mechanics is given by
F(M,P) = ff exp(гap + гrí)((a,r)dadr,
and the operators q,p satisfy the commutation relation (1). Based on these assumptions for functions of polynomial form f(q,p) = qmpn, a number of different relations were obtained [7], one of which can be written as
1 n i
k=0
which we will call the Weyl-MacCoy quantization rule.
1 n'
WMc{qmpn =pn qm} = — ' kyPn-k, (4)
In [8], the correspondence rule was obtained in the following form
1 n n'
f(.Q)pn ^ Y,kl(nLk)[pkf(^)pn-k,
which is represented by repeating anti-commutators [a,b]+ = ab + ba as follows:
f(q)pn ^[[-[f(q),p] + ,P] +,..] +.
In addition to the main works mentioned above, there are publications in which the problem of the correspondence of classical quantities and their quantum analogs is discussed from different perspectives (see, for example,
[9]-[14]).
A critical review of various quantization rules for classical Hamilton functions was carried out in [15].
In this paper, for the Hamiltonian, in general, non-integrable system with two degrees of freedom, we have received the classical normal Birkhoff-Gustavson form for which the corresponding quantum analogs are obtained by the Born-Jordan and Weyl-MacCoy quantization rules. For these quantum analogs, i.e., Schrodinger operators, approximate formulas for energy spectra are found. According to these formulas, for some specific numerical values of the parameters, the energy spectra were calculated and compared with the literature results obtained by direct numerical calculations.
1. A quantum analog of the classical normal form
The paper considers a classical system with two degrees of freedom, whose Hamilton function is
H = \(p\ +pl) + V(qi ,q2),
1 2 1 1 \ 2 (5) y(Ql, Q2) = 2 (q1 + q2) + b (g1 g2 + 3q3) + Cq1q2 + d(q1 + q2 ) '
where the coordinates q1 ,q2 and momenta p1 ,p2 are canonically conjugate variables, b, c, d are dimensionless parameters.
Since system (5) is resonant with a frequency ratio of 1:1, when we bring it to normal form for the initial Hamilton function, we apply the canonical transformation with a valence equal to an imaginary unit [16]:
Qi = b(-Qi + Q2 + P1 - P2), V2 = \(Qi + Q2 + P1 + P2), 2l 12 (6) Pi = ^(Qi -Q2 + P1 -P2), P2 = 2-iQi + Q2-P1 -P2),
and its inverse transformation is written in the form
1 i 1 i
Ql = 2(q2 — ip2 ) + 2(qi — iPl = 2(q2 — ip2) — 2(Qi — iPl)'
11 1 i (7)
Pi = 1^2 + iP2) - 1(Ql + iPl ), P2 = 1^2 + iP2) + 1(Ql + iPl ).
It directly follows from expressions (7) that the variables Ql, Q2 are complex conjugate to the variables Pl, P2 respectively.
Canonical transformations (7) using standard substitution
d
Pv ^Pv = Qv ^ Qv = Qv, V = 1,1
dqv
with a known commutation rule (Planck constant h = 1)
[Pv ,Qv vi V,v=1,1 (s)
(5^v — Kronecker symbol) will be presented by us in the operator form
Ql = a+ + , Q2 = a+ — , Pl = a2 — ial, P2 = a2 + ial, (9) where
= i^l ^ i
(10)
ax = ^ + ^), = ^ + ^),
where the upper "+" symbol denotes Hermitian conjugation. Taking into account expressions (8), it is easy to verify that the operators (10) commute by the rule
and the operators (9) obey the rule
K ,Qt] = 5„ v. (11)
However, the commutation (11) can be directly obtained from the Dirac quantization condition (3), given that the classical canonical transformation (6) has a valence equal to an imaginary unit.
From the expressions (9), (10) it follows that the operators Pv and Qv, (u = 1,2) are the annihilation and birth operators, respectively.
Using the quantization rules (2) and (4), we obtain two expressions of its quantum analogs K6J and K6WMc, respectively, and each expression can be represented as the sum of the diagonal and nondiagonal parts
ftBJ = KBJ + KBJ
6 diag 1 nondiag.
ft WMc = ft WMc + ft WMc
6 diag ' nondiag.
Below we present the diagonal parts only:
KdJg = Q! Px + Q2 P2 + 1 + K.
41
(Qi Pi + Q2P2 + 1)2 + 5
+
+ K42 ( Qi Pi + Q2 P2 + 2Qi Pi Q2 P2 +2)+K43
(QiPi - Q2P2)2 + 5
+
+ Kt
6i
1
(Qi Pi + Q2P2 + 1)3 + 4(Qi Pi )2 +
1 - - 27 - - - - 13'
+ 4(Q2 P2)2 + -¿(Qi Pi + Q2P2) +y - K64 [(QiPi + Q2P2 + l)(QiPi + Q2P2 + 2QiP1Q2K + 3)] +
+ Kt
66
(Qi Pi + Q2 P2 +1) ((Qi Pi - Q2 P2 )2 + 4(Qi Pi + Q2 h )) -
- - Qi Pi Q2 P2
. (12)
^dWaf = Qih + Qi P2 + 1 + K4i
(Qi Pi + Q2 P2 + 1)2 + Ô
+
+ K42 ( Qi Pi + Q2 P2 + 2Qi Pi Q2 P2 + -)+K43
(QiPi - Q2P2)2 + 1
+
+ K61 [(Q1P1 + Q2P2 + 1)3 + 2(Q1 Px + Q2P2 + 1)] -- K64 [(QiPi + Q2P2 + 1)(QiPi + Q2P2 + 2Q1P1Q2P2 + 1)] +
+ k66 [(Q1P1 + Q2P2 +1) ((Q1P1 )2 + (Q2P2)2 - 2Q1P1Q2P2 + 1)]. (13)
We note that the quantum state vectors [2]
IN,L) =
2 J V 2 Pi 10,0) =P210,0) = 0,
q2 2 }Qi 2 >10,0),
(14)
where N is the main quantum number, N = 0,1,2,3,..., and L is the orbital quantum number, which for a given value N takes the following values: L = ±N, ±(N — 2), ±(N — 4), ..., ±1(0), are eigenvectors for the diagonal parts of quantum analogs (12) and (13). The presence of nondiagonal terms in quantum analogs (12) and (13) is due to the fact that in the original classical Hamiltonian system (5) there is a 1:1 resonance ratio between frequencies.
2. The energy spectra of quantum normal forms
Since vectors (14) represent an orthonormal basis, the energy spectra of quantum normal forms (12) and (13) are determined by the following expressions:
EJ = (N,L\KJ\N,L) + Y (N',L'\N,L), (15)
N', L '
£WMc = (N,L\K™?\N,L) + Y (N',L'KWMdcag\N,L). (16)
', '
Using the relations
QlPl\N,L) = \N,L), Q2P2\N,L) = =\N,L),
from expressions (15) and (16) without taking into account the nondiagonal terms, we obtain the formulas for the energy spectra
enl=n + 1 + \K4l (1N 2 + 4N + 7)+ l-K2 (N2 + 1N — L2 + 1)+
\k±3(1l2 + 5) + \kqi (8N3 + 15N2 + 78N + L2 + 60)— 1 8
1k64 (N3 + 3N2 + 8N — NL2 — L2 + 3)+
+ \k66 (N2 + 14N + 8NL2 + 9L2 + 11), (17)
^WMc = N+1 + 2ft4i (2N2 + 4N + 3) + 1ft 42 (N2 + 2N-L2 + 1)+ + 2ft 43 (2L2 + 1) + K6i (N3 + 3N2 + 5N + 3)- 1K64 (N3 + 3N2 + 2N- NL2 -L2 + 1)+
+ K66 (N + NL2 + L2 + 1). (18)
As it can be seen, the energy spectrum in both cases of quantization is degenerate by sign of the orbital quantum moment L. Besides, taking into account the contributions of nondiagonal terms can lead to a shift of energy levels, which differ in the value of the orbital quantum number by four and six units. Therefore, it is expected that approximate formulas (17), (18) with satisfactory accuracy describe the energy spectrum of the lowest states in the vicinity of a stationary point located at the origin.
From the comparison of formulas (17), (18) for energy spectra, a general conclusion can be drawn that the quantization rules of Born-Jordan and Weyl-MacCoy predict different values for the ground state energy, which are determined by the numerical values of the parameters b, c and d.
More specific conclusions can be obtained by comparing the results of numerical calculations using formulas (17), (18) with exact energy levels calculated for any particular values of the parameters of the Hamiltonian, which will be performed in the next section. In cases where the classical
system (5) is integrable, approximate formulas (17), (18), expressed directly through the parameters b, c and d have the following form.
1. If there is a relationship c = 4d, and the parameter b is not equal to or is equal to zero, then the energy spectra are calculated by the formulas:
Ef?L = N+l + d(-N2 + \N-\L2 + 6)-
r, . 85 „ 1037 „ 1479 187 r2 51 r2 561
- d2 (—N3 +--N2 +--N--L2--NL2 +--
8 32 16 32 8 8
-b2 (5N2 +5n-5l2 + 5) + V8 4 24 3)
ll2 , 125 r3 1525 r2 2175 275 r2 75 r2 825
+ db2 (-N3 +--N2 +--N--L2--NL2 +
8 32 l6 32 8 8
l4f 1175 r3 14335 r2 6815 2585 r2 235 r2 2585 \
- b4 (-N3 +--N2 +--N--L2--NL2 +--), (19)
V 864 3456 576 3456 288 288 J , y '
= N + l + d(-N2 + \n-\l2 +3) +
2 (17 „ 51 r2 17 51 r2 51 r9\ + d2 l—N3 +--N2 +--N--L2--NL2 ) -
V 8 8 4 8 8 )
-b2 (*N2 +5n-5l2 + 5)-\8 4 24 6 J
2 25 3 75 2 25 75 2 75 2 - db2 l—N3 +--N2 +--N--L2--NL2 ) +
V 8 8 4 8 8 J
235 235 235 235 235
+ b4 (-N3 +--N2 +--N--L2--NL2 ) . (20)
\864 288 432 288 288 ) '
2. If the parameters b = c = 0, but the parameter d > 0, then the formulas have the form:
3_o l
2
e~Bl = N +l + d( -N2 + 3N--L2 + 4)-
- d2 ( llN3 + 13N2 + 75N- 2L2 - -NL2 + — '4 2 4 2
), (21)
=N + l + d(^N2 + 3N-h2 + 2)- d2 (—N3 + —N2 + l9N - -L2 - -NL2 + —) . (22) V 4 4 4 4 2 J
3. If the parameters b = 0, c = -2d, d^ 0, then we obtain the formulas:
EBl = N+l + d(-N2 + ~4N-h2 + 3)-
2 i 85 „ 1037 „ 1479 187 r2 51 r2 561
- d2 \ —N3 +--N2 +--N--L2--NL2 +--
' 32 128 64 128 32 32
(23)
E™ = N+1 + d(9N2 +9-N-h2 + 3) -
nl t T ^ 4 8 2)
-d2
187 r3 561 r2 391
-N3 +--N 2 +--
32 32 16
N -
153 ~32
153
51
L2 -—NL2 + —
32
4
(24)
3. The comparison of energy spectra
Unfortunately, the exact spectrum obtained, for example, by direct numerical calculations of the Schrodinger equation with its quantum analog of the original Hamilton function (5), in which a well-known replacement is to be
, . d . d _ made px ^ px = -i —, P2 ^ P2 = -ít¡— , Qi ^ Qi = Qi, Q2 ^ $2 = Q2 for
dq-1 dq2
arbitrary values of its parameters, is not available in the literature.
Also, direct numerical calculations using modern computer technologies face the difficulty of solving eigenvalue problems, for example, even with the help of carefully developed software packages based on the diagonalization method, which is also the task of integrating the Schrodinger equation for two or more independent variables.
Below we present the results of numerical calculations of energy spectra for specific numerical values of the parameters b, c, d in cases where the classical system (5) is integrable.
Table 1 shows the values of the lowest energy levels calculated by the approximate formulas (19) and (20) in the first case of integrability, i.e., under the condition c = 4d and b = 0.
Table 1
The comparison of energy levels at parameter values b = 0, c = 0.02, d = 0.005, ( c = 4d)
No en, l E BJ pWMc pBJ pWMc ^nl n l PWMc pWMc °n+1 ^n fbj - EBJ ^n+1 °n
1. e0,0 1.028247 1.015000 0.013247 - -
2. e1,±1 2.055166 2.045000 0.010166 1.030000 1.026919
3. e2,±2 3.095512 3.089363 0.006149 1.044363 1.039953
4. e2,0 3.108259 3.106275 0.001984 0.016913 0.013141
5. e3,±3 4.147469 4.147450 0.000019 1.041175 1.039209
6. e3,±1 4.172475 4.182550 -0.010075 0.035100 0.025006
7. e4,±4 5.211578 5.218625 -0.007047 1.036075 1.039103
8. e4,±2 5.247175 5.273188 -0.026013 0.054563 0.035597
9. e4,0 5.259041 5.291375 -0.032334 0.018188 0.011866
From the table 1, it follows that the Weyl-MacCoy quantization rule leads to a lower energy level for the ground state and a greater decomposition of the levels with respect to the orbital moment at a given value of the principal quantum number N. In the classically integrable case under consideration, there are no exact (analytical or numerical) values of the energy spectrum in the current literature. However, the spectrum is known [17] in the second classical case of integrability, when the parameters of the quantum analog of the Hamilton function (5) are equal b = 0, c = 0, d ^ 0.
Besides, the values of the energy spectrum of a one-dimensional anharmonic oscillator are known and also with great accuracy, in particular, with a fourth degree in potential energy. Knowing this spectrum, it is possible to construct an approximate spectrum of a quantum analog of the original Hamilton function (5), but already a two-dimensional Hamiltonian given the values of parameters b = 0, c = -2d, d ^ 0, for which system (5) is integrable in the classical case.
We will compare below these well-known and very reliable numerical results for the energy spectra with our results, which are calculated by formulas (23), (24) according to the Born-Jordan and Weil-McCoy quantization rules.
For parameter values b = 0, c = 0 and d ^ 0 the Schrodinger equation corresponding to the classical Hamilton function (5) allows separation of variables in polar coordinates, and the energy spectrum is characterized by a radial quantum number n and orbital momentum I. In [17] a method for numerical solving the radial Schrodinger equation was developed and energy levels were calculated for the values of quantum numbers equal n,l = 0,l,2 for a parameter value d = 0.000005. Quantum numbers n, I are connected with our numbers N, L by the following relations: N = 2n + I, \L\ = I.
Table 2 shows the energy levels obtained in [17], as well as their values calculated for the same value of the parameter using formulas (21) and (22) based on quantization of the classical normal form, according to the Born-Jordan and Weil-McCoy rules, respectively.
Table 2 shows that a very good approximation to the exact spectrum is given by the application of the Weyl-MacCoy quantization rule. In particular, the ground state energy obtained using the Weyl-MacCoy quantization rule differs from the result of [17] by 0.5 • l0-7%, and when quantized by the Born-Jordan rule, by 0.00l%. At the same time, for energy of level 14, these errors are equal, respectively, 0.4 • l0-8% and 0.000l%, i.e., the prediction according to the Born-Jordan rule improves.
In the third case (b = 0, c = -2d, d ^ 0) of integrability of the classical system (5), with its quantum-mechanical description, it is necessary to solve the following two-dimensional Schrodinger equation
(H-1 + H2 )^ = 2E^, H% = -^2 + qf + 2dq4, i = l,2, (25)
where the variables are separated. Therefore, its solving is reduced to solving two identical one-dimensional equations for the anharmonic oscillator, and the energy spectrum is found in the form of the following sum 2E = 2E1 + 2E2.
The quantum numbers of an isotropic two-dimensional oscillator (N, L) are connected with the quantum numbers (nl,n2) of one-dimensional oscillators
by the following relations: N = n1 + n2 and L = n1 — n2. We note that the ordering of the values of the energy spectrum levels by the value of quantum numbers (N, L) as compared to another numbering of states has the advantage that the values of the energy spectrum levels, numbered by quantum numbers (N, L), grow with an increase of the main quantum number.
Table 2
The comparison of energy levels and E^ff with their values from [17]
for d= 0.000005
No. 2en,l E BJ E WMc Results [17]
1. 2E0,0 2.0000399985 2.0000199995 2.0000199995
2. 2E1,±1 4.0000799961 4.0000599979 4.0000599981
3. 2E2,±2 6.0001399918 6.0001199946 6.0001199949
4. 2E2,0 6.0001599905 6.0001399933 6.0001399936
5. 2E3,±3 8.0002199853 8.0001999892 8.0001999892
6. 2E3,±1 8.0002599818 8.0002399856 8.0002399859
7. 2E4,±4 10.000319975 10.000299981 10.000299981
8. 2E4,±2 10.000379969 10.000359974 10.000359975
9. 2E4,0 10.000399967 10.000379971 10.000379972
10. 2E5,±5 12.000439962 12.000419969 12.000419969
11. 2E5,±3 12.000519952 12.000499958 12.000499958
12. 2E5,±1 12.000559947 12.000539953 12.000539953
13. 2E6,±6 14.000579946 14.000559953 14.000559953
14. 2E6,±4 14.000679930 14.000659937 14.000659937
15. 2E6,±2 14.000739921 14.000719928 14.000719929
16. 2E6ft 14.000759918 14.000739925 14.000739925
Conclusions
In this paper for a classical system with two degrees of freedom with the Hamilton function (5), a classical normal form is obtained in the Birkhoff-Gustavson approach, for which its quantum analogs are constructed according to the Born-Jordan and Weyl-MacCoy heuristic quantization rules. For these quantum analogs, which are nothing but approximate differential expressions for the exact Schrodinger operator, the eigenvalue problem is solved and the formulas of energy spectra are found.
Using these formulas, in two special cases with specific numerical values of the parameters, the lower energy levels were calculated and the results obtained were compared with the data available in the works published by other authors. It was found that the best and good agreement with the known results of calculating the energy spectrum is obtained using the Weyl-MacCoy quantization rule in comparison with the Born-Jordan rule.
Both the Weyl-MacCoy and Born-Jordan quantization rules are derived from the fundamental, but different postulates of classical and quantum mechanics. For the system under consideration, particular numerical results for the energy spectrum reveal the advantage of the Weyl-MacCoy quantization rule, however, it is probably premature to extend this conclusion to other systems.
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For citation:
I. N. Belyaeva, The quantization of the classical two-dimensional Hamilton-ian systems, Discrete and Continuous Models and Applied Computational Science 30 (1) (2022) 39-51. DOI: 10.22363/2658-4670-2022-30-1-39-51.
Information about the authors:
Belyaeva, Irina N. — Candidate of Physical and Mathematical Sciences, Associate Professor of Belgorod State National Research University (e-mail: ibelyaeva@bsu.edu.ru, ORCID: https://orcid.org/0000-0002-1368)
УДК 519.711.3
DOI: 10.22363/2658-4670-2022-30-1-39-51
Квантование классических двумерных гамильтоновых
систем
И. Н. Беляева
Белгородский государственный исследовательский университет ул. Победы, д. 85, Белгород, 308015, Россия
Аннотация. В статье рассматривается класс гамильтоновых систем с двумя степенями свободы. На основе классической нормальной формы, согласно правилам Борна—Йордана и Вейля—Маккоя, построены её квантовые аналоги, для которых решена задача на собственные значения и найдены приближённые формулы для энергетического спектра. Для конкретных значений параметров квантовых нормальных форм с использованием этих формул были проведены численные расчёты нижних энергетических уровней, полученные результаты были сопоставлены с известными данными других авторов. Обнаружено, что наилучшее согласие с известными результатами достигается с использованием правила квантования Вейля—Маккоя. Процедура нормализации классической функции Гамильтона является крайне трудоёмкой задачей, так как вовлекает сотни и даже тысячи многочленов для необходимых преобразований. Поэтому в работе нормализация выполняется с помощью системы компьютерной алгебры REDUCE. Показано, что использование правил соответствия Борна-Йордана и Вейля—Маккоя приводит практически к одним и тем же значениям для энергетического спектра, при этом их близость увеличивается для больших величин квантовых чисел, то есть для высоковозбуждённых состояний. В работе использовано каноническое преобразование, квантовый аналог которого позволяет построить собственные функции для квантовой нормальной формы и получить таким образом аналитические формулы для энергетических спектров разных гамильтоновых систем. Итак, показано, что квантование классических гамильтоновых систем, в том числе допускающих классический режим движения, с применением метода нормальных форм даёт очень точное предсказание уровней энергии.
Ключевые слова: функция Гамильтона, нормальная форма, правило Вейля-Маккоя, правило Борна-Йордана, квантовая нормальная форма, компьютерное моделирование, энергетические спектры
