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Semiclassical calculation of energy levels for non-separable systems
Volume 47, number t
CHEMICAL PHYSlCS LE-ITXRS
1 April 1977
SEMICLASSICAL CALCULATION OF ENERGY LEVELS FOR NONSEPARABLE
SYSTEMS
3.3. DEL05 and R.-f_ SWiMM
Received 17 December I976
A series expansion for a second isolating integral is calculated for a nonseparable system of two coupled oscdlators. This mtegraJ, together with the hamiltomnn, is used to c&Mate semiclassical energy bvels. The agreement with the exact quantum Ievets is sofficirnt to verify the validtty of the method.
1, introduction SemicIassical methods have been used for many cafculate the spectrum of bound energy levels of one-d~~nsiona~ systems, or of Mary-d~en~ona~ systems for which the Hami~t~u-Jacobi equation is separable. Recently there have been several studies of spectra of energy levels of non-separable systems [l-4],. fn this paper we develop and apply a method for calculating such energy levels by use of an additionapforma1 isolating integral of the motion. For a system of~t degrees of freedom, if there exist it isolating [I?] integrals (constants of the motion), then the trajectories for the system are quasi~e~odj~~ lying on a manifold that is topologicaliy equivalent to an rr-dimensional torus. The motion can then be characterized in terms of action-angle variables such that the harn~~t~~a~ is a function only of the actions. Quantization of action then gives the energy ieveis in terms of n quantum numbers, and the spectrum is said to be regular. In the following sections we: {i) develop a series for an intcgraI of the motion far a model two-dimensional system; (2) use this integral to obtain the caustics for the trajectory (i.e. the boundaries between the classically allowed and forbidden regions); (3) inte-grate along these caustics to obtain the action variables; (4) quantize the actions to obtain the q~nt~m spectrum. The result is shown to be in good agreement with exact quantum calculations. years to
2. The integral of the motion We consider a hamiftonian for a pair of oscillators that are coupled by a non-linear force, H=G (p; “p: +&+a
~~~,y~)~~~(x2~~~=)
=&+I+*
(11
a mode1 whi& has been extensively studied [Z&-4,6]An integral of the motion i7f is a function I@,, pY, x, y) that is constant along any trajectory, so that df/dt=
i&H] =i&!J”=O,
where J? is the LiouvilIe operator f8]- We expand B=Pz
f.+,
(31
so that .&?+J represents the Poisson bracket with Nz, the quadratic part oflY; and Es the Poisson bracket with I$, f the cubic part oFW. Then the integral can also be expanded in degrees [7], r=r,
+13 +I4 +...
(4)
and the terms collected to give equations in each degree, JzzIz = 0,
@aI
J?&
fP?&
=o,
(5bI
e,r,
‘J&J1~ = 0.
(5CI
Among the solutions ta @a) are the functions
76
(21
1 April 1977
CHEMICAL PHYSICS LETTERS
Volume 47, number 1
I=; and
(P_~+w_~x2)+A(-P~y+p_~p~x)+Bx~_~ +Fx4 +K(p,,~-p_~y)~
+ibf_s’_v2
(6b) or any combination? that is independent higher terms are obtained formally from
+w;)‘(P_;+w_y)2
of H2. Then
+ S(3 I2 (P-Z + w_y)
13 = - [e,]-‘fz31r,,
(7a)
I4 =-
(7b)
+ z-(;)’ [J?2]-‘.@3~3.
Each term 1, is a homogeneous polynomal of kth degree in the variables (p,, py, x, y). Although we used a somewhat different method, the Iteration can be carried out most easily by expressing I(p,, p,,, x, y) and L! in terms of the eigenfunctions [8] of z3, :
The solutions to this equation are the classical analogues of the creation and annihilation operators
x x+ =‘wx, In addition, constructed
x Y’ =+w
Y’
a complete set of eigenfunctions from products of these:
can be
@ = 0x+ &_ @:‘! _ @‘:_ , ~2~~=[(k-z)w,+(m-n)wy]
(P_Z+ w_$‘)
(p; +w_$y
(III
where A = 2X/(@ K=
-
w_;>,B=w_:A.
3w--9) A ’ 4(w_Z - w;>
,=-W-3dA
and R, S, and Tare arbitrary set equal to zero.
3. Calculation
F=;hA.
4
*
at this point; they were
of the spectrum
Marcus and co-workers [2] have shown that the energy spectrum can be obtained by quantrzation of action integrals that are evaluated either on a caustic or else on a PoincarC surface of section. As has been shown by Contopoulos [IO], the four corners of the classically allowed region are simultaneous solutions to the four equations (1222)
a.
(10)
If I,, and 2, I,* are expressed in terms of the Q’s, it is easy to see that fn+I is well defined provided that w, and oY are incommensurable. Note also that sinceQ2 has eigenfunctions with eigenvalue zero (Q,+@,_ and @Y,+Y_), it follows that (i) the inverse2 21 is not unique, and can be modified by the addition of any solution to the homogeneous equation (5a); (ii) these homogeneous terms must be chosen such that the projection of L’s Zk onto the null space of P2 is zero. This was not a difficulty in our calculation, which was only carried out to fourth degree. The result is
I@,.
Py. x Y) = 10,
(i2b)
J(P,.
PJ.9 -T Y) = 0,
(13c)
K(P,.
P,,) = 0,
(12d)
where IO is the (constant) value of the integral Z(p_,, pt., x, y), J is the jacobian J = NH. I)/Np,. p_,,l and K is the kinetic energy K = + (p_z +p_z)_ The cailstic is the locus of points satisfying ( 1?a)-( 13~) but not (12d). From these three equations, px and p-v can he eliminated to give an implicit equation for the caustic
F(x -L
t A particular combination gives what Whittaker calls the adelphic integral [9 l-
Y;
E’,1,) = 0,
(13)
which for each x (or y) can be solved forp (or x) by Newton’s method. The two action integrals 77
CHEMICAL
Volume 47, number 1
Ai=
s
PHYSICS
are then easily obtained.
Since quantization
1 April i977
proximately constant along each trajectory: while the second degree terms in (11) vary as much as 47% along a trajectory, the inchrsion of third degree terms reduces this variation to at most 8.6%, and the fourth degree terms reduce the variation to 3.6%. The second part of table 1 contains a tabulation of the energy levels as predicted by several methods. Although the results of the present method do not agree identicaliy with the quantum mechanical results, they are close enough to justify the basic method. The results obtained by the second order form of Born’s method [1 I] are included for comparison_ The results of Marcus and co-workers [2], and the results of Miller and co-workers [3] essentially match the quantum mechanica! calculations. Miller and co-workers [3] also investigated the energy levels as the strength of the coupling is varied. Table 2 compares our results with theirs, and those of Born. The agreement of our calculations with the quantum mechanical calculations is good.
(14)
p-dr
LETTERS
requires
Ai=(iti+~)tt, the procedure is, then: (i) guess approximate values for E and IO, calculate the caustics and the action integrals; (ii) after three such guesses, use two-dimensional linear interpolation to find a closer approximation to the correct quantized result; (iii) iterate. Convergence is rapid. It is clear that this method is very similar to that of Eastes 2nd Marcus [2], except for the way that the caustics are calculated. They obtained the caustics by interpolation of a numerically calculated trajectory, while we obtain them as the roots of a polynomial equation (13). The Poincare surface of section can also be obtained this way, and may provide some advdntages.
4. Results 5. Discussion In testing the validity of this method, we only included terms in our expansion through fourth degree, as in eq. (11). The results for a typical case are presented in table I _The first part of the table contains a tabulation of the relative variation o‘the integral along the trajectory corresponding to the first six energy levels. The results of calculations including second degree terms through fourth degree terms are shown for comparison. This shows that the integral is in fact ap-
The numerical work involved in the present method is substantially less than that involved in the available alternative computational methods [l-4]. The savings,in numerical effort results from the analytic evaluation of an approximate integral of the motion [ll]. It is expected that the accuracy of the result can be improved by evaluation of higher degree terms in the integral, but even the present results verify the essential validity of the method. -
Table 1 Variation of integral, and energies of states a) Quantum numbers
(Ima\ - lmm)lfz,e second
Euncoupledb)
EQhi
Epresent
EBorn
r’_x
II 1’
dqrce
third degree
fourth degree
0 1
0
0.12
0.015
0.0125
l.OOOa
0.9955
0.9954
0.9956
0
0.018
0 2
0.30 0.16
0.013 0.0025
1.7000 2.3000
1.6870 2.2781
1.6853 2.2786
1.6874 2.2789
0
1 0
1 2
0.036 0.015 0.0037
2.4000 3.0000 3.6000
2.3750 2.9583 3.5479
2.3709 2.9541 3.5503
2.3760 2.9604 3.5506
1
0.47 0.37 0.23
0.015 0.086 0.056 0.022
‘) For case with wX * = 0.49. w_;,= 1.69, h = -0.1, n = 0.1. b, i.e. h = 0. ‘1 l-orsecond order calculation.
78
c)
-
Volume
47, number
Table 2 Energies of states
EQhl
0 -0.08 -0.12 -0.16 -0.20
PHYSICS
LETTERS
i April
1977
References as a function
b,
1.0000 0.9975 0.9926 0.9826 0.9621
CHEhIICAL
1
E
present
1 .oooo 0.9974 0.9924 0.9826 0.9640
of coupling
E
a)
CGM
1 .oooo 0.9975 0.9927 0.9836 0.9667
cl
E
[I] M.C. GutzwiUcr,
Bornd)
1 .oooo 0.9975 0.9929 0.9846 0.9710
a) For case with -_; = 0.49, w_; = 1.69, n = --h, N, = 0. N2 =o. b, As calculated by hliiier and co-workers 131. c, Miller and co-workers 131 d, for second order calcuiation~
Acknowledgement We thank E.A_ Remler for many stimulating conversations. This research was supported in part by grants from the Research Corporation and from the National Science Foundation.
J. blah. Phys. 8 (1967) 1979; LO (1969) 1004; 11 (1970) 1791; 12 (1971) 343. [2] R.A. Marcus, Discussions Taraday Sot. 55 (1973) 34; W. Lastes and R.A. Marcus, J. Chem. Phys. 61 (19741 4301; D.W. Noid and R-A. Marcus, J. Chem. Phys. 62 (1975) 2119. I31 W.H. Miller, J. Chem. Phys. 63 (1975) 996; S. Chapman, B.C. Garrett and W.H. h’liller, 3. Chem. [41
Phys. 64 (1976) 502. I.C. Percival, J. Phys. 86 (1973)
L229; J. Phys A7 (1974)794; 11’. Pompluey, J. Phys. B7 (1974) 1909; 1-C’. Perciv.rI and N. Pomphrey, X101. Phys. 31 (1976) 97. foundations of celestial meI51 A. Wintrier, Analytical chanics (Princeton Univ. Press, Princeton. 1947) p_ 96. [61 hl. Henon and C. Heties. Astron. J. 69 (1964) 73. Lcs hlc’thodes Nouvelles de la bl&hanique [71 H. Poincar;, Celeste (Dover, New York, 1957) ch. 5. Phys. Rev. 144 (1966) 170. Pl R. Zwanzig, A treatise on the analytical dynamics of [91 ET. Whittaker, parttcles and rigtd bodres (Dover, New York, 1944) ch. 16. Z. Astrophysik 49 (1960) 273. [lOI G. Contopoulos, of the atom (Frederick Ungar [Ill hl. Born, The mechanics Publishing Co., 1960) 5 42.
79
CHEMICAL PHYSlCS LE-ITXRS
1 April 1977
SEMICLASSICAL CALCULATION OF ENERGY LEVELS FOR NONSEPARABLE
SYSTEMS
3.3. DEL05 and R.-f_ SWiMM
Received 17 December I976
A series expansion for a second isolating integral is calculated for a nonseparable system of two coupled oscdlators. This mtegraJ, together with the hamiltomnn, is used to c&Mate semiclassical energy bvels. The agreement with the exact quantum Ievets is sofficirnt to verify the validtty of the method.
1, introduction SemicIassical methods have been used for many cafculate the spectrum of bound energy levels of one-d~~nsiona~ systems, or of Mary-d~en~ona~ systems for which the Hami~t~u-Jacobi equation is separable. Recently there have been several studies of spectra of energy levels of non-separable systems [l-4],. fn this paper we develop and apply a method for calculating such energy levels by use of an additionapforma1 isolating integral of the motion. For a system of~t degrees of freedom, if there exist it isolating [I?] integrals (constants of the motion), then the trajectories for the system are quasi~e~odj~~ lying on a manifold that is topologicaliy equivalent to an rr-dimensional torus. The motion can then be characterized in terms of action-angle variables such that the harn~~t~~a~ is a function only of the actions. Quantization of action then gives the energy ieveis in terms of n quantum numbers, and the spectrum is said to be regular. In the following sections we: {i) develop a series for an intcgraI of the motion far a model two-dimensional system; (2) use this integral to obtain the caustics for the trajectory (i.e. the boundaries between the classically allowed and forbidden regions); (3) inte-grate along these caustics to obtain the action variables; (4) quantize the actions to obtain the q~nt~m spectrum. The result is shown to be in good agreement with exact quantum calculations. years to
2. The integral of the motion We consider a hamiftonian for a pair of oscillators that are coupled by a non-linear force, H=G (p; “p: +&+a
~~~,y~)~~~(x2~~~=)
=&+I+*
(11
a mode1 whi& has been extensively studied [Z&-4,6]An integral of the motion i7f is a function I@,, pY, x, y) that is constant along any trajectory, so that df/dt=
i&H] =i&!J”=O,
where J? is the LiouvilIe operator f8]- We expand B=Pz
f.+,
(31
so that .&?+J represents the Poisson bracket with Nz, the quadratic part oflY; and Es the Poisson bracket with I$, f the cubic part oFW. Then the integral can also be expanded in degrees [7], r=r,
+13 +I4 +...
(4)
and the terms collected to give equations in each degree, JzzIz = 0,
@aI
J?&
fP?&
=o,
(5bI
e,r,
‘J&J1~ = 0.
(5CI
Among the solutions ta @a) are the functions
76
(21
1 April 1977
CHEMICAL PHYSICS LETTERS
Volume 47, number 1
I=; and
(P_~+w_~x2)+A(-P~y+p_~p~x)+Bx~_~ +Fx4 +K(p,,~-p_~y)~
+ibf_s’_v2
(6b) or any combination? that is independent higher terms are obtained formally from
+w;)‘(P_;+w_y)2
of H2. Then
+ S(3 I2 (P-Z + w_y)
13 = - [e,]-‘fz31r,,
(7a)
I4 =-
(7b)
+ z-(;)’ [J?2]-‘.@3~3.
Each term 1, is a homogeneous polynomal of kth degree in the variables (p,, py, x, y). Although we used a somewhat different method, the Iteration can be carried out most easily by expressing I(p,, p,,, x, y) and L! in terms of the eigenfunctions [8] of z3, :
The solutions to this equation are the classical analogues of the creation and annihilation operators
x x+ =‘wx, In addition, constructed
x Y’ =+w
Y’
a complete set of eigenfunctions from products of these:
can be
@ = 0x+ &_ @:‘! _ @‘:_ , ~2~~=[(k-z)w,+(m-n)wy]
(P_Z+ w_$‘)
(p; +w_$y
(III
where A = 2X/(@ K=
-
w_;>,B=w_:A.
3w--9) A ’ 4(w_Z - w;>
,=-W-3dA
and R, S, and Tare arbitrary set equal to zero.
3. Calculation
F=;hA.
4
*
at this point; they were
of the spectrum
Marcus and co-workers [2] have shown that the energy spectrum can be obtained by quantrzation of action integrals that are evaluated either on a caustic or else on a PoincarC surface of section. As has been shown by Contopoulos [IO], the four corners of the classically allowed region are simultaneous solutions to the four equations (1222)
a.
(10)
If I,, and 2, I,* are expressed in terms of the Q’s, it is easy to see that fn+I is well defined provided that w, and oY are incommensurable. Note also that sinceQ2 has eigenfunctions with eigenvalue zero (Q,+@,_ and @Y,+Y_), it follows that (i) the inverse2 21 is not unique, and can be modified by the addition of any solution to the homogeneous equation (5a); (ii) these homogeneous terms must be chosen such that the projection of L’s Zk onto the null space of P2 is zero. This was not a difficulty in our calculation, which was only carried out to fourth degree. The result is
I@,.
Py. x Y) = 10,
(i2b)
J(P,.
PJ.9 -T Y) = 0,
(13c)
K(P,.
P,,) = 0,
(12d)
where IO is the (constant) value of the integral Z(p_,, pt., x, y), J is the jacobian J = NH. I)/Np,. p_,,l and K is the kinetic energy K = + (p_z +p_z)_ The cailstic is the locus of points satisfying ( 1?a)-( 13~) but not (12d). From these three equations, px and p-v can he eliminated to give an implicit equation for the caustic
F(x -L
t A particular combination gives what Whittaker calls the adelphic integral [9 l-
Y;
E’,1,) = 0,
(13)
which for each x (or y) can be solved forp (or x) by Newton’s method. The two action integrals 77
CHEMICAL
Volume 47, number 1
Ai=
s
PHYSICS
are then easily obtained.
Since quantization
1 April i977
proximately constant along each trajectory: while the second degree terms in (11) vary as much as 47% along a trajectory, the inchrsion of third degree terms reduces this variation to at most 8.6%, and the fourth degree terms reduce the variation to 3.6%. The second part of table 1 contains a tabulation of the energy levels as predicted by several methods. Although the results of the present method do not agree identicaliy with the quantum mechanical results, they are close enough to justify the basic method. The results obtained by the second order form of Born’s method [1 I] are included for comparison_ The results of Marcus and co-workers [2], and the results of Miller and co-workers [3] essentially match the quantum mechanica! calculations. Miller and co-workers [3] also investigated the energy levels as the strength of the coupling is varied. Table 2 compares our results with theirs, and those of Born. The agreement of our calculations with the quantum mechanical calculations is good.
(14)
p-dr
LETTERS
requires
Ai=(iti+~)tt, the procedure is, then: (i) guess approximate values for E and IO, calculate the caustics and the action integrals; (ii) after three such guesses, use two-dimensional linear interpolation to find a closer approximation to the correct quantized result; (iii) iterate. Convergence is rapid. It is clear that this method is very similar to that of Eastes 2nd Marcus [2], except for the way that the caustics are calculated. They obtained the caustics by interpolation of a numerically calculated trajectory, while we obtain them as the roots of a polynomial equation (13). The Poincare surface of section can also be obtained this way, and may provide some advdntages.
4. Results 5. Discussion In testing the validity of this method, we only included terms in our expansion through fourth degree, as in eq. (11). The results for a typical case are presented in table I _The first part of the table contains a tabulation of the relative variation o‘the integral along the trajectory corresponding to the first six energy levels. The results of calculations including second degree terms through fourth degree terms are shown for comparison. This shows that the integral is in fact ap-
The numerical work involved in the present method is substantially less than that involved in the available alternative computational methods [l-4]. The savings,in numerical effort results from the analytic evaluation of an approximate integral of the motion [ll]. It is expected that the accuracy of the result can be improved by evaluation of higher degree terms in the integral, but even the present results verify the essential validity of the method. -
Table 1 Variation of integral, and energies of states a) Quantum numbers
(Ima\ - lmm)lfz,e second
Euncoupledb)
EQhi
Epresent
EBorn
r’_x
II 1’
dqrce
third degree
fourth degree
0 1
0
0.12
0.015
0.0125
l.OOOa
0.9955
0.9954
0.9956
0
0.018
0 2
0.30 0.16
0.013 0.0025
1.7000 2.3000
1.6870 2.2781
1.6853 2.2786
1.6874 2.2789
0
1 0
1 2
0.036 0.015 0.0037
2.4000 3.0000 3.6000
2.3750 2.9583 3.5479
2.3709 2.9541 3.5503
2.3760 2.9604 3.5506
1
0.47 0.37 0.23
0.015 0.086 0.056 0.022
‘) For case with wX * = 0.49. w_;,= 1.69, h = -0.1, n = 0.1. b, i.e. h = 0. ‘1 l-orsecond order calculation.
78
c)
-
Volume
47, number
Table 2 Energies of states
EQhl
0 -0.08 -0.12 -0.16 -0.20
PHYSICS
LETTERS
i April
1977
References as a function
b,
1.0000 0.9975 0.9926 0.9826 0.9621
CHEhIICAL
1
E
present
1 .oooo 0.9974 0.9924 0.9826 0.9640
of coupling
E
a)
CGM
1 .oooo 0.9975 0.9927 0.9836 0.9667
cl
E
[I] M.C. GutzwiUcr,
Bornd)
1 .oooo 0.9975 0.9929 0.9846 0.9710
a) For case with -_; = 0.49, w_; = 1.69, n = --h, N, = 0. N2 =o. b, As calculated by hliiier and co-workers 131. c, Miller and co-workers 131 d, for second order calcuiation~
Acknowledgement We thank E.A_ Remler for many stimulating conversations. This research was supported in part by grants from the Research Corporation and from the National Science Foundation.
J. blah. Phys. 8 (1967) 1979; LO (1969) 1004; 11 (1970) 1791; 12 (1971) 343. [2] R.A. Marcus, Discussions Taraday Sot. 55 (1973) 34; W. Lastes and R.A. Marcus, J. Chem. Phys. 61 (19741 4301; D.W. Noid and R-A. Marcus, J. Chem. Phys. 62 (1975) 2119. I31 W.H. Miller, J. Chem. Phys. 63 (1975) 996; S. Chapman, B.C. Garrett and W.H. h’liller, 3. Chem. [41
Phys. 64 (1976) 502. I.C. Percival, J. Phys. 86 (1973)
L229; J. Phys A7 (1974)794; 11’. Pompluey, J. Phys. B7 (1974) 1909; 1-C’. Perciv.rI and N. Pomphrey, X101. Phys. 31 (1976) 97. foundations of celestial meI51 A. Wintrier, Analytical chanics (Princeton Univ. Press, Princeton. 1947) p_ 96. [61 hl. Henon and C. Heties. Astron. J. 69 (1964) 73. Lcs hlc’thodes Nouvelles de la bl&hanique [71 H. Poincar;, Celeste (Dover, New York, 1957) ch. 5. Phys. Rev. 144 (1966) 170. Pl R. Zwanzig, A treatise on the analytical dynamics of [91 ET. Whittaker, parttcles and rigtd bodres (Dover, New York, 1944) ch. 16. Z. Astrophysik 49 (1960) 273. [lOI G. Contopoulos, of the atom (Frederick Ungar [Ill hl. Born, The mechanics Publishing Co., 1960) 5 42.
79