TAP CHi KHOA HOC OHSP TPHCM Hoang Do Ngoc Tram
i
THE FK OPERATOR METHOD FOR TWO-DIMENSIONAL SEXTIC DOUBLE WELL OSCILLATOR
HOANG DO NGOC T R A M ' ABSTRACT
The FK operator method is used for solving the Schrodinger equation ofa two dimensional sexlic double well oscillator We obtain the exact numerical energies for any quantum slates with Ihe precision of up to six decimal places. The FORTRAN program for aulomalically calculating Ihe solutions is made and tested for Ihe slates ofthe principal quantum number up lo hundreds.
Keywords: operator mettiod, Schrodinger equation, energy, sextic double well oscillator.
T 6 M TAT
Phuang phap loan tir FK cho dao dong tir dgng hi thi dot v&i si hang phi dieu hoa bgc sdu hai chieu
Phuang phap loan li FK dugc sir dyng di gidt phuang trinh Schrodinger cho dao dgng tu dgng ho Ihi dot vai so hgng phi diiu hoa bdc sdu hai chiiu Chiing toi Ihu dugc nghiim so cho bdi loan vai dg chinh xde din sdu chu si thdp phdn cho trang thdi lugng lir bat ki va gid iri bat ki ciia Idn so dao dong Chuang Irinh tinh todn tu dgng trln ngon nga lgp trmh FORTRAN duge xdy dung va dugc klim chimg cho cdc trang that co s6 lugng lu chinh ten din hdng trdm.
Tir khda: phuong phip toto tii, phuong ttinh SchrSduiger, ntog luong, dao dona hi ho the doi bac siu.
1. Introduction
The anharmonic oscillator is one of tiie siraplest qumtiim models which fiiids extensive application in various areas of physics as weU as chemistty: atomic m d molecular physics, qumtiim chemistty, condensed matter physics, particle physics statistical physics, qumtiim Held tiieoiy m d cosmology. However, solutions of flie above problems cannot be found usmg exact calculation mettiods. Hence, developing approximate calculation mettiods for ttiese systems interests ramy physicists [41 Among anharmonic oscillator models, ttie double well oscillator, obttiined by chmgmg ttie harmonic term i<i,V into - i ^ V , c m be used for modeling of two-state systems such as ttie interpretation of tiie infrared spectta of flie NH3 molecule, infrared m d Ramfa spectta of ttie hydrogen-bonded systems, inversion characteristics of isomers sttuetural phase ttansitions, polarizability of perovskite ferroelecttics, fomiation of noble-gas monolayers on a graphite substtate, macroscopic qumtiim coherence in
• Ph.D.. HCMC University of Education: Email tramhdn&icmup.edu.vn
TAP^Hi KHOA HOC BHSP TPHCM Si 6(84) nam 2016
superconducting Josephson devices, switching fad storage devices in computers, and so on [1, 3, 10]. Various methods have been applied for finduig ttie energy of the system in ttie ease of one-dimensional space [2, 3, 9, 10]. In addition, tiie work [3]
showed fa interesting point tiiat ttie exact malytical solutions exist for tile case of one- dimensional sextic double well oscillator in some consttained conditions. For flie case of higher dimensional spaces, ttie less attention has been given because ofthe presence of mgular-momentum sttites that make die problem more complicated [2].
The FK operator mettiod (FK-OM) [5, 6] is fa ab Initio mettiod for solving ttie Schrodinger equation of non-perturbative systems. It allows to obtain exact numerical solutions (energies and wave-functions) for systems witti arbittary intensity of extemal field. This mettiod was development successfully for various systems in atomic physics, condense matter physics, field tiieory, and so on [6-8].
In this work, we apply the FK-OM to solving the Schrodinger equation of a two- dimensional sextic double well oscillator (2D-SDW0) in order to obtain flie exact numerical solutions. These results are also the base for the follow-up research to find if ttie problem has exact faalytical solutions similarly to ttie case one-dimensional space, fad if have, what conditions it must satisfy to have these solutions.
The paper is divided into fliree main sections. In section 2 we present ttie FK-OM md apply ttie mettiod to tiie problem of 2D-SDW0. Section 3 is for ttie obtained results fad discussion. Section 4 concludes the paper.
2. FK operator method for two-dimensional sextic double well oscillator The 2D-SDW0 potential has ttie form:
yi^,y) = -~(x'+y-)+^(x'+/)\ (1)
in which the harmonic term is negative -m0-12<0; here m, ra md 1 are the mass, the oscillation frequency fad ttie coefficient of sextic anharmonic term of the 2D- SDWO, respectively.
For convenience, the dimensionless Schrodinger equation has been used:
\^\'^,.,~('''+y')+\(.x''+/)'W(x,y) = E'V(x,y), (2)
in which the units of mass, energy md fi-equency are ilWJTin, ilHh" lm' md yj^-h^ lm', respectively.
We will apply the FK-OM with four basic steps to obtain the exact numerical solution for the problem as follows: (1) rewrite the Schrddinger equation in the algebraic representation of the two-dimensional Dirae creation and annihilation operators. Note that the considered system is two-dimensional on the surface Oxy, so the projectile of angular momentum on the axis Oz is conservative. Hence, we will use
' -8.
TAP CHi KHOA HOC DHSP TPHCM Hoang Do Ngoc Tram
such new creation md annihilation operators tiiat flie operator L. is diagonalized; (2) use ttie idea of ttie pertiirbation ttieory to fmd the zero-order approxunate solutions, in which ttie Hamiltonim is divided into two parts. The main part contains only tiie terms of neuttal operators which have ttie same number of creation md annihilation operators. The eigen-functions of fliis part are fliose of harmonic oscillator. The rest temis belong to the pertiirbative part; (3) establish flic basic set of eigen-fimctions in ttie form of ttie wave-flinetions of ttie two-dimensional harmonic oscillator. This set is also ttie wave-function of Z. because tile creation md annihilation operatore are chosen m order ttiat this operator is diagonalized. Note tiiat in step (1), we put a free parameter a into ttie creation md annihilation operators. So tiie two divided parts of ttie Hamiltonim depend on tiie value of a but tiie total Hamiltonim does not, which helps to regulate ttie rate of convergence of ttie metiiod via choosing appropriate value of a;
(4) Use ttie perturbation ttieory schemes to obttun exact nuraerieal solutions. The calculation results will be presented in bellows.
First, we will ttmsform ttie Schrddinger equation (2) into tiie algebraic form. We use ttie two-dimensional Dirae creation md annihilation operators defined as follows:
^s (3) dy
in which a is a free parameter. These operators satisfy flie following commutntive relation:
[i,a*] = [A,6-] = 1, (4) other commutators equal zero.
The projectile of angular moraentinn on Oz-axis has tfie form:
For diagonalizing tills operator, we choose new creation md annihilation operators so ttiat L cm be rewritten under ttie form of neuttal operator:
"* = ; ^ ' ^ * +'•**)• " =-7={a-/i),
- . 1 . - 1 - («)
>• "n^^" -'**)• v = -j=(a + ib).
TAP CHi KHOA HOC BHSP TPHCM SS 6(84) nam 2016
These new operators also satisfy ttie commutative relations similar to tfie formula (4):
[i;,«*]=[v,r]=i. p) Now, flie projectile of flie mgular momenttmi on Oz-axis cm be rewritten as
follow:
L=U*U-i*V. (gj Thus, we obtiiin the Hamiltonim in algebraic representation of creation fad
annihilation operators (6):
H = -^^[M*-N^M)-^[M^^N^M]^^[M^^N^M)- (9, in which il5'*=2i:V,;v = 2«-,-2v*v-h2 and M^liX. These operators are the
elements ofa closed algebra with tiie commutative relations as follows:
[M,fry AM, [M,M'] = 2N, [ W , M - ] = 4W*. (10) which are tfie tools for latter algebraic calculation.
Next, we will establish m orthogonal basic set of wave fiinetion for calculation of mati-ix element of Hamiltonim. The eigen-functions of two-dimensional harmonic oscillator will be used:
'''•"=>'7iT("*r(^*ri<'W). (11)
in which |o(a)) is vacuum state defined as follows:
<:|0(ff)) = 0, i;|0(a)) = 0, {0(a:)|0(a)) = l. (,2) _ The wave-functions (11) are also the eigen-fiinetions of the mgular momentum
4 with flie eigen-value m which is flie qufatura magnetic number:
i,|»,,n,) = m|K,,K,), (m = 0,±I,±2,..). (13) For convenience, we use two basic sets of wave fimction depending on the value
of m as follows:
- For m > 0: we use two qumtiim numbers m md n = «j, tiien the wave flinctions (11) become:
TAP CHl KHOA HOC DHSP TPHCM Hoang Do Ngoc Tram
- For m < 0: we use two qufattim numbers msnin = n„ then tfie wave fimctions (11) become:
in which n = 0,l,2,...;|m| = 0,1,2,....
For fiirtiier calculation, we use the following action formulae:
M^\n,m) = l,^(n+\)[„+\m\+\)\n + \,m),
M\n,m) = 2,jn(n+\m\)\n-\,m), (lg) N\n,m) = 2{2n+\m\ + \)\n,m).
Finally, we obtain tfie non-zero mattix elements of Hamiltonifa for calculation the exact numerical solutions as follows:
//:, = (2« + H - n ) | £ l ^ + - i _ [ 6 „ ( „ + | ^ | ) + ( 2 „ + | „ | + 2 ) ( 2 „ + | „ | + 3 ) ] | ,
/':...=(^[«(" + H)+(2"+H+2)(2«-lH+3)]-^l^]7(„ + ,)(„+m+,),
^;.«=^(2"-^•H+3)^/(«•H)(n-^H-H)(«-^2)(«-^|m|-^2), (17)
"''"' ° 8 ? ^ ' ' ' * ' ' ' " * H + 'Xn•^2)(«•^-H-^2)(^-^3)(«-^|m|-l-3).
The ottier non-zero mattix elements cm be deduced based on ttie symmettic property: .ff.".„ =//."„..
3. Results and analysis
The computational program m FORTRAN 90 permits to obttiin exact numerical energies md wave-functions of 2D-SDW0 for my stiite fad fay oscillation frequency This program is tested for ttie qumtum nmnber of up to 500. Some results are shown ta flie Table 1 witti ttie precision of up to she decimal places. For tiiis problem ttie conveigence zone of ttie free parameter a are rattier wide. TTie precision of obttiined solutions Cfa be inereased if ttie value of tiiis parmieter is i n v e s t e d more carefu^y as m the work [7, 8]. The program witti ttiese unprovements will be published in ti^^
journals specified for publishing codes. "sneo m me
TAP CHi KHOA HOC BHSP TPHCM
So 6(84) nam 2016
Table I. The energies oflD-SDWO in different states and with different values of oscillation fi-equency The energies in bold text are predicted the exact analytical solutions of the problem
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
m = 0, ft) = 2.
-1.414214E+00 1.414214E+00 5.315649E+<)0 1.052921E+01 I.675396E-I-01 2.384386E-)-0I 3.170412E-1-0]
4.026593E-f01 4.947621E-f01 5.929236E-)-01 6.967924E-I-0]
8.060717E-I-01 9.205067E+01 1.039876E-I-02 1.163983E-I-02 1.292654E-K)2 1.425733E-I-02- 1.563079E-t^02 1.704564E-I-02 1.850070E+02 1.999488E-)-02
m = 0, fi; = V5
-2.ooaoooE+ao
2.000000E'4-00 6.60524 lE+00 1.223973E-l^01 I.880020E-tOI 2.617903E-1-01 3.429697E-1-01 4.309356E-f0l 5.2S2089E+01 6.253985 E-l-01 7.31I776E-1-01 8.422677E-KI1 9.584280E-1-0I 1.079448E+02 I.205140E-t02 1.335338E-1-02 1.469892E+02 1.608665E-I-02 1.75I533E-f02 1.898383E+02 2.049I10E-I-02
|w) = 3, a) = 3 -6.000000E-fOO 3.52I549E-07 6.000000E-^00 1.246l83E-f01 ).958590E+0I 2,739994E+0I 3.587924E+0I 4.4989 lOE-l-OI 5.469567E-I-01 6.496834E+0I 7.577999E+0I 8.71066 lE-KIl 9.892689E-101 l.n22I8E-f02 1.239744E+02 1.37169IE-f02 1.507920E-H02 1.648303E-)-02 1.792723E-f02 1.94107 IE-^02 2.093248E-1-02
|m| = 100, <»=I0O -3.847569E-1-05 -3.845569E-(-05 -3.843569E-1-05 -3.841S70E-KI5 -3.839S7IE-I-05 -3.837S72E-I-05 -3.835574E-I-05 -3.833576E-H)S -3.829580E-I-05 -3.827583E-I-05 -3.825586E-^05
-3.819597E-I-0S
-3.81I615E-I-05 -3.809621E-F05 -3.807626E-I-05 • ttie st^es i r / h ' ^ ^ ^ ^ W e r a in one-dirafasional space, the auttiors showed that
^ ^ u r o f e l t t s i r r ' ' "• *= " " °''^'' ^ ' " ^ ' ^ ' " ^°'"«°"» have ttie smie prinTd in b" d e x " - e * s „ T r ; ' * " f ' '" *^ ••"""= «"^" ^""'^ *^ - - « -
^—c.be:tr^:---^:--.--^^ea
.0 flie energy £ = ±I.4I42I4 = ±V2; flie ease «i = 0 , « = V ? , md „ = 0 , corresponding to flie energy £ = ±2.0; md the case H = 3, «i = 3. md „ - 0 2 beconfirmed1n;ffo:ow-uTt ^ b " " ' ' " " ' " ^ ' ° " ' ° ^ ° - ™^ ''^'='""'™ « "
TAP CHi KHOA HOC BHSP TPHCM Hoang Do Ngoc Tram
4. Conclusion
In this work, usmg the FK-OM, the exact numerical solutions for the 2D-SDW0 are obtained with the precision of up to six decimal places for m y state and any value of oscillation fi-equency. The program e m be upgraded to reach higher precision results. Some results under the form of ±E are expected being the exact analytical of the problem, which need iurther research.
Acknowledgment: This research is fimded by Vietnam National Fotmdation for Science and Technology Development (\.4FOSTED) under grant number 103.01-2013.3 i and by HCMC University of Education under grant number CS20I5.19.69.
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(Received- 04/5/2016: Revised. 19/5/2016: Accepted: 13/B/2016)
