IOSR Jurnal f Applied Physics (IOSR-JAP) e-issn: 78-4861.Vlume 7, Issue 5 Ver. II (Sep. - Oct. 015), PP 33-38 www.isrjurnals Quantum Harmnic Oscillatr, a cmputatinal apprach Sarmistha Sahu, Maharani Lakshmi Ammanni Cllege fr Wmen, Bangalre Abstract: Why shuld the wavefunctin have a Guassian Functin fr a Harmnic Oscillatr? What is the lwest energy? What des the Zer pint energy signify? When is a wavefunctin scillatry? Des the wavefunctin have any bearing with the turning pints? What is parity? When is parity even r dd? Many such questins can be answered by the cmputatinal methd. The cmputatinal methds used fr slving the secnd degree differential equatin (Schredinger s time independent equatin) is by Runge-Kutta furth rder methd using Micrsft-Excel. Fr harmnic scillatr, the accuracy f the results is gd. An initial guess f the wavefuntin at a pint can give the value at successive pints using a small step size. Keywrds: Zer pint energy, scillatins, turning pint, Guassian functin. I. Intrductin Obtaining the eigen value and eigen functins by slving the Schrdinger equatin is pretty invlved. Slving a secnd degree differential equatin by cmputatinal methd using Runge Kutta 4 th rder is anther easy way. The animated wavefunctin speaks vlume abut the characteristics f eigenfunctins. It als explains the discrete eigenvalues. Cncept f parity becmes bvius and, cmparisn with the classical scillatr, pictrially, is satisfying. The crrespndence principle fr large quantum number becmes visual. View the spread sheet II. The ptential f a harmnic scillatr is V shwn in Fig 1. Methdlgy kx The parablic ptential f a harmnic scillatr is 1 Fig 1. Parablic ptential Eigenvalue Prblem The time independent Schredinger equatin fr a nn-relativistic quantum system, in ne dimensin, and in atmic units ( m 1, 1 ), can be written as DOI: 10.9790/4861-0753338 www.isrjurnals.rg 33 Page
( E V ( x)) x Fr a harmnic scillatr V 1 x. With k 1, the equatin is..1 1 ( E x )) x Guess slutin with a Guassian functin gives () x Ce x.. The Guassian functin satisfies the requirement f the wavefunctin at x r x. But, C is t be determined! Bundary cnditins are essential in determining the slutin. Since the energy f the scillatr shuld be equal t r larger than the ptential, the turning pints at x 1 and x gives [1] E = V(x 1 ) =V(x )..3 The turning pints are symmetrically placed frm the rigin. We can write x 1 = -x 0 and x = x 0, then E kx0 x0( E), where E..4 k Fr a bund state, the energy eigenvalues are discrete and each f the eigen slutin crrespnding t the eigen value describes sme physical statinary state. The simplest numerical methd is t start with a trial energy and keep n changing the value till it satisfies the bundary cnditin 0 fr x n..5 With ψ as the wavefunctin, ψ and ψ the first derivative and secnd derivative respectively., the furth rder k1 ' 1 m1 ''( x, ) ( E x ) k 1 ' m1h m 1 ''( x h, k1h) 1 ( E 1 ( x h ) )( k1h) Runge-Kutta cefficients [] are k 1 3 ' mh 6 m 1 3 ''( x h, kh) 1 ( E 1 ( x h ) )( kh) k4 ' m3h m4 ''( x h, k3h) 1 ( E ( x h) )( k3h) Hence, the wavefunctin and its derivative at (x+h) is 1 6 1 3 4 1 6 1 3 4 ( x h) ( x) ( k k k k ) h '( x h) '( x) ( m m m m ) h Cmputatin using Micrsft excel gives the fllwing. The eigenfunctin btained with E =.5 and the crrespnding graphs is shwn in Fig. DOI: 10.9790/4861-0753338 www.isrjurnals.rg 34 Page
Fig. Wavefunctin f harmnic scillatr with E =.5 The cntinuity f the derivative f the eigenfunctin is shwn in Fig. 3 Fig. 3 The derivative f the eigenfunctin is als cntinuus. III. Results f harmnic scillatr The eigen values are discrete, E = 0.5, 1.5,.5,..energy units nly satisfy the bundary cnditin 0 fr x and x Fr eg., E = 0.5 satisfies the bundary cnditin 0 fr x and x as shwn in Fig. 5 (a) But, fr E greater than 0.500 the wavefunctin tends t minus infinity (as shwn in Fig. 5 (c)) while fr E lesser than 0.500, the wavefunctin tends t plus infinity (as shwn in Fig. 5 (b)).these wavefunctins are nt physically acceptable. All acceptable eigenfunctins have eigen values that are half integers. Thus, E n = (n+ 1 ) units where n = 0,1,,.. n refers t the ndes f the scillatry functin within the ptential well. Fr example, at E =.5 units, there are nly tw ndes. Odd n wavefunctins are assymetrical, hence refer t dd parity while fr even n the wavefunctin are symmetrical,thus have even parity (Fig 4 & 5(a) respectively) Fig 4. Odd parity fr dd n DOI: 10.9790/4861-0753338 www.isrjurnals.rg 35 Page
0 at x = 0 fr dd n 1 at x = 0 fr even n Fr even n Let n m where m 0,1,,... When m is even 1 and when m is dd 1 ' 1 at x = 0 fr dd n ' 0 at x = 0 fr even n Fr dd n let n m 1 where m 0,1,,... When m is even ' 1 and when m is dd ' 1 Accuracy f the eigenvalue depends n the step-size h. Uncertainity decreases with decreasing step-size. Refer t (Fig 5(a) & 6 (a). Fig 5(a) Wavefunctin tends t zer at ± and has even parity fr even n. Fig 5(b) Nt a meaningful wavefunctin. DOI: 10.9790/4861-0753338 www.isrjurnals.rg 36 Page
Fig 5(c) Nt a meaningful wavefunctin Fig. 6(a) Meaningful wavefunctin with Energy 0.50000 Fig 6 (b )Nt a meaningful wavefunctin, h= 0.05. Fig 6 (c) Nt a meaningful wavefunctin h=0.05 Eg., E = 1.500 ± 0.001 fr h = 0.1 as shwn in Fig 5 (a),(b),(c) and E = 1.5000 ±.0005 fr h = 0.05 as shwn in Fig 6 (a),(b),(c). The nrmalizatin cnstant increases the amplitude/intensity f the wave functin, but des nt affect therwise. Like in Classical thery, fr large n (=100) the prbability f btaining the particle at the turning pints is mre as shwn in the Fig. 7. This is the Bhr s crrespndence principle.[3] DOI: 10.9790/4861-0753338 www.isrjurnals.rg 37 Page
Fig 7. Bhr s crrespndence principle fr n = 100 Thus, the animated wavefunctin answers all the questins and is a nvel methd t explain difficult tpics t yung students f Physics in the classrm. Fr watching the animatin, g t the link View the spread sheet and use the excel sheet. Change the eigenvalue E quickly and watch the mdificatins in the wavefunctin in the graph. The wavefunctin explains it all. Acknwledgement I am grateful t my students wh have prmpted me t delve deep int a tpic fr writing gd bks. References [1]. Eyvind Wichman, Quantum Physics, Berkeley Physics Curse, Vl 4 McGraw Hill Cmpanies Inc.(011) []. RC Verma, PK Ahluwalia, KC Sharma, Cmputatinal Physics An intrductin, New Age Internatinal Publishers pp (1999) [3]. Sarmistha Sahu, Cncise Physics, Vl 5 Statistical Physics and Quantum Mechanics, Subhas Stres (013) [4]. C N Banwell Fundamentals f Mlecular Spectrscpy Tata McGraw-Hill Publishing Cmpany Limited, 3 rd ed (1983) DOI: 10.9790/4861-0753338 www.isrjurnals.rg 38 Page