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Path integral for the quantum harmonic oscillator using elementary methods S. M. Cohen Citation: American Journal of Physics 66, 537 (1998); doi: 10.1119/1.18896 View online: http://dx.doi.org/10.1119/1.18896 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/66/6?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in The zeta function method and the harmonic oscillator propagator Am. J. Phys. 69, 232 (2001); 10.1119/1.1311784 Comment on “Continued fractions and the harmonic oscillator using Feynman’s path integrals,” by L. Q. English and R. R. Winters [Am. J. Phys. 65 (5), 390–393 (1997)] Am. J. Phys. 67, 928 (1999); 10.1119/1.19152 Introduction to Classical and Quantum Harmonic Oscillators Am. J. Phys. 66, 551 (1998); 10.1119/1.18902 Integral equations and the simple harmonic oscillator Am. J. Phys. 66, 254 (1998); 10.1119/1.18855 Continued fractions and the harmonic oscillator using Feynman’s path integrals Am. J. Phys. 65, 390 (1997); 10.1119/1.18545 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.239.69.216 On: Wed, 15 Apr 2015 16:08:03 Path integral for the quantum harmonic oscillator using elementary methods S. M. Cohen Department of Physics, Portland State University, Portland, Oregon 97207 ~Received 12 September 1997; accepted 12 November 1997! We present a purely analytical method to calculate the propagator for the quantum harmonic oscillator using Feynman’s path integral. Though the details of the calculation are involved, the general approach uses only matrix diagonalization and well-known integrals, techniques which an advanced undergraduate should understand. The full propagator, including both the prefactor and the classical action, is obtained from a single calculation which involves the exact diagonalization of the discretized action for the system. © 1998 American Association of Physics Teachers. I. INTRODUCTION Since their introduction,1 Feynman path integrals have become a powerful method of calculation for quantum mechanical problems.2,3 Though until recently exact solutions were available for only the simplest cases, great advances in developing methods of solving these integrals have been made in the last 15 years.4 Yet even before these advances, the approach bore fruit in many ways. For example, the derivation5 of the ‘‘Feynman rules’’ was an extremely important contribution which greatly simplified calculations in perturbation theory. In a recent article, English and Winters6 have presented a method of calculating the Feynman path integral for the prefactor of the propagator of the quantum harmonic oscillator. The motivation for their work was ‘‘to introduce a formulation of quantum mechanics which is usually considered beyond the scope of most undergraduate courses.’’ We agree with these authors that it is of interest to make alternative approaches to quantum mechanics accessible to the undergraduate. We believe that path integrals have great beauty in the simplicity of their basic formulation. They also clarify various aspects of quantum mechanics, such as the uncertainty principle. The clarification in this particular case follows immediately from the central idea upon which the path integral formulation is based: that all paths in configuration space contribute to the evolution of the wave function. Thus there is an intrinsic uncertainty as to the evolution of any system ~we cannot know the trajectory the system follows!, and this uncertainty is explicitly illustrated in this approach. In this note, we give an alternative presentation which we believe is somewhat more direct than that of English and Winters. The method used by these authors required the use of a symbolic computational program, and an intermediate result written in terms of continued fractions. ~But see our Appendix for a discussion of how the approach of these authors may be completed analytically.! Our method does not require the use of a computer and is straightforward, formally, so it should be accessible to students. An understanding of Gaussian integrals, and of matrices and their eigenvectors and eigenvalues, are the only prerequisites to following this approach. Although this problem has been addressed in numerous other works,2,3,6,7 our presentation is new in some important ways. First, we discretize the action from the very beginning, allowing us to obtain a final result which is exact for arbitrary N ~the number of intervals chosen for the discretization—see below!. These results are thus directly transferable to the case of a polymer chain with nonvanish537 Am. J. Phys. 66 ~6!, June 1998 ing bond lengths confined in a harmonic potential. Additionally, we show how the classical action arises naturally, along with the prefactor, from a single calculation. This differs from previous approaches in which only the prefactor was calculated, the appearance of the classical action being assumed due to a theorem given by Feynman.2 II. FORMAL EVALUATION OF THE PATH INTEGRAL The quantum propagator, K(b,a) for a particle beginning at position x(t a )5a and ending at x(t b )5b, is given as2 K ~ b,a ! 5 E D @ x ~ t !# exp S E i \ tb ta D L @ x ~ t ! ,x˙ ~ t !# , ~1! where L @ x ~ t ! ,x˙ ~ t !# 5 21 mx˙ 2 2 21 m v 2 x 2 ~2! is the classical Lagrangian, and the symbol * D @ x(t) # represents integration over all paths in configuration space beginning at a and ending at b. As is common practice, these integrals may be done by first partitioning the time interval into N pieces of width e each, so that T5t b 2t a 5N e . At the end of the calculation the limits N→`, e →0, are taken, such that T5N e is held constant. Then, with x j 5x( j e ), we may write K ~ b,a ! 5 lim N→` e →0 3 S m 2 p i\ e E E ` 2` ••• D N/2 ` 2` dx 1 dx 2 •••dx N21 N 3e im/2\ e ( j51 $ ~ x j 2x j21 ! 22e2v2x2 j% . ~3! The argument of the exponential contains the quadratic form N Q5 ( j51 @~ x j 2x j21 ! 2 2 e 2 v 2 x 2j # 5x 20 1x 2N 2 e 2 v 2 x 2N 22x 1 x 0 22x N x N21 1Q 8 , ~4! where we may write, Q 8 5xW T AxW . Here, xW T 5 ~ x 1 x 2 •••x N21 ! ~5! is the transpose of xW , and © 1998 American Association of Physics Teachers 537 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.239.69.216 On: Wed, 15 Apr 2015 16:08:03 A5 S 22 e 2 v 2 21 21 22 e v 0 0 0 ••• 21 0 ••• 21 22 e 2 v 2 21 ••• 0 0 21 22 e 2 v 2 A A A 2 2 D K ~ b,a ! 5 lim N→` e →0 . ~6! ( j51 l j z 2j , ~7! K ~ b,a ! 5 lim where the constant does not depend on the z j , then the ~coupled! integrals in Eq. ~3! will have been reduced to N 21 separate Gaussian integrals. First, we will find a transformation of variables, ~8! N/2 exp H F GJ im x 2 1x 2 2 v 2 e 2 x 2N 2\ e 0 N ( j51 FE S ` ) j51 exp 2` D G im l z 2 dz j . 2\ e j j ~14! The integrals are now simple Gaussians, as advertised above, yielding N→` e →0 xW 5OyW , D ~ ON21,j x N 1O1,j x 0 ! 2 lj N21 3 N21 Q 8 5constant1 m 2 p i\ e N21 2 If, using a change of variables from the x j to new variables z j , we can rewrite Q 8 into the form S S m 2 p i\ e N21 2 ( j51 D H F 1/2 exp im x 2 1x 2 2 v 2 e 2 x 2N 2\ e 0 N ~ ON21,j x N 1O1,j x 0 ! 2 lj 5e i/\ S cl lim N→` e →0 S m 2 p i\ e D GJ S ) D N21 j51 21/2 lj 1/2 ~ det A! 21/2[F ~ T ! e i/\ S cl. ~15! such that We will show below that ~9! OT AO5L, with L a diagonal matrix, L i j 5l j d i j , and O will be orthogonal since A is symmetric and real. Then we may write S cl5 lim N→` e →0 N21 2 N21 Q5x 20 1x 2N 2 v 2 e 2 x 2N 22x N N21 22x 0 ( j51 ( j51 ON21,j y j ( j51 l j y 2j . x N ON21,j 1x 0 O1,j , z j 5y j 2 lj ~11! yielding N21 ( j51 2 ( j51 ` 2` ••• ` 2` dx 1 dx 2 •••dx N21 ⇒ ~12! E E ` 2` ••• ` 2` dz 1 dz 2 •••dz N21 ~13! in Eq. ~3! along with the transformation of variables. We obtain 538 ~16! To implement the transformation of variables, Eq. ~8!, we must find the matrix O which diagonalizes A. As is well known, O is the matrix of the eigenvectors of A. It is not difficult to show that a complete set of eigenvectors, which we shall denote as eW j , is given in terms of their components by A S D 2 pi j sin , N N ~17! with 1<i, j<N21. The corresponding eigenvalues are ~ ON21,j x N 1O1,j x 0 ! 2 . lj Since O is orthogonal, det O51, and the Jacobian of both transformations, Eqs. ~8! and ~11!, is unity. Hence we have the replacement E E G III. DIAGONALIZATION OF THE MATRIX, A ~ eW j ! i 5 l j z 2j 1x 20 1x 2N 2 v 2 e 2 x 2N N21 ~ ON21,j x N 1O1,j x 0 ! 2 lj ~10! Completing the squares, we change variables once again to Q5 ( j51 is indeed the classical action, as it must be; and we will find det A and thus the prefactor F(T), as well. N21 O1,j y j 1 F m x 2 1x 2 2 v 2 e 2 x 2N 2e 0 N Am. J. Phys., Vol. 66, No. 6, June 1998 l j 522 v 2 e 2 22 cos S D S D pj pj 54 sin2 2 v 2e 2, N 2N leading directly to the result N21 det A5 N21 ) l j 5 j51 ) j51 FS S D )F 4 sin2 ~ 24 ! N21 5 2 sin u 21 pj 2 v 2e 2 2N DG M /2 j51 sin2 u 2sin2 S DG pj M , ~18! ~19! where we have written sin u for v e /25 v T/2N, and M 52N. The product appearing in the final form of this equation is given in Hansen,8 S. M. Cohen 538 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.239.69.216 On: Wed, 15 Apr 2015 16:08:03 M /2 ) j51 F sin2 u 2sin2 S DG pj M 5 ~ 21 ! M /22 12M sin~ M u ! cot u , The evaluation of a 0 is identical to that of a N , apart from the term proportional to 1/N; that is, ~20! a 05 9 and we find det A5 sin~ 2N u ! , sin~ 2 u ! ~21! S cl~ N ! 5 F mN 2 2 v 2 T 2 2 x 0 1x N 2 2 x N 2T N N21 2 5 ~ ON21,j x N 1O1,j x 0 ! 2 lj ( j51 G mN a N5 2T S O2N21,j v 2T 2 12 2 2 . N lj j51 ( But, N21 ( j51 O2N21,j lj ~23! D N21 5 5 5 2 N 2 N N21 ( j51 ( j51 1 2N 4 sin2 M /221 ( j51 ( ~24! j51 S D S D S D S D S D S D sin2 N21 ( j51 S D S D 52 1 M Mf csc f csc cos 2 2 SD pj N pj 2sin2 u 2N K N ~ b,a ! 5 ~25! S HS D J SD S cl~ N ! 5 mN sin~ 2 u ! 2 p i\T sin~ 2N u ! mN 2T S M 2k f 2 u 5arcsin f f 1 1 csc2 2 ~ 21 ! k sec2 , 4 2 4 2 ~26! D v 2 T 2 cos~~ N22 ! f ! 2cos~ N f ! mN a N5 12 2 2 . 2T N 2 sin f sin~ N f ! Am. J. Phys., Vol. 66, No. 6, June 1998 ~30! D 1/2 e i/\ S cl~ N ! , HS 12 D cos~~ N22 ! f ! 2cos~ N f ! 2 a 2 sin f sin~ N f ! D v 2 T 2 cos~~ N22 ! f ! 2cos~ N f ! 2 2 b N2 2 sin f sin~ N f ! J 2 sin f ab , sin~ N f ! ~32! S D vT , 2N ~33! and S D v 2T 2 f 5arccos 12 . 2N 2 ~27! ~31! with 2 S S D S D pj N 52N sin f csc N f . pj cos f 2cos N ~ 21 ! j sin2 1 12 where in our case we need k50,2. Then we have 539 ( j51 S D S D pj N mN sin f 52 . pj T sin~ N f ! cos f 2cos N ~ 21 ! j sin2 Using the above results, we can determine the propagator for an arbitrary number, N, of divisions of the time interval T. This expression may be useful for students and others doing numerical work with path integrals, as a check of their discrete-time algorithms. The result is 4p j 12cos M , 2p j cos f 2cos M 2 p jk M 2p j cos f 2cos M cos N21 ON21,j O1,j lj IV. EXACT PROPAGATOR FOR N DISCRETE TIME INTERVALS with, as before, M 52N, sin u5vT/2N, and also cos f51 2 v 2 T 2 /2N 2 . This sum may also be found in Hansen,10 M /221 m T ( j51 In writing the last equality, we have again referred to Hansen,11 N21 p j ~ N21 ! N p j 4 sin2 2sin2 u 2N sin2 ~28! ~29! 5a N x 2N 1a 0 x 20 1a 0N x 0 x N . Now, N21 mN a 0N 52 T ~22! we may calculate D In the same way, we find correct for all N> v T/2. Furthermore, taking Oi j 5 ~ eW j ! i , S mN cos~~ N22 ! f ! 2cos~ N f ! 12 . 2T 2 sin f sin~ N f ! ~34! Note that we have used x 0 5a, x N 5b. Finally, the true propagator is obtained as S. M. Cohen 539 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.239.69.216 On: Wed, 15 Apr 2015 16:08:03 K ~ b,a ! 5 lim K N ~ b,a ! N→` S mv 5 2 p i\ sin v T 3exp H D N21 ) j51 1/2 Aj 5 B j j51 J ) im v @~ a 2 1b 2 ! cos v T22ab # , 2\ sin v T 5 ~35! Thus we have having used the fact that for large N, H 5 and vT . N In conclusion, we have presented a strictly analytical method by which the full propagator for the quantum harmonic oscillator may be obtained using Feynman’s path integral approach. Though the details are involved, the general approach should be accessible to advanced students. In particular, our presentation may be of interest to those instructors of graduate-level quantum mechanics who would like to introduce path integrals into their courses. 5 5 S m 2 p i\T D S m 2 p i\T D S m 2 p i\T DH F~ T !5 S J 1/2 lim N→` 1/2 lim N→` 1/2 H H ~A5! F S DG DG v 2T 2 2N 2 N v 2T 2 sin N arccos 12 2N 2 sin arccos 12 F F G F S DG vT N N vT sin N N vT sin~ v T ! sin J Finally, then ACKNOWLEDGMENTS We would like to thank Pui-Tak Leung for helpful comments and for bringing Ref. 6 to our attention. [email protected] j arccos~ g /2 !# [email protected] arccos~ g /2 !# [email protected]~ j11 ! arccos~ g /2 !# [email protected] arccos~ g /2 !# [email protected] arccos~ g /2 !# . [email protected] N arccos~ g /2 !# F~ T ! vT , u> 2N f> N21 mv 2 p i\ sin~ v T ! S J J 1/2 1/2 1/2 ~A6! . D 1/2 ~A7! . This is the desired result, once again obtained by purely analytical means. APPENDIX 1 In Ref. 6, it is shown that the prefactor, F(T), may be written as a product of factors, S m F~ T !5 2 p i\T D 1/2 F lim N N→` N21 ) j51 Aj Bj G 1/2 , ~A1! where the A j and B j satisfy the same recursion relations, A j 5 g A j21 2A j22 , ~A2! B j 5 g B j21 2B j22 , ~A3! with g 522 v 2 T 2 /N 2 , and starting conditions, A 21 521, A 0 50, B 21 50, and B 0 51. As these authors observe, A j11 5B j . What we would like to point out is that these relations brand these objects as Chebyshev polynomials of the second kind.12 Specifically, B j 5U j SD g [email protected]~ j11 ! arccos~ g /2 !# 5 . 2 [email protected] arccos~ g /2 !# ~A4! Therefore, the product appearing in the formula for F(T) is just 540 Am. J. Phys., Vol. 66, No. 6, June 1998 R. P. Feynman, ‘‘Space-time approach to nonrelativistic quantum mechanics,’’ Rev. Mod. Phys. 20, 367–387 ~1948!. 2 R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals ~McGraw–Hill, New York, 1965!. 3 L. S. Schulman, Techniques and Applications of Path Integration ~WileyInternational, New York, 1981!. 4 C. Grosche and F. Steiner, ‘‘How to solve path integrals in quantum mechanics,’’ J. Math. Phys. 36, 2354–2385 ~1995!, and references therein. 5 R. P. Feynman, ‘‘An operator calculus having applications in quantum electrodynamics,’’ Phys. Rev. 84, 108–128 ~1951!. 6 L. Q. English and R. R. Winters, ‘‘Continued fractions and the harmonic oscillator using Feynman’s path integrals,’’ Am. J. Phys. 65, 390–393 ~1997!. 7 J. T. Marshall and J. L. Pell, ‘‘Path-integral evaluation of the space-time propagator for quadratic Hamiltonian systems,’’ J. Math. Phys. 20, 1297– 1302 ~1979!. 8 Eldon R. Hansen, A Table of Series and Products ~Prentice-Hall, Englewood Cliffs, NJ, 1975!, p. 497, Eq. #91.1.18. 9 This result has been obtained previously in H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics ~World Scientific, Singapore, 1990!, p. 85, Eq. #2.140. 10 Reference 8, p. 272, Eq. #41.2.19. 11 Reference 8, p. 272, Eq. #41.2.25. 12 I. S. Gradshteyn and I. M. Rhyzik, Table of Integrals, Series, and Products ~Academic, Orlando, FL, 1980!, pp. 1032–1033. S. M. Cohen 540 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 132.239.69.216 On: Wed, 15 Apr 2015 16:08:03