verage value of position for the anharmonic oscillator: Classical versus quantum results R. W. Robinett Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 682 Receive 8 January 996; accepte June 996 The evaluation of the average value of the position coorinate,, of a particle moving in a harmonic oscillator potential (V()k 2 /2) with a small anharmonic piece (V()k ) is a stanar calculation in classical Newtonian mechanics an statistical mechanics where the problem has relevance to thermal epansion. In each case, the calculation is most easily one using a perturbative epansion. In this note, we perform the same computation of in quantum mechanics using time-inepenent perturbation theory an the laer operator formalism to show how similar results are obtaine. We also inicate how a semiclassical calculation using a classical probability istribution can also be use to obtain the same result. 997 merican ssociation of Physics Teachers. I. INTRODUCTION The simple harmonic oscillator is one of the most often iscusse problems in all of classical, statistical, an quantum mechanics. In aition to its irect connections to important physical systems, one of its most useful features is that at every stage of evelopment, from elementary classical mechanics to quantum fiel theory, it is eactly soluble an so can be easily use as a close-form analytic eample. Many interesting physical phenomena, however, rely irectly on small eviations away from the simple form of the quaratic potential of the harmonic oscillator. s an eample, even introuctory tets 2 4 often stress the fact that the thermal epansion observe in many materials is ue to a lack of symmetry about the minimum in the intermolecular potential. Such a feature of realistic potentials can be moele by the aition of a small anharmonic term of the form V()k to the usual harmonic piece, V()k 2 /2. The average value of position,, in such an oscillator can then be calculate using classical Newtonian mechanics see, e.g., Ref. 5 or statistical mechanics see, e.g., Ref. 6 with the familiar result that increases linearly with amplitue square, energy, or temperature in contrast to the case of the pure oscillator where. Many tets on quantum mechanics consier the effects of small anharmonicities on the energy spectrum of the harmonic oscillator. quaratic term of the form V()a 4 is often iscusse see, e.g., Refs. 7 in the contet of perturbation theory where there is a nontrivial first-orer effect; such calculations are also often use as an illustration of the power of operator methos an raising/lowering operator techniques are use to evaluate the require matri elements. In contrast, the V()k term is presente see, e.g., Refs. as an eample where the first-orer correction vanishes ue to the symmetry of the problem an the perturbation an a secon-orer calculation is require. In a few cases, the first-orer correction to the wave function itself has also been eplicitly presente Refs. an, but no physical information is etracte from it. In this note, we eten these very familiar analyses to evaluate, using perturbation theory an operator techniques, the average value of position for the stanar oscillator problem perturbe by a small cubic anharmonic term an make comparisons with the classical an statistical mechanical results. In aition, we also present a erivation of the same effect using classical probability istributions, P CL (), which are also iscusse in many elementary tets an make a nice connection between the classical Newtonian solutions an the probabilistic quantum mechanical erivation using P QM ()() 2. In Sec. II we very briefly review the classical an statistical mechanical results an their relevance to thermal epansion, while Sec. III presents the main quantum mechanical an semiclassical results. II. CLSSICL RESULTS In classical Newtonian mechanics, a potential energy given by 9 m. J. Phys. 65, March 997 997 merican ssociation of Physics Teachers 9
U 2k 2 k correspons to a force Fkk 2 so that Newton s law reas ẍt 2 t 2 2 t, where k/m as usual. perturbative solution of the form t t t 2 2 t, where (t) cos(t) is the lowest-orer result with amplitue, gives the ifferential equation ẍ t 2 t 2 cos 2 t for the O term. This can be easily solve for (t) using stanar techniques to give t cost 2 2 2 4 5 cos2t. 6 The average value of position can be efine via t tt tt t using, for eample, as the perio of the unperturbe motion or in the limit that. In either case, the averaging process singles out the c component only, giving 2 2. In orer to make connections with thermal epansion, we note that the total energy of the unperturbe oscillator system is given by E tot k 2 /2. For the one-imensional oscillator with two quaratic egrees of freeom, this energy will also correspon to E2[k B T/2]k B T, where k B is Boltmann s constant. In this way we fin that Eq. 8 implies that k E k k BT, so that the average separation increases with temperature. This last result can be seen more irectly see, e.g., Ref. 6 by using the Boltmann istribution which is proportional to epu(), where /k B T in the thermoynamic average value given by e U e U k 4 e k2 /2 e k2 /2 k k k BT, where we have use the epansion e U e k2 /2 k. 7 8 9 III. QUNTUM MECHNICL ND SEMICLSSICL RESULTS In quantum mechanics, information on the average value of position is most easily etracte from the position-space stationary state wave functions, in this case labele n (). The average value of position in the unperturbe system is given by n n, 2 which vanishes because of the parity properties of the energy eigenfunctions. The first-orer shift in the energy eigenvalue also vanishes since EE n E n n V n k n n for the same reason. The first-orer wave function is given by where n n jn c jn j, c jn j V n E n E j V jn E n E j. 4 5 Using the anharmonic piece, V()k, as the perturbing potential, we require the values of V jn k j n. 6 To evaluate these, we use the formalism of raising an lowering operators which satisfy  n n n an  n n n in terms of which the position coorinate is given by  Â. The matri elements are given by 9,,2, /2 j n nnn2 j,n n /2 j,n n /2 j,n nn2n j,n. 7 8 9 One then has for the first-orer wave functions Refs. an n n k /2 nnn2 n 9n /2 n 9n /2 n nn2n n. 2 Using the perturbe wave function, we fin the average value of position to first-orer in is given by n n n   n. 2 We once again make use of raising an lowering operators to fin to O 9 m. J. Phys., Vol. 65, No., March 997 R. W. Robinett 9
Fig.. Quantum mechanical soli an classical ot ash probability istributions, P QM () () n () 2 an P CL () vs. The state corresponing to n2 is shown an the vertical ashe lines inicate the locations of the classical turning points at given by E n (n/2)k 2 /2. n  n n  n 6 k /2 n 2 22 so that the average value of position can be written as m 2 n 2 k E n, 2 since km 2. If we then associate the unperturbe energy eigenvalues, E n (), via E n 2k 2, we o inee reprouce the result of Eq. 8, namely, 2 2. 24 25 To better visualie how the probability is shifte aroun in response to the anharmonic perturbation to yiel a nonvanishing value of, we first plot in Fig. a stanar image of the quantum mechanical probability ensity P QM () n () 2 soli curve for the case n2. For later comparison, the corresponing classical epression, P CL (), ot ashe curve an the classical turning points vertical ashe lines are also shown. The ifference in probability ensities corresponing to the unperturbe oscillator an the result incluing the effect of the cubic perturbation to first orer given by Eq. 2 is given by P QM n 2 n 2 n nnn2 n 9n /2 n 9n /2 n nn2n n. 26 This is seen to be an o function of for all values of n an is plotte in Fig. 2 for the same n2 case. The classical connection is better realie by suitably averaging locally over the obvious rapi oscillations in the quantum case, an in Fig. we show the same probability ensity, but now Fig. 2. The ifference in probability ensities efine in Eq. 26, P QM () vs between the first-orer wave function () n () of Eq. 2 an the eroth orer result () n (). We use the same value of n2 as in Fig.. binne into finite intervals. The shift in probability ensity from left to right near both the classical turning points is quite similar to what woul be epecte from classical arguments. One final way to erive these results is to use semiclassical probability istributions. These are often iscusse in elementary tets on quantum mechanics see, e.g., Ref. 4, where the unperturbe harmonic oscillator is the stanar eample as in Fig., but which can be applie to many other systems as well. 5 We recall that the average value of a function of the position coorinate can obtaine by generaliing Eq. 7 to yiel f f tt 2 f f PCL, v 27 where we have use the connection v()/t an efine the classical probability istribution via Fig.. The ifference in probability ensities, P QM (), between the eroth- an first-orer results vs integrate over small bins in the position coorinate; we use the same value of n2 as in Figs. an 2. Note that the probability has been shifte slightly from left to right near both classical turning points as epecte from a classical picture. 92 m. J. Phys., Vol. 65, No., March 997 R. W. Robinett 92
P CL 2 v. 28 The integral is now taken between the classical turning points, i.e., in the range (, ), which is over only half the perio, /2. The local spee is relate to the potential energy functions via E v 2 U. For the harmonic oscillator, the stanar result is P CL 2 2, which satisfies PCL 29 as it shoul; this is the classical result reprouce in Fig.. For the perturbe oscillator, the turning points are etermine by the roots of the cubic equation 2k 2 E 2 k 2 k or 2 2 2. 2 The two roots which represent the actual turning points of the classical motion in the slightly moifie well are given by an, where an has the series epansion 5 28 2 4. 4 The thir etraneous root, which is ue to the eventual turnover of the potential cause by the cubic piece, is given by /22 2 O ( 2 ) an properly is pushe to infinity when. With these changes in turning points an potential, we can write the epectation value of the position coorinate in the form v v 2 2 2 2 2 2 2 2 2 2 I R, 5 where we have efine R()I()/ in the epectation that the integral giving information on the shift in the average value of position will have an epansion in starting with. Our first inclination might be to attempt to epan both require integrals using the stanar calculus result for the ifferentiation of efinite integrals, namely, b b f, f b, f a, a a b a f,. 6 The singular nature of the integrans near the classical turning points, however, makes such an epansion far from obviously convergent. To eplicitly check the behavior of I for, we perform the necessary integrals numerically in this case, using MTHEMTIC for several values of an the results are shown in Table I. For smaller values of, the numerical integration results are not reliable an even for. we are somewhat in the noise. Over this range, however, it certainly seems that the integral is well escribe by the epansion I/2 an this implies, of course, that the epectation value of position satisfies 2 2 2, 7 as epecte from all our earlier results. Using values of in the range.., it also seems likely that the net term in the epansion is 9 2 /4. We can attempt to etract these results in a more analytic fashion using some rather formal properties of the integrals involve. For eample, we can argue that the new classical probability istribution can actually be written in the form P CL 2 2 2 8 with the same normaliation to orer or as before. This can be justifie by calculating? PCL 2 2 2 2 2 2 2 2 y y 2 2y FF2FO 2 O 2, 9 where an y an we have assume a series epansion for F; one can perform this normaliation integral 9 m. J. Phys., Vol. 65, No., March 997 R. W. Robinett 9
Table I. Results of the numerical integration of Eq. 5 for various values of. I//2 R/2/ 2. 6 7 6. 2.26 6 2.26. 2.26 4 2.26. 2.26 2 2.26.5.65 2.6 numerically as mentione above for several small values of an fit to a lower-power polynomial in an one inee fins the result 2 O 4. This implies that the normaliation of P CL () is unchange in this case, at least by a first-orer change in the potential; this is similar to the quantum result where first-orer changes in () only change the normaliation of () 2 to secon orer. Using this notation, we fin that the average value of position in the perturbe oscillator is given by PCL GG 2 G, 2 2 2 2 2 2 2 2 2 y y 2 2y y 4 where we also assume that G has a series epansion. We can etract information on G by writing G 2 2 2 26 2 6 2 2 2 2 26 2 2 2 2 H H 2. 2 2 4 The first integral can be performe yieling H 22 2 2, 42 since efines the classical turning point where the integral vanishes. The secon integral can be evaluate with since this term is alreay eplicitly of orer ; this results in H 2 2 2 4. 4 This implies that G()H 2 ()/4 an from Eq. 4 we fin 2 4 2 2, 44 as epecte. Once again, one can perform this integral numerically an fit to powers of an this result is confirme. The assumption that we can systematically epan all of these semiclassical quantities in powers of is, perhaps, only justifie to leaing orer in as that correspons, in some sense, to the use of first-orer perturbation theory. Higherorer terms in, which woul correspon to secon-orer an higher perturbation theory woul arise from truly quantum mechanical effects coupling to ifferent states, weighte by energy ifferences an cannot necessarily be epecte to be reprouce by purely classical arguments. CKNOWLEDGMENTS We are grateful for conversations with G. Fleming, an to P. Gol for proviing us with a copy of Ref... B. Pippar, The Physics of Vibration Cambrige U.P., Cambrige, 978, Vol. ; Cambrige U.P., Cambrige, 98, Vol. 2. 2 D. Halliay, R. Resnick, an J. Walker, Funamentals of Physics Wiley, New York, 99, 4th e., p. 54. H. D. Young, University Physics ison-wesley, Reaing, M, 992, 8th e., p. 42. 4 R.. Serway, Physics for Scientists an Engineers Sauners, Philaelphia, 992, r e., upate version, p. 5. 5 C. Kittel, W. D. Knight, an M. Ruerman, Berkeley Physics Course, Mechanics Eucational Services Inc., McGraw-Hill, New York, 965, pp. 227 229. 6 C. Kittel, Introuction to Soli State Physics Wiley, New York, 97, 4th e., p. 222. 7 D. S. Saon, Elementary Quantum Mechanics Holen-Day, San Francisco, 968, pp. 97 99. 8 R. L. Liboff, Introuctory Quantum Mechanics ison-wesley, Reaing, 99, 2n e., pp. 555 556. 9 T.-Y. Wu, Quantum Mechanics Worl Scientific, Singapore, 986, pp. 26 26. C. Cohen-Tannouji, B. Diu, an F. Laloë, Quantum Mechanics Wiley, New York, 977, pp. 4 6. L. Pauling an E. B. Wilson, Introuction to Quantum Mechanics McGraw Hill, New York, 95, pp. 6 62. 2 E. E. nerson, Moern Physics an Quantum Mechanics Sauners, Philaelphia, 97, pp. 4 46. C. S. Johnson, Jr. an L. G. Peerson, Problems an Solutions in Quantum Chemistry an Physics Dover, New York, 986, pp. 74 75. 4 See Ref. 8, pp. 92 95 5 R. W. Robinett, Quantum an classical probability istributions for position an momentum, m. J. Phys. 6, 82 82 995. 94 m. J. Phys., Vol. 65, No., March 997 R. W. Robinett 94