THE THEORY OF THE ANHARMONIC OSCILLATOR BY K. S. VISWANATHAN, F.A.Sc. (Memoir No. 97 of the Raman Research Institute, Bangalore-6) Received May 22, 1957 l. INTRODUCTION ANHARMONICITY of vibration plays an important role in several branches of physics. Its importance was first recognised in acousticsa; the presence of overtones of the fundamental mode of a tuning fork and the alteration of its pitch with intensity changes were successfully accounted for as the effects of the anharmonicity of the oscillator. More recently, the subject has acquired a fresh interest in relation to the subject of molecular spectroscopy. 2 The thermal expansion '~ of crystals also owes its origin to the anharmonic nature of the vibrations inside a crystal lattice. It is the object of the present note to give an exact treatment of the classical problem of the anharmonic oscillator and also to draw attention to certain of its features which by an appeal to the Correspondence Principle lead to the results of wave mechanics, in Section 2, exact expressions have been given for the shift in the mean position of the oscillator from the origin, its frequency and the amplitudes of the different harmonics, in section 4, the eigenvalues of the oscillator are determined by using the W.K.B. method. It is shown that these are the solutions of a transcendental equation involving elliptic functions and that the first correction to any energy level of the system from its harmonic oscillator value is identical with the one obtained from the perturbation theory. 2. THE CLASSICAL PROBLEM Let m denote the mass of the oscillator and x be its displacement. The potential energy of the system may be expressed as
V = 89 ~ + 1. k~x8" (1)
We stop with the cubic term of the expansion since this is sufficient to explain most of the observed features of the spectra of diatomic molecules. Further, we take the constant k2 to be negative in accordance with the conventions of molecular spectroscopy. A2 203 204 K.S. VlSWA~ATaAN
The equation of conservation of energy for the oscillator is
89mx 2 q- 89klx 2 -f- ~; k2x n = h (2) where h is the energy of the system.
Or, dx dt- = (A -- to~x 2 + 4ax3)89 (3) where
2h m 2-kl and 4a= k~ A=m ; --at 3m"
Before proceeding further, we consider the roots of the expression
f(x) = (4ax 3 -- to2x ~ -t- A) (4) which define the regions wherein the right-hand side of (3) is real. The dis- criminant of the cubic, namely (G 2 -4- 4H3), is equal to
64Ae~ (4Ae~ - ~) and this will be negative as long as A~2 Z 1 iN" (53
When this condition is satisfied all the three roots off(x) = 0 are real, and two of them will be positive and the remaining one negative. Denoting these roots by Xl, x2 and xa where xl > x2 > xs, we have f (x) = 4a (x -- xl) (x -- x~) (x -- x3). (6) NOW {f(x)}t is real in two cases, ,,iz., (i) when x > Xl, or (ii) when x lies in the region x2 >~ x ~> x3. For x > x~,* the force acting on the molecule becomes repulsive and it increases with increasing distance of the oscillator from the origin. The vibrations in this case are unstable and the oscillator flies off to infinity within
* The solution in this case is given by x = x~ -~ (Xl - x2) nc 2 (u, k). Since cn (u, k) has zeros at points u = (2n d- 1) K, where n is an integer, the displacement is infinite at these points. Thus within a period t = ~r/~ after crossing the point xl, the molecule dissociates. The Theory of the Anharmonic Oscillator 205 a short period after crossing the point x 1. In other words, when the inter- nuclear distance of a diatomic molecule exceeds its equilibrium value by an amount x~, the molecule will dissociate. The vibrations in the region x~ .~ x >~ xa are stable and retain in a measure the characteristics of a harmonic oscillator. Writing
T~ s 3 '
the roots of f(x)= 0 are approximately
~2 xl=~(1 --16r=--512~)
x==~(l+2r+lOr 2+64r s+224r 0
xs = -- "v"A.(1 -- 2r + lOt = -- 64r 8 + 224~). (7) tO
The above relations were derived by using the bionomial expansion in the expressions for the roots of the cubic, assuming that ,r is small. One can however easily verify by substitution that they satisfy the equation f(x) = 0 upto terms of the order of ~-9. Equation (3) can now be written as 1: t - 2~/a {(x - xO (x - x,) (x - x3)}v (8) 88
The right-hand side of the above equation is an elliptic integral and its value~ is
2~/~, • a/x~ - x~ sn-~ (4xx-x, -x ~ X~ - x~ "
Hence writing u = a/~ (xl -x3) t and /:= (x~- x.) (xl - x.)' we get the wellknown solution x = x~ + (x2 -- xa) sn 2 (u, k). (9)
Given the energy of the oscillator, the above equation determines its complete course in time. 206 K.S. VISWANATHAN
In view of their periodicity properties, the Jacobian elliptic functions admit of expansions as Fourier series. For the function sn ~ (u, k), we have the following expansion~ (see Whittaker and Watson, p. 520, example 5):
-- 2 ~__i (l nqq2~Z)c~ (lO) 11~-1 where y=~u; (11)
K and E are the complete elliptic integrals of the first and second type which in terms of hyper-geometric functions are defined by
K=~.F(89189 k 2)
E =2F(--89189 (12) and q is a parameter which is related to the modulus k of the elliptic function by means of the formula q = e -t- 2, 5 + 15e 9 -t- 150e lz -~- 0 (E xT) (13) where again 2E= -1 -- ~/~ 1 +~/~ and k 1 = (1 -- k2)t. Substituting (10) in (9) we get
x = x8 + (x2 -- x3) [ K (K -- E) (Kk) 2
oo --2~r2 Z ( 1 nq;2n)COs2ny]. (14)
Equation (14) shows that the displacement of the oscillator is no longer a simple sinusoidal function, but contains a constant term and harmonics of the fundamental mode. The coefficient of t in 2y namely (~r/K) ~/a (xl -- x3) The Theory of the Anharmonic Oscillator 207 gives the circular frequency of the system. Denoting this by ~', we have there- fore 71' = K Ca (xz -- xa) (15)
The shift xa + (xg, -- xa) (K -- E) (16) Kk ~ The amplitude of the nth harmonic is proportional to n qn (17) (1 -- q~n)" In particular the intensities of the first three harmonics are in the ratio q2 . 4q~ 9qG (1 _q2)2 (1 -- q~)2 : (1 -- q-~" (17a) Equations (15) and (16) express the frequency of the oscillator and its mean shift as functions involving elliptic integrals. We shall now obtain approximations for these which are useful for appreciating their physical magnitudes as well as for comparison with earlier work. The following expansions now hold good for k S, K, (K- E) and ~/a (xx -- x-~ when r ~ 1 : k 2 = 8r (1 -- 4r + 50r 2 -- 336r 8 + 3694r 4) K =2(1 +2r+r ~+78r 3) v'~ (x, - x~) = oJ (1 + 2r -- 14r 2 + 48r 3) (K -- E) = 27rr (1 -- r + 41r ~ -- 80.5ra). (18) Substituting (7) and (18) in (15) and (16), one finds after some simplifica- tion that the change in frequency of the oscillator and its mean displacement from the origin are approximately given by -- 15Aa ~ (~- - ,o) - o,~ (19) 3Aa 225 A=a a 3. SOME REMARKS We shall now consider how the frequency and mean shift of the oscil- lator get modified when the energy of the system is altered. Remembering that the quantity A occurring in equations (19, 20) is a constant multiple (2/m) of the energy h of the system, we get on differentiation of (19) with res- pect to h that d~ --15a ~ (21) dh ~rmoJ5 where 2~" Thus to an approximation which ignores terms of the order of T4, we see that the frequency of the oscillator is a decreasing function of the energy, the rate of decrease being a constant characteristic of the oscillator. From (21) it follows that the change in the frequency of the system as its energy is increased from E1 to r is -- 15a ~ (~2 -- ~1). ~r/'//r 6 Writing E. = (v q- 3/2) h v and q = (v -b 89h v, we see that this is equal to -- 15~ --hv ~-moJ 5 or --5 k~*'hv "48 Ir (mkln)t which is the value given by the perturbation theory. 8 Hence in this case even a semi-classical theory is able to give the correct results yielded by wave- mechanics. Differentiating now (20) with respect to h', one finds that d (x> 6~ 900a 3 -~ dh -- ~-~ q- (m----~)~ "" (22) which shows that the mean shift of the oscillator increases with rise in the energy of the system. The Theory of the Anharmonie Oscillator 209 As before, the magnitude of the shift in the vth state may be obtained by writing ~v = (v + Jr) h v in (20). We have 3ah 225 h2a ~ 89 (23) (X) V= ~mm3-- (V -~- 89-'[- 2 Ir.2m2m8 (V -~ . From the above, it is possible to obtain the stretch (or contraction) in the bond-length of a diatomic molecule during a transition between two differ- ent quantum states specified by the numbers v and v' respectively. This is given by 3ah 225 h2a a rrmoJ3 (v'-- v) + ~- ~r~m2oj~ (v' -- v) (v' + v + 1) (24) 4. THE ENERGY LEVELS The Schrodlnger equation for the oscillator is given by d2u 87r2m dx ~ + -hT (, -- V) u = 0 (25) where V has the form given in (1). A method for determining the eigenvalues of an equation of the above type was developed by Wentzel, Kramers and Brillouin, and this was later refined by Langerd According to the latter, if x3 and x~ are two consecutive simple zeroes of ~ -- V (x) such that ~ > V for x > x3, and e < V for x > x~, the eigenvalues of (25) are determined by -( 'j's~r~m (, --V)}89 (nq-89 d( h 2 , dx. (26) @3 Writing A = 2~[rn, the above equation becomes (n + 89rr ------2hm I (27) where s I = I ( 4ax3 -- ~ + A)~ dx. f#3 The integral I has been evaluated in Appendix I and substituting the value given for it there in (27), one gets 8m (n + 89h = i3 ~/~ {- (K -- E) [Z(x2 -- x~)~] + ~1 (~1 + n~ K} (28) where ~t = (x~ - x~) and ~/2 - (xt - ~;~, 210 K.S. VISWANATHAN The quantities occurring on the right-hand side of the above equation are all functions of E and hence by solving this transcendental equation for different integral values of n, the eigenvalues ~1, ~,- ...... ~n of the oscil- lator may be determined. In practice, equation (28) can be solved by expressing the right-hand side as a power series in the parameter ~- which will result in a series expansion in and then inverting the series. For example, by making use of the expansions (7) and (18) and further of the following relations oj4 Z'(x~. -- xs) z = ~ ; (29) 71 (71 + *72) -- x/A~ (1 + 12~- -- 14~-z + 240"r 3) 2a it can be seen that (,28) is equivalent to rrA (1 +~ ~'2 (n + 89h = 2m 9 +0 (~)f (30) Or, writing a =- k~/12m, co = 2Try and A = 2~/m we have 5 k,7 ~2. (n + 9 hv = , + 96try (m--~Ot Hence 5 k22h2v (n + 89 (31) , = (n + 89h v 96 zr (mk?)89 which is the eigenvalue of the oscillator as obtained from the perturbation theory correct to the second order. By expanding (25) still further than in (27), one can obtain higher order corrections. The author's grateful thanks are due to Professor Sir C. V. Raman for his kind interest in this work. SUMMARY Exact expressions have been obtained for the frequency of an anhar- monic oscillator, the shift in its equilibrium position from the origin and for the amplitudes of its different harmonics. It is shown that the frequency of the oscillator is a decreasing function of the energy and that some of the results of wave-mechanics can be obtained from the classical theory by substi- tuting in the classical energy of the oscillator the different energy values of a harmonic oscillator. The eigenvalues of the oscillator are determined by using the W.K.B. method. The classical theory of a Morse oscillator has similarly been worked out in Appendix II. The Theory of the Anharmonie Oscillator 211 REFERENCES 1. Lord Rayleigh .. Theory of Sound, 1948, pp. 76-78. 2. Herzberg, G. .. Spectra of Diatomic Molecules, Second Edition, pp. 90-102. 3. Fermi, E. .. Molekiile und Kristclle, p. 140. 4. Magnus, W. and Oberhet- Formulas and Theorems for the Special Functions of Mathe- tinger matical Physics, 1949, 108. 5. Whittaker and Watson .. Modern Analysis, Fourth Edition (Chapters XXI and XXII). 6. Ta-You-Wu .. Vibrational Spectra and Structure of Polyatomic Molecules, 1939, 48. 7. Langer, R. E. .. Phys. Rev., 1937, 51~ 669. 8. Morse, P. M. .. Ibid., 1929, 34, 57. APPENDIX 1 J m~ Evaluation of the integral I = S V'4ax a - oJ~x~ + A dx. Obviously we have m2 y (4aX 8 -- oJ~x~ + A) dx. (1) I = 2 V'a. {(x xl) (x -- x2) (X -- x3)~ m8 If we substitute (x -- xs) = y~ in the above equation and denote by Qm the integral y x'dx - (x' - 7,) 0 where nl = (xz -- x3) and n~ = (xl -- xa), equation (1) takes the form 1 = (PQ6 + qQ4 + rQ2 + sQo) where p = 4 ~/~ : q = Ca (12x.~ -- ~) ; r = a/~ ( 12x3 ~ 2~__x3) ; and s=~-~1 (4ax33 _ r + A). Since x3 is a root off(x) = 0, the constant s in the above is equal to zero. The integrals Qra can be evaluated by reduction formulae. For example, d 3x~Q x3Q ' {xSv'~} =-~-~ + 2~/U~ where Q = V(x ~ - ~h) (x ~ - %). If we integrate this equation between 0 and ~Tt~, the left-hand side vanishes and we get 1 Q~ = 5 (4aQr - 3bQ~) (3) 212 The Theory of the Anharmonic Oscillator 213 where a = (7h + "qa) and b = ~7:72. Similarly Q4 = Jr (2aQ~ - bQo). (4) Substituting (3) and (4) in (2) we get after some reduction that 4 I = -- ]3 ~ {2Q~ (xl 2 + x2 2 + x3 ~) - abQo}. (5) Now #~ dx Qo__ f V_0 = ~/~"K (6) 0 Next by the substitution x = z a/~, we find that 1 ~ql f z 2dz Q2 -- ~ {(1 - z 2) (1 - k2z2)} t 0 where k 2 _ ~7_A1 7/2 K Or Q2 = ~/~nl oI s.~ (u, k) du = ~/~ (K -- E). (7) Substituting (6) and (7) in (5), one gets I = 4 v~, {- ~(x.~ - x3) ~ (K -- E) + nl (~1 + 70 K} (8) APPENDIX II The potential (1, Section 2) is valid, when the vibrations are small but nevertheless depart from harmonicity. Apart from its applications in acoustics, it is useful in studying anharmonicity in complex molecules when one splits their vibrations into those of an assembly of anharmonic oscillators. For diatomic molecules however Morse 8 suggested a potential given by V = D (1 -- e-eZ)= (1) where D is the dissociation energy of the molecule and fl is a constant, which holds good approximately upto the dissociation range. The quantum mecha- nical energy values of an oscillator moving in a potential field of the type (1) was given by Morse. We give below the classical theory of such an oscillator which merits attention in view of its extreme simplicity and for purposes of comparison with the quantum theory. The equation of energy for the Morse oscillator is 89m~ = + D (1 -- e-~Z) = = (2) m where as before h denotes the energy of the system. Hence ~c ~ = c + by + ay ~ (3) where 2- 4D. a----2D c----- h--D); b= m , m m ( and y = e-/~z. (4) c can immediately be seen to be negative. The turning points for the oscillator in its course are -- lift log (1 + a/hTD) and -- 1/fl log. (1 -- v'h~). We shall measure time from the instant the oscillator is at its extremity in the left. From (3) and (4) we get 1 1 ay (5) Or 1 by+2c = cos -1 -- (6) y ~/~ -- 4ac" 214 The Theory of the Anharmonic Oscillator 215 Hence we have 1 D 1 x=fllog ~+~log The above equation determines the course of the oscillator with time. From (7) it is immediately obvious that the frequency of the gravest mode is given by ~ :~42(D-- h) (8a) where v=~ m; v will be the (harmonic oscillator) frequency of the system when the anhar- monic terms in the expansion of (1) as a power series in x are neglected. The eigenvalues of a Morse oscillator are En = (n q- 89 v -- z (n q- 89 h v (9) where fl~h hv z = ~ = ~. (I0) The frequency of emission of the oscillator during a transition from the nth state to the one immediately below it is given by -- vnem (1 2nz) -~v{z+41--4ZE"hv} =v{zq-41--~"} (lla) from (10). Similarly the frequency of absorption from the nth state is given by Equation (8 b) shows that the frequency of the classical oscillator is a decreasing function of its energy. The dependence of the frequencies of 216 K. S. VISWANATHAN emission or absorption on the transitional energy levels is likewise described by (I 1 a) and (11 b). Apart from the first term which is a consequence of the quantum phenonmenon that has no classical analogue, namely, that the molecule possesses a zero-point energy and on the non-linear dependence of the energy levels on the quantum numbers, the formulae (11) possess a striking resemblance to the classical formula (8b). In fact (vnab-k vnem)/2 depends on the energy of the nth level in the same way as the classical fre- quency does with the energy of the system. When the energy equals its dis- sociation value the discrete spectrum stops for the oscillator. 8 Classically its frequency then vanishes. In order to find out the shift in the mean position of the oscillator and the amplitudes of the different harmonics, we express x as a Fourier cosine series in the form oo x = ao + S an cos nu (12) #='J, where u=2u ~t. Then ao-----~log ~+~ log -- cos u) du -1 ~ log ~ D + ~1 log 89{1 + 41 _ ~} 1 J (13) From the above it follows that the shift in the mean position of the oscillator increases with the energy of the system and tends logarithmically to infinity when the energy approaches its dissociation value. Next we have an = ~fl y log -- 4g osu) cos nu The integral mayI be evaluated by expanding the logarithmic function as a power series, and integrating term by term the resulting series which can be seen to be uniformly convergent for all values of u in (--~r, zr). Thus one gets The Theory of the Anharmonic Oscillator 217 an ~ ,ffi,,, (~) 1 2s (15 a) = -- 2s22S_l 2s2_ n for even values of n and an=--~,__._l,2.D~. (2s+1) 2 ~s 2s+l--n 2 for odd values of n. The above formula~ give the law of dependence of the amplitudes of the various harmonics on the energy, of the oscillator. One notices that the amplitude of the nth harmonic is of the order of (h/D)n/2.