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.
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https://www.ias.ac.in/article/fulltext/seca/046/03/0203-0217
(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.
To download the article click on the link below:
https://www.ias.ac.in/article/fulltext/seca/046/03/0203-0217
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