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The double well harmonic oscillator is thoroughly treated in many text books of guantum mechanics. The theory of the double well harmonic oscillatcr is applied to explain certain molecular phenomena .The solutions of the Schroedinger wave eguation for such an oscillator is generally in the form of confluent hypergeometric functions.In order to find the orthgonality of the sofutions,one must find the generating functions for the wave functions.In case of integral and pcsitive indes,the wave function in the form of hypergeometric functions really has generating function.In this report, the relation between the wave function and the generating function is shown and proed.
With the aid of the generating function, we are able to perform the perturbation calculations by modifying the double well potential with some simple perturbation potentials. The results of the calculations are shown in the form of matrices. It is also shown that if he perturbation potertial is made equal to zero, and the double well shrinks to one well,the perturbation results reduce to that of the one well harmonic oscillator.
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