The problem of small oscillations can be solved through the study of molecular vibrations which further, can be introduced by considering the elementary dynamical principles. The solution for the problem of small oscillations can be found out classically, as it is much easier to find its solution in classical mechanics than that in quantum mechanics. One of the most powerful tools to simplify the treatment of molecular vibrations is by use of symmetry coordinates. Symmetry coordinates are the linear combination of internal coordinates and will be discussed later in detail in this chapter.
When a molecule vibrates, the atoms get displaced from their respective equilibrium positions. Consider a set of generalized co-ordinates q_1,q_2,q_3……… q_n (the displacements of the N atoms from their equilibrium positions) in order to formulate the theory of small vibration. As these generalized coordinates do not involve the time explicitly, so classically, kinetic energy (T) is given by
2T = ∑_(i,j)▒〖k_ij (q_i ) ̇ 〗 (q_j ) ̇ (2.01) where k_ij = (∂^2 T)/(∂(q_i ) ̇∂(q_j ) ̇ ) (2.02) and potential energy, V is given by
2V (q_1,q_2,q_3…q_n )=2V_0+2∑_i▒(∂V/〖∂q〗_i ) q_i+∑_(i,j)▒((∂^2 V)/(〖∂q〗_i 〖∂q〗_j )) q_i q_j+ higher order terms (2.03)
In the expression of potential energy (V) given by equation (2.03), the higher order terms can be neglected for sufficiently small amplitudes of vibration. To make coinciding with the equilibrium position, the arbitrary zero of potential must be shifted to eliminate V_0. Consequently the term (∂V/〖∂q〗_i ) becomes zero for the minimum energy in equilibrium. Therefore, the expression of V will be reduced to
2V = ∑_(i,j)▒((∂^2 V)/(〖∂q〗_i 〖∂q〗_j )) q_i q_j (2.04)
2V = ∑_(i,j)▒f_ij q_i ...
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...ke the vanishing determinant, a fixed value of λ = λk, is chosen accordingly. Therefore, at λ = λk, the coefficients of the unknown amplitude Aj in equation (2.12) will become fixed and then it will be possible in obtaining the solution Ajk (the additional subscript k will be used to indicate the correspondence with the particular values of λk). Such a system of equations does not determine the Ajk uniquely but gives their ratios. A convenient mathematical solution designated by the quantities mjk are defined in terms of an arbitrary solution A_jk^' by the formula m_jk = (A_jk^')/[∑_j▒(A_jk^' )^2 ]^(1⁄2) (2.14)
These amplitudes are normalized
∑_j▒(m_jk )^2 =1 (2.15)
Thus the solution of the actual physical problem can be obtained by taking
A_jk= N_k m_jk (2.16) where N_k are the constants and can be determined by the initial values of q_j and q ̇_j.
When explaining the topic, I was completely lost and had trouble catching up but as soon as there was a demonstration, I soon caught on and was able to complete each equation with confidence.
5. Collected Papers, Charles Hartshorne and Paul Weiss, (edd.) (Cambridge: The Belknap Press of Harvard University Press, 1960). Volume and page number, respectively, noted in the text.
Fletcher and Rossing, The Physics of Musical Instruments (2ndEdition), Springer, New York (1998). Chapter 16 Lecture Notes on Woodwind Instruments.
The amazing transformation the study of physics underwent in the two decades following the turn of the 20th century is a well-known story. Physicists, on the verge of declaring the physical world “understood”, discovered that existing theories failed to describe the behavior of the atom. In a very short time, a more fundamental theory of the ...
Rubba, J. (1997, February 3). Ebonics: Q & A. Retrieved July 12, 2010, from http://www.cla.calpoly.edu/~jrubba/ebonics.html
The weakest feature of the paper is that although the formulas, presented by authors, are in general correct, but they do not support the conclusions the author extract from them, and mistake is hidden in the interpretation.
2. The motion of these quanta are governed by a set of materialistic principles constituting the
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3. The two most widely spaced dots were marked on the tape - the zero
numbers 1-13. The analysis of the system will involve the use of the Energy Rate
2) Fundamentals of Physics Extended: Fifth Edition. David Hanley, Robert Resnick, Jearl Walker. Published by John Wiley & Sons, Inc, New York, Chichester, Brisbane, Toronto, Singapore. 1997.
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... the vibrations should be equal to the frequency of the IR in order for the radiation to be absorbed. That would modify the amplitude of the molecular vibrations.