Harmonic Oscillator Models
It is as difficult to overestimate the role of harmonic oscillator models in physics in general and in quantum mechanics in particular as the influence of Beatles and Led Zeppelin on modern popular music. Harmonic oscillators are ubiquitous and appear eve
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Harmonic Oscillator Models
It is as difficult to overestimate the role of harmonic oscillator models in physics in general and in quantum mechanics in particular as the influence of Beatles and Led Zeppelin on modern popular music. Harmonic oscillators are ubiquitous and appear every time when one is dealing with a system that has a state of equilibrium in the vicinity of which it can oscillate, i.e., in a vast majority of physical systems—atoms, molecules, solids, electromagnetic field, etc. It also does not hurt their popularity that the harmonic oscillator is one of the very few models which can be solved exactly. Consider a particle moving in a potential V.x; y; z/, which has a minimum at some point x D y D z D 0. Mathematically speaking, this means that at this point @V=@x D @V=@y D @V=@z D 0, while the matrix of the second derivatives ˇ Lij @2 V=@ri @rj ˇxDyDzD0 , where r1 x; r2 y, and r3 z; is positive definite. If you still remember the connection between the potential energy and the force in classical mechanics, you should recognize that in this situation, point x D y D z D 0 corresponds to the particle being in the state of stable equilibrium. Stable in this context means that a particle removed from the equilibrium by a small distance will be forced to move back toward it rather than away from it. Expanding potential energy in a power series in the vicinity of the equilibrium and keeping only the first nonvanishing terms, you will get V.x; y; z/
1X Li;;j ri rj : 2 i;j
Respective classical Hamiltonian equations 3.2 and 3.3 yield for this potential: X dpi D Li;j rj dt j
© Springer International Publishing AG, part of Springer Nature 2018 L.I. Deych, Advanced Undergraduate Quantum Mechanics, https://doi.org/10.1007/978-3-319-71550-6_7
(7.1)
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7 Harmonic Oscillator Models
pi dri D dt me
(7.2)
(i D x; y; z). They can be converted into Newton’s equations by differentiating Eq. 7.2 (with respect to time) and eliminating the resulting time derivative of the momentum using Eq. 7.1: dri2 1 X D Li;j rj : 2 dt me j
(7.3)
The presence in matrix Lij of nondiagonal elements indicates that the particle’s motion in the direction of any of the chosen axes X; Y, or Z is not independent of its motion in other directions. In layman’s terms, it means that it is impossible to arrange for this particle to move purely in the direction of any of the axes. Nevertheless, solutions of these equations still can be presented in the standard time-harmonic form ri D ai exp .i!t/ with amplitudes ai and frequency ! obeying equations: 1 X Li;j aj D ! 2 ai ; me j
(7.4)
which is an eigenvalue equation for the matrix Li;j =me . It is obvious that this is symmetric (Li;j D L;j;i ), real-valued, and, therefore, Hermitian matrix. Thus, based on the eigenvalue theorems discussed in Sect. 3.3.1, this matrix is guaranteed to have real eigenvalues and corresponding orthogonal eigenvectors. The equation for the eigenvalues is found by requiring that Eq. 7.4 has nontrivial solutions: det me ! 2 ıi;j Li;j D 0 and
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