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The derivation of the harmonic approximation of the Hamiltonian of a model of coupled three-dimensional harmonic oscillator is presented. It is shown how the splitting of the total Hamiltonian into the intrinsic and collective Hamiltonians leads to the description of the mechanism for energy dissipation in physical systems.

Investigation into the mechanism of energy dissipation in heavy-ion reactions has been carried out by different authors from different approaches, an example is the quantum dynamical model of Diaz-Torres, Hinde, Dasgupta, Milburn and Tostevin, which is based on the dissipative dynamics of open quantum systems in this model both deep-inelastic process and quantum tunneling were treated with a quantum mechanical coupled-channels approach [

Mshelia, Scheid and Greiner formulated a nuclear energy dissipation theory to account for energy dissipation that occurs in heavy-ion collisions [

In this paper, we consider some of the salient features in the complex model of Ibeh and Mshelia which consists of three-dimensional coupled oscillators located at the corners of a tetrahedron, three oscillators at the corner of the triangular base representing intrinsic motion while the one at the apex represents the collective motion [

By the symmetry consideration of the arrangement of four particles in space

three particles are positioned at

For harmonic vibration recall that in classical mechanics [

in this the only term of interest is the third term, which is sufficient for small amplitude of vibration, so that the harmonic potential energy is approximated to

in which the

The quantities

where the constants,

Similarly the quadratic kinetic energy of the system is

where,

In Ibeh and Mshelia [

where the intrinsic and collective Hamiltonians are explicitly stated as:

and

The collective Hamiltonian is assumed to be that of a free particle with mass, M and described by coordinates

The total Hamiltonian in Equation (6) is given in terms of Equations (7) and (8) as

From the above consideration we observe that the kinetic energy matrix is diagonal while the potential energy matrix is non-diagonal due to the products

The normal frequencies are determined by the secular equation

where the coefficients

Using the matrices of the kinetic and potential energies according to Equation (9), we obtain from Equation (10) the following twelve eigenfrequencies:

where, the constants appearing in the eigenfrequencies above are defined as follows:

Note the two double degenerate frequencies namely Ω_{1} = Ω_{3} and Ω_{2} = Ω_{4} and the two non-degenerate eigenfrequencies Ω_{5} and Ω_{6} describing the motion in which all the four particles vibrate about their common equilibrium configuration. The six eigenfrequencies: Ω_{7}; Ω_{8}; Ω_{9}; Ω_{10}; Ω_{11} and Ω_{12} which vanish are assumed to consists of the three zero eigenfrequencies Ω_{7}, Ω_{8}, Ω_{9}, corresponding to the eigenmodes describing a uniform translational motion of the system as a whole, while the remaining three zero eigenfrequencies Ω_{10}, Ω_{11} and Ω_{12} have no direct bearing on the theory of energy dissipation in this work. The corresponding transformations to normal coordinates are obtained as:

In terms of the normal coordinates the quantum mechanical total Hamiltonian is

We now obtain solutions of the time-independent Schrödinger equation with the decoupled total Hamiltonian H given by Equation (13).

Since H describes a free translational motion of the centre of mass and the decoupled harmonic oscillators in the g_{1}, g_{2}, g_{3}, g_{4}, g_{5} and g_{6} degrees of freedom the eigenvalues and eigenfunctions are obtained as,

and

where, the quantum numbers are

The normalized, bound state, wave functions of the harmonic oscillators are written as

The quantity

the normalization constant

The total wave function in Equation (16) is normalized as follows:

The intrinsic Hamiltonian can be stated in terms of intrinsic coordinates defined as following

The resulting eigenvalue equation of the intrinsic Hamiltonian is

where

Solving Equation (23) results in the eigenvalues

And the following set of eigenfrequencies and eigenfunctions:

where, the constants appearing in the eigenfrequencies above are defined as follows:

the normalized intrinsic oscillator eigenfunctions:

where, the intrinsic inverse oscillator lengths and normalization constants are respectively,

From Equation (26) the total intrinsic wave-function becomes

The fact that the intrinsic Hamiltonian eigenfunctions obtained form a complete set, by use of the completeness relation the total wave function

thonormal set of oscillator functions

since the

On the other hand, the normalization of _{1}, g_{2}, g_{3}, g_{4}, g_{5} and g_{6} gives

The relationship between the left hand and the right hand of Equations (30) and (31) is given by the transformation

When values are substituted the Jacobian is

Comparing Equations (31), (32) and (33) the normalization condition

Equation (34) gives a measure for the probability for intrinsic excitation from collective motion [

The collective amplitude is the expansion coefficient of the total wave function

This work has shown that the harmonic approximation of the Hamiltonian of coupled oscillators leads to a Schrödinger equation which describes the coupling of collective degree of freedom, represented by free motion with intrinsic degrees of freedom, represented by three coupled oscillators. This model explains the mechanism for energy dissipation in a physical system, based on the coupling of intrinsic and collective degrees of freedom. The model can be extended to nuclear fission and heavy-ion reactions, where the collective degree of freedom is the relative coordinates of the two heavy-ions and the intrinsic degrees of freedom are the single-particle degrees of freedom [

Furthermore, of current interest and one which is an extension of the above model is the cluster model consisting of a dinuclear system which is not easily solvable analytically because it includes other degrees of freedoms such as butterfly, belly-dancer-type motions, γ-and β-vibrations, etc., of individual nuclei, this model is based on the assumption that cluster-type shapes are produced in the mass asymmetry of nuclear molecules. Theoretical and experimental evidences exist that show that this model is capable of explaining many of the features of deformed heavy nuclei [

GodwinJoseph Ibeh,Elijah DikaMshelia, (2015) The Harmonic Approximation in Heavy-Ion Reaction Study. Applied Mathematics,06,1831-1841. doi: 10.4236/am.2015.611161