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Communications in Theoretical Physics

Squeezing Transformation Involved in Solving the Dynamics of CO 2 Molecule

Hong-Yi Fan, Jia-Fan Fan

Published Apr 1, 1993• 0 citations
Publisher

Abstract

For the CO 2 molecule we find a frequency-jump-squeezing related unitary operator which plays a role in diagonalizing the Hamiltonian of internal motion of CO 2 . The coordinate representation of the unitary operator is derived which provides us with an alternative approach to derive the wave function of CO 2 's internal motion.
DOI
10.1088/0253-6102/19/3/361

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For a typical triatomic linear molecule with different atomic masses one finds a unitary operator which is closely related to squeezing transformations caused by frequency jumps. The coordinate representation of the unitary operator is identified, which provides an alternative approach for calculating the wavefunction of internal motion of the molecule. The author also shows that the unitary operator is equivalent to a two-mode squeezing-rotating combination operator multiplied from the right by two single-mode squeezing operators. The discussion is mainly based on the technique of integration within an ordered product (IWOP) of operators.
We find the unitary operator for diagonalizing the Hamiltonian of a triatomic linear molecule. The unitary operator is expressed in a coordinate representation that brings convenience for deriving the density matrix \ensuremath{\rho}(x1{\mathrm{x}}_{1}x1​,x2{\mathrm{x}}_{2}x2​,x3{\mathrm{x}}_{3}x3​;${\mathrm{x}}_{1}^{\ensuremath{'}}$,${\mathrm{x}}_{2}^{\ensuremath{'}}$,${\mathrm{x}}_{3}^{\ensuremath{'}}$,\ensuremath{\beta}). The unitary transformation approach is simple because the derivation can be reduced to the calculation of two independent harmonic-oscillator density matrices and one free-particle density matrix. The average of the potential energy and the kinetic of the molecule is calculated by using the density matrix.
The two quantum invariant operators are found from the time-dependent Hamiltonian of the harmonic oscillator with an auxiliary condition. The solution of the Schr\"odinger equation for the system, such as the eigenfunctions, eigenvalues, and minimum uncertainty, is derived by utilizing these invariant operators. The coherent states of this system are not the squeezed states, and the eigenfunction of the invariant operator is not the eigenfunction of the Hamiltonian of the system unless it is in the invariant representation. The squeezing function, which is an eigenfunction of the Hamiltonian of the system in the invariant representation and which also gives the minimum uncertainty, is obtained by a set of unitary transformed operators, i.e., squeezing operators.
The theoretical basis of a mechanism for squeezing through a frequency jump of a harmonic oscillator is examined. It is found that a related squeezing transformation exists, but its interpretation in terms of a frequency change is not possible. A reinterpretation of the results in terms of a scaling transformation in coordinate or momentum variables is given. Moreover, a sudden jump is not essential, and squeezing can evolve continuously. The equations of motion for such a process are obtained and the interaction responsible for squeezing is obtained.