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International Journal of Quantum Chemistry

Squeezing transformation caused by frequency jumps for solving the problem of two coupled oscillators

Hong‐Yi Fan

Published Sep 1, 1990• 2 citations
Publisher

Abstract

Abstract For the Hamiltonian equation image a unitary operator that is closely related to the squeezing transformation caused by frequency jumps is found and its coordinate representation provides an alternative approach for calculating the wave function of the eigenstate of ℋ︁, whereas its normally ordered form is convenient for deriving the resolvent kernel in a coherent‐state basis.

DOI
10.1002/qua.560380306

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Abstract For the Hamiltonian equation image a unitary operator that is closely related to the squeezing transformation caused by frequency jumps is found and its coordinate representation provides an alternative approach for calculating the wave function of the eigenstate of ℋ︁, whereas its normally ordered form is convenient for deriving the resolvent kernel in a coherent‐state basis.
We derive the density matrix ρ(x1, x2; x1', x2', β) for two coupled harmonic oscillators by taking a unitary transformation approach. The corresponding unitary operator U is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of integration within ordered product and the coordinate representation of U we manifestly show that the ground state of the coupled oscillators is a squeezed state. Based on the derived ρ(x1, x2; x1', x2', β) we calculate the average coupling energy.
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.