各向异性n维耦合谐振子能量本征值的代数解法

An algebraic appoach to energy eigenvalue of anisotropic n-dimensional coupled harmonic oscillators

耦合谐振子是量子光学中的重要问题之一,许多实际物理问题的解决都依赖于耦合谐振子的模型,因此研究耦合谐振子求解的简便方法显得十分必要。运用数学上二次型正交化理论构造了一个形式上的变换矩阵,使既有坐标耦合又有动量耦合的各向异性n维耦合谐振子的Hamiltonian对角化,求出了其本征值。并应用此方法求解了三维耦合谐振子的本征值,验证了该方法的正确性。由于该方法不需要求出变换矩阵的具体形式,使得运用此方法求解具有对称形式的Hamiltonian的本征值问题变得简单、易计算出结果,该方法更具有普遍性,是一种十分有效的代数方法。

Study of the coupled harmonic oscillator is an important problem in quantum optics, and many actual physical problems are dependent on the model of the coupled harmonic oscillator, so the easy way to solve the coupled harmonic oscillator appears to be necessary. Through structuring a formal matrix by quadratic orthogonal mathematical theory and letting the Hamiltonian diagonalization of the n-dimensional anisotropic harmonic oscillators both coordinate and momentum coupling, its eigenvalues are obtained. The energy eigenvalue of three-dimensional coupled harmonic oscillator is solved by the method. The method does not need to derive the concrete form of the transformation matrix, which make it simple and easy to calculate the results to the eigenvalue problems of the Hamiltonian with symmetrical form.